CARDIFFU N I V E R S I T Y P R I F Y S G O LC a eRDY[§>O n a D eriva tion of th e B o ltzm ann E q u a tio n in Q u a n tu m Field T heo rybyG regor LeilerThesis submitted for the degree of Doctor of Philosophy in MathematicsCardiff University School of MathematicsSeptember 2010, CardiffUMI Number: U585413All rights reservedINFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted.In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed,a note will indicate the deletion.Dissertation PublishingUMI U585413Published by ProQuest LLC 2013. Copyright in the Dissertation held by the Author.Microform Edition © ProQuest LLC.All rights reserved. This work is protected against unauthorized copying under Title 17, United States Code.ProQuest LLC 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 48106-1346Research advisor Prof. Stefan HollandsExternal examiner Internal examiner Examination chairDr. Christopher J. Fewster Dr. Christian D. Jakel Dr. Karl M. SchmidtExamination date: 6th October 2010D E C L A R A T IO NThis work has not previously been accepted in substance for any degree and is not concurrently submitted in candidature for any degree. ^Signed .. (Gregor Leiler) Date . ̂ t. . ^ ^S T A T E M E N T 1This thesos os being submitted in partial fulfillment of the requirements for the degree of PhD in Mathematics.Signed .. (Gregor Leiler) Date . / . . ^ ^S T A T E M E N T 2This thesis is the result of my own independent work/investigation, except where otherwise stated. Other sources are acknowledged by explicit references. ^Signed . . . (Gregor Leiler) DateST A T E M E N T 3I hereby give consent for my thesis, if accepted, to be available for photocopying and for interlibrary loan, and for the title and summary to be made available to outside organisations.Signed . (Gregor Leiler) Date ^ ^ST A T E M E N T 4: PR E V IO U S L Y A P P R O V E D B A R ON A CCESSI hereby give consent for my thesis, if accepted, to be available for photocopying and for interlibrary loan a fte r ex p iry o f a b a r on access previously approved by th e G ra d u a te D evelopm ent C o m m itte e . ^Signed .. . ........... . . . (Gregor Leiler) Date 'f f . .Hardly any equation in theoretical physics has evoked as much discussion and controversy as the Boltzmann equation.N. M. HugenholtzAbstractThe Boltzmann equation (BE) is a commonly used tool for the study of non-equilibrium many particle systems. It has been introduced in 1872 by Ludwig Boltzmann and has been widely generalized throughout the years. Today it is commonly used in physical applications, from the study of ordinary fluids to problems in particle Cosmology where Quantum Field Theoretical techniques are essential. Despite its numerous experimental successes, the conceptual basis of the BE is not entirely clear. For instance, it is well known that it is not a fundamental equation of physics like, say, the Heisenberg equation (HE). A natural question then arises whether it is possible to derive the BE from physical first principles, i.e. the Heisenberg equation in Quantum Field Theory.In this work we attem pted to answer this question and succeeded in deriving the BE from the HE, thus further clarifying its conceptual status. In particular, the results we have obtained are as follows. Firstly, we establish the non-perturbative validity of what we call the “pre-Boltzmann equation” . The crucial point here is that this latter equation is equivalent to the Heisenberg equation. Secondly, we proceed to consider various limits of the pre-Boltzmann equation, namly the “low density” and the “weak coupling” limits, to obtain two equations that can be considered as generalizations of the BE. These limits are always taken together with the “long time” limit, which allows us to interpret the BE as an appropriate long time limit of the HE. The generalization we obtain consists in additional “correction” terms to the usual Boltzmann collision factor, and can be associated to multiple particle scattering. Unlike the pre-Boltzmann equation, these latter results are only valid pertubatively. Finally, we briefly consider the possibility to extend these results beyond said limits and outline some important aspects in this case.Verba volant, scripta manent.AcknowledgmentsThis is the space which the author of a written work commonly uses to thank those who, in one way or another, have helped him or her complete the work, and this is indeed what I will do. But before coming to that I want to reclaim the ownership of a space that is too frequently too “academically correct” to feel authentic. First and foremost I want to use this space to express my own pride and satisfaction with this work in a way which will outlast a polite conversation with someone “inteterested” .I started my doctoral studies with a specific goal: To obtain a solid and as broad as possible understanding of Quantum Field Theory rather than the focused and specific knowledge which is inevitably associated with a research project. And this had to be from the viewpoint of physical applications, while still catching at least a glimpse of the many mathematical problems behind it. After all has been set and done I can genuinely say that I have obtained exactly what I had wanted. Working on the Bltzmann equation allowed me to get a feel for the breadth and scope of applications of QFT, which goes way beyond the “Shut up, renormalize and calculate Feynman diagrams!” approach that subtly underlies much of the physics literature on the subject.Of course, little did I know about what I was really embarking on. I had no idea about how hard and frustrating it would turn out to be, a feeling I found particularly common amongst fellow QFT students. I remeber reading about “QFT being the hardest subject in physics” and thinking it surely cannot be particularly harder than General Relativity. It is. By far. Had I known all of this before, I would have probably not started it. Which is why, despite the inevitable conflicts two intellectually proud persons are bound to have, I want to thank my research advisor, Stefan Hollands, for what he did. It is not an understatment to say that without his support I would have not been able to complete my studies. At least not as successfully as I did. And my understanding of QFT (physical and mathematical) would have not increased by “at least an order of magnitude” , as a past lecturer remarked after I presented my work at a seminar. And I also want to thank Christian Jakel for contributing to my QFT understanding, and for the numerous “off-topic” conversations which we had.But my studies were not research only. W ithout mentioning anyone but my last year’s flatmate and fellow QFT student Heiner Olbermann, I want to thank all my friends— undergrads, postgrad students and postdocs—you know who you are, for the “non- academic” part of the last few years. It was a lot of fun.C ontentsC o n ten ts iL ist o f F igu res iii1 In tro d u c tio n 12 T h e B o ltzm an n eq u a tio n 82.1 The Boltzmann equation as a continuity e q u a tio n ....................................... 82.2 Equilibrium distributions and the quantum Boltzmann e q u a tio n ............... 122.3 Relativistic gen era liza tio n ................................................................................. 152.4 Conceptual p ro b lem s.......................................................................................... 183 C lassical an d q u a n tu m field th eo ry 213.1 Classical Hamiltonian field th e o ry ................................................................... 213.2 Quantization of a Hamiltonian field th e o ry ................................................... 233.2.1 Heuristic quantization of free f i e ld s .................................................... 243.2.2 Problems with the heuristic quan tiza tion .......................................... 263.2.3 The axiomatic a p p ro a c h ....................................................................... 273.2.4 Basic features of the P(^p)2 m o d e l....................................................... 314 N o n -p e rtu rb a tiv e p re -B o ltzm an n e q u a tio n 494.1 General fram ew ork ............................................................................................. 504.1.1 Decomposition of the equations of motion ....................................... 524.1.2 A specific p ro jec to r................................................................................. 534.1.3 Generalization for a time dependent H am ilton ian ........................... 584.2 Number operator densities for the P{^>)2 m o d e l .......................................... 604.2.1 Estimates on the number densities...... ................................................ 624.2.2 Perturbative expansion e s t im a te s ....................................................... 654.3 The pre-Boltzmann equation ............................................................................. 69iCO NTEN TS ii4.3.1 D erivation ................................................................................................ 724.3.2 Alternative form of the pre-Boltzmann e q u a tio n ............................. 785 Scaling lim its o f th e p re-B o ltzm ann eq u a tio n 815.1 Collision kernel ev a lu a tio n ................................................................................ 835.1.1 Step 1 ....................................................................................................... 845.1.2 Step 2 ....................................................................................................... 865.1.3 Step 3 ....................................................................................................... 885.2 The collision kernel in terms of retarded p ro d u c ts ....................................... 935.3 Weakly interacting and dilute systems .......................................................... 965.3.1 The thermodynamic l i m i t .................................................................... 975.3.2 The Boltzmann equation for dilute g a s e s ..............................................1005.3.3 The Boltzmann equation for weakly interacting system s.................... 1075.3.4 Physical in te rp re ta tio n ............................................................................. I l l5.4 Beyond the weak coupling l i m i t .......................................................................... 1135.4.1 Self consistent mass renormalization.......................................................1165.4.2 A hierarchy of equations—work in progress.......................................... 1266 C onclusions 130A W ick reo rd e rin g for no rm al o rd e re d p ro d u c ts 136B Form al so lu tion o f an o p e ra to r d ifferen tial equation 140C E xpansion o f o p e ra to rs on Fock space 142B ib liography 145List o f Figures2.1 Explanation of the ” signs in the Boltzmann e q u a t io n ......................... 105.1 Example of divergent contribution to 2 ) ................................................1215.2 Examples of one particle irreducible g raphs......................................................1215.3 Graphical representation of the 2-point and 4-point functions...................... 1225.4 On-shell divergences in M.{2 2): schematic rep resen ta tion .......................123iiiChapter 1 IntroductionThe study and description of non-equilibrium phenomena in physics has always represented a big challenge for physicists and mathematicians alike. This is particularly true for thermodynamical systems which at equilibrium can be completely described by a few macroscopic parameters. For instance, a sufficiently dilute gas in thermal equilibrium is completely described by just three parameters: The pressure, the volume and the temperature of the system. This simplicity is reflected in the mathematical description of the system. If P denotes the pressure of the gas, V its volume and T the temperature, then its equation of state is given byP V = N k BT ,where kB is a universal constant known as the “Boltzmann constant” and N is the number of molecules of a specific gas. Not only is the above relation extremely simple, but it also holds for any gas, irrespectively of the species of the gas, as long as it is sufficiently dilute.This simplicity remains even if we abandon a thermodynamical description1 of macroscopic systems. Without entering the domain of quantum mechanics (yet), and by adopting a heuristic, physical point of view, a statistical description of a gas at equilibrium is given in terms of its “equilibrium distribution function” / p/ p oc e_u;p,ksT ,thermodynamics is a “phenomenological” theory of matter in the sense that it draws its concepts directly from physical experiments. This is in sharp contrast to “fundamental” theories that start with a set of assumed first principles and try to explain Nature starting from the latter. Deriving a successful phenomenological theory from first principles is then one of the conceptual question that ought to be answered, and is the key idea of this work.12with Up = p 2/2 being the energy of a single molecule. The physical meaning of the equilibrium distribution function is that of a “particle number density” . That is, / Pdp represents the number of particles of a gas in thermal equilibrium with momentum in the interval [P - dp/2 , p + dp/2]. Again, however, what is important to note is that any gas at equilibrium will have the above distribution function2, which is known as the “Maxwell-Boltzmann” distribution function.This is in stark contrast with the remarkable complexity of non-equilibrium systems which, in general, do not share any common unifying condition. In his attempt to shed some light on the problem, in 1873 Ludwig Boltzmann [1] derived an equation that now carries his name, i.e. the Boltzmann equation (BE), which allowed to study the approach to equilibrium of a certain class of systems(Jl + m ' + F - V p ) /p(<,x) = J d3p2d3q,d3q2 ^ ’(p + p, - £, - £2) •R9• M (p, P! q,, q2) | [ /q,(t, x)/q2(«, x) - f j t , x)/P2(<, x) .In the above equation the 4-dimensional “delta function” enforces energy and momentum conservation, and for simplicity we are also using a relativistic notation to denote both energy and momentum, i.e. p = (u;,p).It is a non-linear integro-differential equation for the particle distribution function /p(£,x), which now depends on the position and time in addition to the momentum. From a physical point of view it relates the “thermalization” of a system to the interaction between particles as described by the “scattering amplitude” M., and as we will extensively point out throughout this work, its derivation requires a number of assumptions. This latter aspect in particular entails that it cannot be used to study every possible out-of-equilibrium system. In addition, its mathematical form remarkably complicates its solution. Stated otherwise, it is a very complicated equation that can only be used to study a relatively small number of physical phenomena. This is what we meant by saying that non-equilibrium systems are many and do not share any common features.The experimental success of the BE was such that it has been promptly generalized so it would be suited for the description of (general) relativistic effects as well as to account for the quantum nature of the phenomena. This is reflected also by its wide use to study many different physical applications that includes the simple diffusion of classical gasses, the computation of viscosity coefficients, see [2], and its application to2The equal sign is obtained by multiplying the exponential “Boltzmann weight” by a constant thatdepends on the specific gas we are considering.3exotic matter states like the quark-gluon plasma, see e.g. [3, 4], as well as the study of the relic abundances of the lightest elements or the understanding of the observed m atter- anti-matter asymmetry. For the latter two applications and for a generic introduction to the use of the BE in cosmology see [5].Boltzmann’s original derivation adopted a heuristic viewpoint which could be understood as being halfway between the phenomenological one of classical thermodynamics, and a fundamental one that would start from first principles as we understand them today. As such, it is not entirely satisfactory from a conceptual point of view. Not only that, one could easily claim that it raises more questions than it answers! This is the meaning of Hugenholtz’s words [6] that open this work. Here we will not spend much time on all the problems that the BE raises and simply mention what we believe to be the most important one. Since it is not a fundamental (or “exact”) physical equation, when does it provide a valid description of the physical phenomena? In principle this question has long been answered and we now know that the BE provides a valid description when studying either a very weakly interacting system or a very dilute one. This is hence a feature we expected to play a crucial role in this work, and indeed it did.These conclusions are the result of numerous investigations, both for classical and quantum systems. In the classical case, a lot of work has been devoted to gain a deeper understanding of the BE, and this has led to the so called BBGKY3 hierarchy of equations. The BBGKY equations are a chain of equations for the correlation functions ps, which give the probability of finding s particles of our system having specified positions and momenta. It can be then shown that the BE can be obtained by an appropriate truncation of the BBGKY hierarchy. For a short introduction on the BBGKY equations we refer the reader to [7], and to [8] for a detailed exposition. In [6], Hugenholtz derives the non-relativistic quantum BE for a Fermi gas, i.e. a system of particles obeying the Pauli exclusion principle. His derivation employs a perturbative expansion in terms of multiple commutators. Central in his derivation is the “A2t limit” , in which the coupling constant A is scaled to zero and the time t to infinity in such a way that the product A2t remains finite. That is, he obtains the QBE as a “long-time and weak coupling” approximation to the full two point function. A somewhat different derivation is given in [9], where the perturbative expansion is replaced by the assumption of “restricted quasifree- ness” , i.e. the assumption that in the weak coupling limit the four-point function and the eight-point function can be factorized in terms of two-point functions. Again the weak coupling and long time limits are crucial to obtain the Boltzmann equation. The review paper by Rau and Muller [10] was also particularly illuminating as it introduced us to3BBGKY stands for Bogoliubov, Born, Green, Kirkwood and Yvon.4the projection operator technique by employing it to derive various transport equations and the QBE in particular. For further references on the derivation of the BE we refer the reader to this latter paper ant the numerous references in it which, however, were not of direct relevance for us.We also mention the work by Erdos and Yau [11] where they study the time evolution of a quantum particle in a Gaussian random environment. What they show is that in the weak coupling limit the Wigner distribution of the wave function converges to a solution of a linear Boltzmann equation globally in time. This investigation is not particularly relevant for this work, but it shows again that the weak coupling limit is essential to establish a QBE. A common feature shared by all these derivations is that the scattering amplitude in the resulting QBE appears in the Born approximation, that is to the lowest perturbative approximation, a result we improve by obtaining sub-leading corrections in the form of the full scattering matrix. Closely related to this last work is also the work of Eng and Erdos, see [12], where they study the same quantum particle in a Gaussian random environment but consider a different scaling limit, namely the low density limit. The difference between the two scaling limits is that now the collision kernel is given by the full scattering cross section of the obstacle potential. The relevance for our work in this case is the expectation of the emergence of the full scattering matrix when considering the low density limit, which is indeed what we obtain.As a last note in this brief review of the literature we want to emphasize that this is in no way an extensive review. These are merely the studies that most affected this work. On the other hand, even this short a review would not be complete without mentioning the Kadanoff-Baym equations [13], which are considered the quantum mechanical generalization of the BE, and rely on Green’s function techniques. For a detailed exposition of these equations we recommend the textbook [14] by the same authors. The Kadanoff- Baym equations can be used to derive the QBE after some additional assumptions are made, see e.g. [15], and have even been used to derive a BE on curved spaces in [16]. Finally, for a quantitative comparison between the use of the BE and the Kadanoff-Baym equations for an interacting scalar quantum field see [17].In this work we have asked the following question: “Is it possible to relate the Boltzmann equation to the physically fundamental Heisenberg equation?” . We have been able to give a positive answer to this question, and to the best of our knowledge this is the first time that a “direct link” between the two is established for an interacting matter model in quantum field theory. To achieve this result we have first of all appropriately modified the so called “projection operator technique”, see [10, 18]. With this tool we were able to decompose the Heisenberg equation into a part which is “parallel”5to a subspace of the algebra of observables, and the complementary, “orthogonal” part. Applying this technique to the Hamiltonian of the P{ oFigure 2.1: In the presence of a positive gradient in the particle density and for particles with positive velocity, more particles will leave a given volume V (the shaded circle) than enter it, hence resulting in a net decrease of the particle number within V.where we are denoting v := dt x and F := dt p. W ith this we then writeNin(t) - N U t ) = - j [(v, F ) f p(t, x)] -n d5E , (2.3)dVwhere n is the unit outward2 vector normal to the surface at any given point, and d5£ is the appropriate integration measure on the surface d V . We now use Gauss’ theorem and rewrite the previous integral as an integral over the volume V to obtainNin(t) - N U t ) = - / v • [(v, F ) /P(f, x)] =7 r 1 (24)= - I [v • Vx/ P(t,x) + F • Vp/p(t,x)J d x d p ,Vwhere, with a slight abuse of notation we writev - ( s r 5 ; H v-'v->The collision-less Boltzmann equation^ /p (£ , x) = — — • V x/p(*, x) — F • Vp/p(t, x) , (2.5)then follows by combining equation (2.2) with (2.4) and arguing that since the volume V was completely arbitrary, the resulting identity between integrals can only hold if2This is a convention and we might as well have chosen a normal unit vector pointing towards the inside of the volume. The end result, however, would not change as in that case we would not be computing the change in N(t) due to particle leaving the volume V but with particles entering it. We would hence need to replace the minus in (2.3) with a plus, thus giving the same result.2.1 The Boltzm ann equation as a continuity equation 11the corresponding identity (2.5) between integrands holds, and we have also identified p =: rav.This is the collision-less Boltzmann equation. From the above derivation it is manifest that it has the form of a “continuity equation” , i.e. an equation akin to the, say, mass (charge) continuity equation. The physical interpretation of the latter is mass (charge) conservation and we can hence interpret the BE as a particle number continuity equation. Indeed, this is precisely how we derive it and its meaning is that of local conservation of the number of particles, where “local” now refers to locality both in position and momentum (velocity). A quick inspection of Figure 2.1 reveals that the minus signs on the RHS of the above equation are indeed correct. Moreover, we physically interpret the two terms on the RHS as follows. The first term corresponds to freely moving particles (in the presence of a gradient), whereas upon identifying the quantity F as an “external” force field, we identify the second term as modifying the particle velocities because of the external field F.Once the collisionless BE has been established it does not take much to realize that the final momenta of two colliding particles will be different from their pre-collision momenta. This means we ought to modify the above equation by including a collision term as! / „ ( . , x ) - . V , /„ ( l ,« ) - F V p/p (f ,x ) + ( ^ i ) ^The collision term will clearly have to include all the details of the specific model (dynamics) as the collision-less equation has been derived on purely kinematical grounds. And it was Boltzmann’s crucial contribution to the problem of approach to equilibrium to give an explicit expression for the collision term, see [1]. We will not derive such a collision term here but only state the end result and, most importantly, clearly point out the assumptions that are crucial for the derivation. The end result is(d t + to ' Vx + F ' VpV p^ ’x ) = f d3P2 d3

q2) is the scattering amplitude for the process p + p 2 qi + q2-It is important at this point that we note what axe the assumptions that go into the2.2 Equilibrium distributions and the quantum Boltzm ann equation 12derivation of the above collision term. They are:1. A process and its inverse have the same scattering amplitude, that isM ( p , P 2 -» Q i .q j) =M{ p , p 2) := M {p,p 2 q , , q 2) .2. Only binary collisions are taken into account—the gas is sufficiently dilute.3. The effect of external forces on the collision is negligible if compared to the interparticle forces.4. The momenta of the particles are uncorrelated at all the times— “molecular chaos” assumption or “Stosszahlansatz” .The first assumption above is justified (classically) by appealing to the electromagnetic origin of the interparticle forces, which is “PT ” invariant3, hence implying the above assumption. The second assumption is a reasonable one for a classical gas and the third assumption is verified in practical situations where the external fields are not strong enough to influence the interparticle dynamics. Finally, the really fundamental assumption, and the most difficult to justify as well, is the last one, i.e. the assumption of molecular chaos. We will leave a critical evaluation of this assumption until Section §2.4 as it is always assumed in all the generalizations of the Boltzmann equation that we will present below, whereas the remaining assumptions will be modified4 leading to appropriate modifications of equation (2.6). Finally, we remark that we have only considered one species of particle without spin here. If one considers a gas of different species of molecules then it is necessary to introduce a particle number distribution function / ,, i = 1 , . . . , N for each of the N particle species present. In that case, the collision term will couple the different distribution functions if different molecule can scatter off each other.2.2 Equilibrium distributions as link to the quantum Boltzmann equationWithout going into details, we note tha t the BE allows us to find the “equilibrium” distribution function. Intuitively, we understand equilibrium as a state where no macro-3PT invariance of a theory means that it is invariant under the action of space and time reflections x —► —x and t -> —t.4The third assumption above will still be understood. Except for a physical situation where quantum gravitational effects become relevant, we are not aware of any case where it would not hold.2.2 Equilibrium distributions and the quantum Boltzm ann equation 13scopically relevant quantity changes anymore, in the sense that they do not change with time. At equilibrium, hence, there can be no gradients in the particle density as a higher density would, in the absence of some barrier, lead to the diffusion of the gas towards the lower density regions. That is, a non-equilibrium situation due to a change of density of the gas. Because of this, the equilibrium distribution function cannot depend on x. Similarly, an external force would cause acceleration of the particles, which is clearly incompatible with the heuristic understanding of equilibrium we mentioned above. It follows that in equilibrium the Boltzmann equation becomes<2'7>where the last equality follows because we understand equilibrium as time translation invariant state, i.e. a state where quantities do not depend on time. In Chapter 4 of [7] it is possible to find the proof to the followingP roposition 2.1. Let fpê be the equilibrium solution to the Boltzmann equation (2.6), i.e. a solution o f (2.7). A necessary and sufficient condition satisfied by f p 9̂ is= 0 , (2.8)for all kinematically allowed momenta p 2, Qi and q2.The proof of this claim is a direct consequence of the proof of Boltzmann’s “H theorem”, which cam be found in [7] as well. We do not state or prove it here as we axe only interested in the consequences of (2.8). In fact, tadung the logarithm of that condition we immediately obtainlog/<'«> + log /£■> = log f g ) + log/£«> .which has the form of a “conservation law”. That is, we have that the sum of two quantities before the collision equals the sum of the corresponding quantities after the collision. This means that log/p6̂ is a conserved quantity. For a spin-less particle the conserved quantities are the energy, the three components of the momenta and a constant. Hence log f p 9̂ is a linear combination of p2, the three components of p and the arbitrary constant:log f p ^ ] = A ( p - Po)2 + log c ,2.2 Equilibrium distributions and th e quantum Boltzm ann equation 14from which one can argue that (see [7])f to ) x e-uVJ twhere wp = p 2/2 m is the energy of the particles, /? = 1 /T is the inverse temperature, and we have set ks = 1. This is known as the “Maxwell-Boltzmann” distribution function and it is a well known expression.Above we have not specified the exact proportionality constants as they are not important for our goal, which is the generalization of the Boltzmann equation to the quantum case. In fact, in the above “derivation” , it was implicitly assumed that the particles were classical (distinguishable) particles. We would like to understand, however, what a corresponding quantum Boltzmann equation would look like. That is, how should we modify the above derivation if we wanted to derive it for indistinguishable (Bose or Fermi5) particles. From the derivation we have given above it is not at all obvious how one should proceed in order to obtain the corresponding quantum equation. A heuristic and pragmatic way to do that is then obtained by “reversing” the above argument that allowed us to obtain the Maxwell-Boltzmann distribution in order to find out what the collision term looks like.The equilibrium distribution functions for Bose and Fermi particles are known (see [7]). They aref(eg) _ 1J p ± J »where the + refers to fermions and the — to bosons. Proceeding now in a reversed order we first find+ /to)= los ( - 7 ^ - ) ’/ pwhere the + now refers to bosons and the — to fermions. This trivially follows from the explicit computation that leads to the above expression, but we emphasize it in order to avoid confusion later. We then note tha t ujp is a conserved quantity, and hence writel0g ( An) ) + g ( An) ) l0g ( An) ) + 0g ( An) ) ’/ Pi / P2 ll 25The distinction between Bose and Fermi particles - bosons and fermions — can be stated by saying that Fermi particles obey “Pauli exclusion principle” whereas Bose particles do not. The Pauli exclusion principle is discussed in essentially all the books on quantum mechanics, see e.g. [20].2.3 R elativistic generalization 15which we use to arrive at the desired result/ £ • > / « (l ± /) (l ± f g > ) = f g >/£»>(l ± /£*>)(l ± /£■>).This result then immediately suggests how to modify the RHS of equation (2.6), i.e. the Boltzmann collision factor. Using the shorthand notation f p. = f ( t , x, p j , the modified (quantum) Boltzmann equation is then( § t + m ' Vx + F ‘ VpV p = / d3p2 d3qi d3q2 tf(4)(2 + £i “ 2i “ 2z) 'R (2.9)• |Ad(p, Pi *■> Qi j Q2)| [/q i/q2(l ^ /p )( l ^ /pa) ~ /p /p 2(l ^ /qi)(^ ^ /q2)] •As an aside we note that the above equation reduces to the classical one (2.6) in the case where the particle densities are very small (for very dilute systems), i.e. when we have / p < l . Incidentally, a very dilute system was one of the assumptions that led to equation (2.6), albeit dropping this assumption in the above derivation would have not led to the above equation.Before considering the generalization of the above equation to the relativistic case, we want to stress that the “argument” that led to the QBE is nothing but a heuristic and pragmatic way of arguing why the collision factor should take that form. The QBE was introduced in [21] by Uehling and Uhlenbeck but it was given with no proper explanation, and an argument as the one we have presented here seems indeed plausible. That the above one is indeed the appropriate collision factor for the QBE has later been derived at various levels of mathematical rigour—for a formal derivation of the QBE for a Fermi gas we refer the reader to, say, [9] while a rigorous derivation for the same model but on a lattice can be found in [6].2.3 Relativistic generalizationSo far we have always been dealing with slowly moving particles but we would like to understand how to deal with relativistic particles as well. Furthermore, we would like to be able to include the creation and annihilation of particles. Clearly, the appropriate framework is then (relativistic) quantum field theory.The first step consists in an almost trivial generalization of the LHS to the relativistic case. The particle distribution function will now be a function of the position and 4-2.3 R elativistic generalization 16vectors x and p and the generalization of the LHS of the BE is simplyand where is the 4-force. More importantly, however, we want to generalize the collision factor, and we do so in two ways. First, we want to allow for scattering processes other than 2 —► 2 scattering so as to allow particle creation, annihilation and particle decay. And since we want to consider particle decay, we are forced to introduce distribution functions for all the particle species that we are considering. Secondly, while the electromagnetic field is PT invariant, we also want to account for situations where a PT-violating interaction acts. That is, we would like to consider phenomena where a process is different from its inverse. Denoting the distribution function of a particlespecies a with f a and considering a specific scattering process ip+a+b-\ —> i+ j + . . . ,the appropriate relativistic BE for the particle species ip is(P>‘^ + F'‘J^)MX'P) = j dI1 (*.«.&•••■.<.*•••) (27r)4«5(4)(Pm-Pout)R(*>.?>... —> a ,6 , . . . ) | 2 f i f j • • • (l ± /*>)(l ± / a ) ( l ± /» )---- 1- (2 1 °)- | M( tp ,a ,b , . . . —► i, i , • • •) | Ufafb - ■ ■ (l ± f i ) ( l ± f j ) ■ ■ ■ j ,wheredn(a , 6, c , . . . ) = d n ad n 6d n c • • • , d n a = — \ ,2cj(paJis the appropriate Lorentz invariant phase space measure and where the 4-dimensional delta again enforces the total 4-momentum conservation, and w(pa) = (p2 + ra2)1/2 is the relativistic energy of a particle of the particle species a. In the above equation it is understood that the scattering amplitudes A4 are to be computed using the appropriate quantum field theoretical rules pertaining to the specific model one is dealing with. We finally note that the above equation is only valid as a first approximation because, in principle, the RHS should contain a “sum over all the possible interactions” that could change the particle number. In practice, however, physicists most often ignore all but one process, thus using the BE as displayed above. The reason for this is that it can be often argued on physical grounds th a t the dominant (quantitative) contribution to some physical phenomenon comes from a single scattering process. Clearly, the same comment also applies to the classical and quantum Boltzmann equations (2.6) and (2.9) respectively.2.3 R elativistic generalization 17The generalization to curved spaces is finally obtained by arguing that the only background field that influences the particle motion is the spacetime metric and that the particles move on the geodesics of spacetime hence leading to the following replacementwhere are the Christoffel coefficients associated with the background metric of the spacetime. Above we use Einstein’s “summation convention” and accordingly consider a sum the repeated indices, which denote the components of the corresponding tensorial quantities in a coordinate basis. We note that the above term is not generally covariant, but the sumis. The reason for this is that under a change of coordinates the “non-tensorial” term arising due to the Christoffel symbol will be cancelled by an identical contribution coming from the first term. In fact we have„q d _ _a_- _ r9&__d_ ^ _dx° dx° ̂ {dxPdx* ^= r J j Mp ) + =where the “tilde” quantities refer to the quantities expressed in the new (tilde) coordinates, and f ( x y p) = f i x , p) as it is a scalar quantity. The last “natural” generalization should concern the scattering amplitude in that one should use the principles of quantum field theory on curved spaces to compute them. We note, however, that “in practice” , when the Boltzmann equation is used in a cosmological setting the scattering amplitudes are computed using the usual Feynman rules (see e.g. [5]).The “general relativistic and quantum” BE as presented in this section is one of the tools to obtain quantitative prediction of Early Universe theories like Baryogenesis and Nucleosynthesis, see e.g. [5]. In the above form, the BE barely resembles the original one from equation (2.6). Its meaning, however, remains the same, i.e. the local conservation of the particle number. It might appear puzzling that the meaning is still that of particle number conservation as we are explicitly allowing the creation and annihilation of particles. This issue is quickly resolved be allowing for a slight generalization of our initial formulation of the problem, i.e. particle number conservation in some volume2.4 Conceptual problem s 18V, where we would now have to include “sources” and “sinks” of particles to account for particle creation and annihilation due to the relativistic equivalence between mass and energy. We thus still have a (now generalized) continuity equation. Despite the (physically motivated and) reasonable arguments that lead us to the above equation and the experimental confirmation of its validity, the conceptual status of this equation is far from clear. In the next section we will address precisely the many conceptual problems associated with it.2.4 Conceptual problemsThe derivation of the BE immediately raises a number of non trivial questions. One highly unsatisfactory aspect is the Stosszahlansatz, i.e. the assumption of molecular chaos. Specifically, it raises a key conceptual issue as it is assumed to be valid at all times and not just, say, at some initial time to- While it is very natural to assume for a system to be “chaotic” at some initial time, the question whether it will remain such is not, in principle, for us to argue about but is fixed by the underlying dynamics of the system. A more satisfactory approach then consists in assuming that particle velocities are uncorrelated at to and study under what conditions would they remain uncorrelated at later times. This will clearly require the detailed study of a specific model, and in this sense the molecular chaos assumption was a “necessary” one given that the BE was derived in a model independent way. We are then interested whether it can be justified or not for specific models.Another issue that even the above heuristic derivation raises lies precisely in the necessity of making a number of assumptions, of which the Stosszahlansatz is but one. This means that the BE is not an exact6 equation, so the question immediately arises about when does it provide an accurate description of some physical system? For instance, one of the assumptions in deriving equation (2.6) was that all the collisions were binary hence making it inapplicable in those cases where, say, three-body collisions are the core physical mechanism in action. This assumption has in fact been dropped later on when we obtained the relativistic BE (2.10), but in doing so it was not clear whether such a modification is consistent with the original derivation. In fact, we have not even provided a heuristic derivation of the equation (2.10) but merely “generalized” the previous results.Far from being the first to be interested in the conceptual problems arising from6In this work we will refer to an equation as being “exact” if it is assumed to be a first principle of the theory. Examples of exact equations are the Newton’s equation in classical mechanics, the Heisenberg or Schrddinger equations in quantum mechanics and Einstein’s field equations in general relativity.2.4 Conceptual problem s 19the BE (see e.g. [6, 9, 22, 23], the review [10] and references therein) the issue of its validity has in principle long been settled, at least from a physical point of view. In the kinetic description of a gas there are two fundamental “time scales”7 associated with the collisions between particles. The first one is the average time between two collisions, r ^ / , while the second one is the typical time it takes for “the collision to happen” , that is, the time after which the two particles don’t “feel” each other’s interaction anymore. We denote this time scale with It is a well known fact that the BE holds when the single collision can be considered instantaneous when compared to the average time between two subsequent collisions. That is, when Tint x) a ( d M t ^ T) = d M t ' x ) ' (3‘3)We can now compute/define the Hamiltonian of the system through the “Legendre1 As is customary in the Hamiltonian formulation of both classical and quantum field theory, we now separate the “time” and the “spatial” parts of the argument of (p and 7r3.2 Q uantization o f a H am iltonian field theory 23transformation” 2H := J ddx |7t(£,x) c^y>(t,x) — JSf j =// 2 2N % \ddx |7 r2( t , x ) + + r a V 2(*,x) + ] T - ^ n(*,x)j(3.4)Physically, the Hamiltonian H specifies the “energy content” of the theory, and is needed to generate the equations of motion. To specify the latter, however, we need an additional ingredient—the “Poisson bracket”—which we will use to define the time evolution of a time dependent functional F. If F[7r, p] and G[7r, p] are two functionals of ip and 7r, we define their Poisson bracket as< * G > / d xSF SG SF SG6 p (t,x ) <$7r(t,x) o(t’x)=( 2 ^ /Ra(p) e- i"(P),+ixP + a (p)* ei" (p),- ixp (3.7)with w (p) = (p 2 + m2)1/2. That the classical field is real suggests that the corresponding quantum field ought to be hermitian, which implies that a(p)* is the hermitian adjoint of a(p ). We also note that because we are considering a free theory, the operators a(p) and a(p)* are time independent. These “creation” (a(p)*) and “annihilation” (a(p)) operators must satisfy the commutation relations[a (p ) ,o (k ) ‘ ] = <5w (p - k) , [a (p ) ,a (k )] = [a(p)*,a(k)*] = 0 , (3.8)in order to be consistent with the CCRs (3.6). These operators can be realized on our Hilbert space as follows. We formally assign to each p G IRd a “1-particle vector” , |p), and the inner product is defined formally by (k|p) = k — p). Furthermore, we introduce a “vacuum” vector |0) and “n-particle” vectors|Pl. P2. • • - - Pn> = -Jf T 2 IP*,) ® |P»2) ® • • • ® |P»„) ,ibm ITwhere the sum is over the set of all permutations of n elements. The inner product between these vectors is the one naturally inherited from the single particle vector inner product defined above. We then define the action of a*(p) on such vectors bya(p)*|0) := |p) , a (p )* |p 1, p 2>. . . , p n) := |p , P i , P a , . . . ,p „ ) ,which implies that the action of the hermitian conjugate operator a(p) isna(p)|0) = 0 , a(p)|p„ p2, . . . , p„) = 53 ̂ ( P - P<)IPi. • • • .Pf. ■ ■ ■. Pn) . (3-9)1 = 1where p f means that the corresponding element should be omitted. It is the action of the creation and annihilation operators on the vacuum that motivates their name: they respectively create and annihilate a particle of momentum p.FYom this discussion it is clear that the quantization of the free field essentially corresponds to the quantization of an infinite collection of harmonic oscillators, one for each momentum p. Stated differently, the quantum theory of a free field is the theory of arbitrarily many free relativistic (non interacting) spinless particles that obey the Bose3.2 Quantization o f a H am iltonian field theory 26statistics.3.2.2 P rob lem s w ith th e heuristic quantizationThe quantization prescription we have outlined in the previous section is plagued with mathematical problems. For instance, we have evidently not given a proper definition of the Hilbert space, the scalar product, the Hamiltonian etc. but only a formal one (as is done in much of the physics literature, see e.g. [31, 32, 33]). This is not merely an aes- thetical point, but it conceals some genuine problems that arise when we take the formal expressions too literally. One of the most important issues has to do with the definition of composite fields like arbitrary powers of the field ^ ( x ) , which are needed e.g. to give mathematical sense to the Hamiltonian H. In fact, naively computing, say, the square of a field yields a divergent expression. The reason for this pathological behaviour is that quantum fields are distributional in nature, and hence a careful definition is required.In the case of a free theory, there is a simple solution to the problem of pointwise multiplication of fields. In this case, we can split the (free) field into a creation and an annihilation partip0(t, x) = a(£, x) + a*(t, x) , (3.10)where it is clear how the two parts are to be defined, see equation (3.7). One then observes [34] that any product of the form (a*((pi))m(a(qJ))n is a well defined distribution valued in the quadratic forms on a suitable domain. If F and G are two functionals of the field 0. This is a highly non-obvious fact, because, although we have that AP(£) > 0 for any £ € R, the potential V is not positive definite as an operator but in fact even unbounded from below. The latter is an unavoidable consequence of the above normal ordering prescription without which the expression for V would be ill-defined. The point is, however that the sum H = Ho + XV is bounded from below by a constant —0(X )L (times the identity operator I) [35], i.e. we have that H > —0(X )L • I in the sense of operators, see theorem (3.12). This is essentially because one can show that for states ^ € W for which (^ , V ^) becomes very negative, the contribution (4/, Hq̂ ) becomes very positive, and in effect overcompensates the negative contribution from the potential V.3.2 .3 T he axiom atic approachA proper mathematical construction of the interacting theory described by the Hamiltonian (3.4) will be given in the next subsections. But before we do that, we want to specify some general properties such a construction should satisfy. In particular, we briefly present the axiomatic approach to QFT within the setting of the “Wightman axioms” [25, 30]. These are a set of axioms that were introduced in an attempt to separate the essential features of any (rigorous) quantum field theory, which we want to state in mathematically precise terms, from the specific properties pertaining to the various models. Rigorous QFT then amounts to an exploration of the consequences of these axioms4. They are:A. H ilb e rt space a n d P o in ca re g ro u p . We have a Hilbert space H which carries a unitary representation of the covering group of the Poincare group ^3. In H there is precisely one state Q, the physical vacuum, which is invariant under all U(g), g € P, i.e. the vacuum state is invariant under the action of the Poincare group ^3. Finally, the spectrum of the energy-momentum operators P M is confined4There are two alternative formulation of the axioms for QFT, the “Haag-Kastler” axioms [36] for local algebras of field operators, and the “Osterwalder-Schrader” axioms [37, 38] for the Schwinger functions of Euclidean Quantum Field Theory. We present the Wightman axioms only as they are closer to the approach adopted in this work.3.2 Quantization o f a H am iltonian field theory 28to the (closed) forward light cone. This is the spectrum condition.B. Fields. Fields are “operator valued distributions” [29] over Minkowski space. This means that (/) = f •The set of all fields contains with each ip also its hermitian conjugate y?*, defined as a sesqulinear form on @ through = (^ iM x)\ip 2 ).C. T ran sfo rm atio n P ro p e r tie s . The fields transform under asU(aJa) • • • <8> ^(Z)] is the “n-particle” subspace. We note that Ho does not represent the physical vacuum of the interacting theory, i.e. the ground state of the interacting Hamiltonian H, but is an auxiliary quantity in our construction. It would be the true vacuum only if we were trying to construct the free theory. Pn projects onto the subspace of totally symmetric rank n tensors over £2(Z), the 1-particle Hilbert space of square summable sequences.That is, Hn is spanned by vectors of the form= Pn k-(1)® • • • ® ^ p, 0P G ^2(Z). We find it convenient to define their scalar product as' (315)pezwhere denotes the complex conjugate of tpp. Furthermore, if ^ ^ i , • • •) and3.2 Quantization o f a H am iltonian field theory 33$ = (4>o, $ 1, . . •) sure in W, their scalar product is simplyoo{* \* )n := , n=lwhere (^nl^nJ-Wn can be obtained by combiningn( î 0 • • • 0 4>n \i 0 • * * <8) 0„) :=t=lwith equation (3.14). Finally, we also find it convenient to introduce an orthonormal basis on 7 i i t the “momentum eigenbasis” , consisting of sequences|p) = ( . . . , 0, . . . , 0, 1, 0, . . . , 0, . . . ) ,where the 1 appears exactly in the p-th place. We then see that we can write= (p |i>)p •Equation (3.14) can equivalently be written aswhich we will use to emphasize the momentum dependence of an n-particle vector.We next define the creation, a* (77), and annihilation, 0(77), operators for a sequence rj € ^ (Z ) as follows:a*(rj)̂ fn : = P n + i [v 0 ^1 <0 * * * ® ^ n ] ,n (3.16)a ( f ) * n := A ) p Pn-l [̂ 1 0 • • • 0 > < 0 • • • 0 1>n] ,1 = 1where $ means that the corresponding element is to be omitted. It is easy to check that the above definition is such that creation and annihilation operators are each other’s adjoint. The names of the above operators come from the physical interpretation of their action: The creation operator acts on an n-particle “state” (vector) to create an ( n + 1)- particle state, with the new “particle” having “wave function” rj. A similar interpretation pertains to the annihilation operator, i.e. it decreases the number of particles in a given state by 1. An immediate consequence of the above definition is a{rj)Clo = 0. Finally, it is a m atter of straightforward computation to evaluate the action of the commutator3.2 Quantization o f a H am iltonian field theory 34of the annihilation and the creation operator on a vector \£n. In operator form, the commutator is[«(»»).«*({)] = in\Ot> , [»( • • • > Pn+l) := /—TT L S(P’ P»)*n(Pl> • • • >Pf, • • • , Pn+l) .V n + 1 “ (3.19)(aP4'n)„_i(Pu • ■ •. Pn-l) := V n *n (p , p x, . . . , P„_i) .On a spacetime with compact spatial section—in our case this is S1—this form of the creation and annihilation operators is equivalent to equation (3.16). On the other hand, for spacetimes with non-compact spatial section (like IR, see section §3.2.1), it is only the “smeared”9 form (3.16) that makes mathematical sense. Using the Definitions (3.19), the commutation relations (3.17) are then rewritten as[°P>ak] = L 6 (p ,k) , [ap,o k] = [a*,ak] = 0 ,as can be seen by (careful) direct computation using (3.19) or, more simply, by using the definition of the scalar product (3.15) in the commutator (3.17) while using the densities (3.18). The above expression is valid in the sense of quadratic forms on their natural domain. This form of the canonical commutation relations for the creation and annihilation operator is commonly found in standard books on quantum field theory and we will often use it in the main body of this work.With the creation and annihilation operators we can now construct the time zero (free) field operators ipoiv) and n0(r/)—they are the starting point for the construction of the interacting quantum field theory. For a positive value of the “mass” m > 0, we9With a slight abuse of language we use the terminology appropriate for the continuum case also in the present context.3.2 Quantization o f a H am iltonian field theory 35set £2(Z) is the multiplication operatorM )(p ) = UpVp ,and cjp = y /p 2/L 2 + m 2. As one can show, ipo(r)) and 710(77) axe essentially self-adjoint on To C W, the set of “finite particle number” vectors, i.e. the vectors that have at most a finite number of non-zero components in the direct sum (3.13). We refer the reader to section X.7 of [48] for a proof of the following fundamentalT h eo rem 3.1. The operators (po{rj) and 770(77) are essentially self-adjoint on To, and for real 77, £ € ^ (Z ) they satisfy the canonical commutation relations[¥’o(’?).7ro(?)] = iL {n \O fi . [¥>(>(»?), ¥>o(f)] = [7ro(r/)>7ro(?)] = 0 ,in the sense of quadratic forms on their natural domains.Alternatively we could construct the free fields using the creation and annihilation operators in their “density” form (3.19). Rather than expressing them in momentum space, in position space they have the familiar form«■<*> - Z(S)i73 E k"” + f-*-] •-M - i5̂ 75 E (t/V "" - •V*‘\ ■ <3 201v ' p€ZMore precisely, the time zero fields are operator valued distributions (OVDs): For a real valued test function rj2nJ dx oo V A as a self-adjoint operator on @(V) C H.3. Prove that H = Hq + XV is essentially self-adjoint on @{Hq) fi @(V) and bounded below.Because we can show that H is essentially self adjoint, we can define in a mathematically unambiguous way the time evolution operators eltH. Using the “(^-bounds” of Glimm and Jaffe (see e.g. [42, 49, 50]) we can moreover define the spacetime field operators y?(£,x) = ettfV (0, x)e~ttH as operator valued distributions on i.e., if / € Co°(IR x S), then • • • .x »>Pn) = ( n 2(mij P/a) exp(" * Y Pixj + * Y pix i ) ■ (3-22)' j = l V Pj) / j eXl j eYlThis definition is easily understood by recalling the field expansion (3.20). Similarly one could extend the definition to include the conjugate momentum field 7r(x) and derivatives of the field d-K(p(x). The important thing to keep in mind is to move the creation operators to the left of the annihilation operators. The normal ordering prescription defines the pointwise product of fields as a quadratic form on 7o x Jo- This quadratic form is actually smooth in the variables X*, meaning that the map(x i , • • • ,Xn) ( tf1|:i and \J>2- We can therefore consider x i , . . . ,x„ —> x in (3.21) and construct powers of the field. In addition, the expectation value of any normal ordered product of field in the state Qo (the free theory vacuum state) is automatically zero because of the action of the annihilation operator.T he free H am iltonian We are now ready to define the (free) hamiltonian Ho. According to the correspondence principle we ought to replace the classical expression defining the dynamics of the classical theory with a corresponding quantum expression. Using normal ordering we can now meaningfully define Ho as a quadratic form on To x T$ as follows, see [35]:2irHo := ^ J dx :L _27ro(x) + L_2(^x(/?o(x))2 + mVo(x ) : • (3.23)0In fact we have(see [35])Theorem 3.2. As a quadratic form on To x To we haveH « = l Y w r a> P ■ ( 3 -2 4 )pezMoreover, Ho leaves each subspace Hn C H invariant, and on Hn is the multiplication operator n(tfo’J'n)n(Pi, • • • , ? „ ) = ( $ 3 “'Bl)« 'B(p„ ■■■’Pn)-*=13.2 Q uantization o f a H am iltonian field theory 38Its action on the (free theory vacuum) vector Oo isHo Oo = 0 , and Ho is essentially self-adjoint on To-From now on, we let Hq denote the self-adjoint operator with domainoo 2®{H0) = { # = (4>o> #1, • • •) e n : X ) (wPi + • • • + “ p J 'M P i- • • • • p») < ° ° ) •n = l W"W ith the free Hamiltonian as a self-adjoint operator we can now define the “time evolved” free fields (po(t, x) asy>0(t,x ) := eitHo eipx+ < * > -* « ],|p| 0. Thensup (o;~e+1/2 *)A) € e2(Zn) .Intuitively, one would expect limA-*.oo = V. And indeed, in the proof of Theorem X.45 of [48], it is shown that VA\I> —► W for € T q.Introduce the total number operator TV, which we define for all E H asA * = ( 0 , * i , 2 t f 2, . . . , n t f n, . . . ) .It is easy to see thatE a p“p = iV -pezAlso, following Glimm and Jaffe [35], let us introduce the operators TVr asE Ŵ ° P •pezWe immediately see that for r = 0 this is just the number operator TV, and for r = 1 it is simply the free Hamiltonian Ho. For a kernel B £ £2(Zr+s), we define a “generalized Wick monomial”T* - = i k e k . ) ( i K ) ( n ak3) , (3.27)Pt.kjGZ ieX jeY3.2 Q uantization o f a H am iltonian field theory 40where X — {1 , . . . , r} and Y = {1, . . . , s}. The Wick monomials Tb are to be understood in the sense of quadratic forms on To x Fq. The proof of the following bounds involving the Wick monomials can be found in the literature, see Theorems 1.2.2 and 1.2.3 of [35], and we only state the result as aT h eo rem 3.4 (Nr estimates). The following bounds holdr a r+8K n ^ M n v * ) ! $ ■and in particularr+auZT'2 B/a(Z*)->ia(Z2r+*)\\n ; ^ tb n ; ^ \ \ < \ \ H ^ r/t = lHere NP1̂ 2 is the operator that gives zero on flo and is the inverse to N ^ 2 on Qq •2. In addition, we have||7b (w + / ) “p/2|| < 1 1 ^ 1 1 ^ . ^ ^ , .The relevance of Wick monomials lies in the fact that the important operators in our model can be expressed in terms of (linear combinations of) the former. In fact, using equation (3.24) together with the cutoff potential (3.26) and the normal ordering prescription (3.21), it is not difficult to see that the formula for the (full unshifted) Hamiltonian H = Hq 4- XV is given byH = a£ak+kez+'aE(2^t5 e ^ E * (E k ,.E -.)n ^ n ^ , n=3 v } iex, jeY kj€Z iex jeY jeY V iex v k* x u y = { i,...,n } q^ezclearly in the sense of quadratic forms on F q x T q . The central issue in the construction of the P2 ( oThe ZAspaces mentioned above are nothing but the usual Holder spaces L^Q, du), i.e. the Banach spaces with the normQOn Q-space we have canonical (complex) multiplication and differentiation operators= - i el«>-l2/2 ^ _ (e—lftP/2^,) t =OQkwhere 9k € C, k > 0. They satisfy the standard canonical commutation relations [Qp>Zy = i<5(p,k). The Hilbert space 71 is related to the space of square integrable functions on Q by an isometry W : W. L 2( Q, dv). This isometry relates the operators i \ , Qk to the creation and annihilation operators introduced above byw'QkW=z i (°k+a_k) ’ w , P k W = ~ a_k) •The vector Qo € % is mapped to WQo = 1, the identity function in L2. This together with the relations just given uniquely determines W: Its action on states with higher particle number gives products of Hermite polynomials in the variables 9k- The advantage of the Q-space representation is tha t the interaction V turns into a multiplication operator. In fact, the Hamiltonian reads in this representationW'HW = i Y , W kd^l2 + IQkl2 - 1) + AK(Q0,<3i, • ■ •) • (3.29)kezHere we are again understanding V as V = limA-*oo we have set Q_k — Qk = 9k for k > 0 (and similarly for P_k), and wherei« w ...)-•£« £ (3M)»*=3 |ki|,...,|kn| 0. Then Ho + V is essentially self-adjoint on @{Hq) n @(V) and is bounded below.Remarks. We remind the reader that a “semigroup” T(t) can, loosely speaking, be considered as the “exponential” of a closed and densely defined operator A. They thus provide a generalization of the relationship between unitary groups and self-adjoint operators. We also point out that a “hypercontractive” semigroup has by definition the following properties• e~tA is //-contractive, i.e. ||e-tiV ||p < || tp\\p for every ^ G / / fl L2, all p G [1,00] , and all / > 0;• for some b > 2 and some constant C t, there is T > 0 so that ||e-T>V||& < C t||^||2 for all ^ G L2(Q,di/).One can show (see section §X.9 of [48]) that in the above definition b plays no special role and that it is possible to replace b and 2 with arbitrary p and <7, provided of course that the constant will now depend on both p and q.The proof of Theorem 3.5 is fairly long so in addition to merely outlining it, we will break it into steps. The central idea, however, is as follows. Instead of dealing with the operators H0 and V directly, one considers the semigroups generated by these operators,i.e. one considers their “exponentiated” form. It is then possible to prove that e~tH is exponentially bounded, and hence tha t H = H0 + XV is bounded from below. This is then used to prove that H is essentially self-adjoint. For this argument to work, it is crucial that e~tH° is a “smoothing” map, which is encoded in the requirement that e~tHo be a hypercontractive semigroup. On the other hand, e~tV is a function in LP and it increases the singularities of the vector it acts on. This latter fact can be seen using the Holder inequality: say 77 G / / , we then havewith r -1 = p~l + 0. This is to say, as a map e~tv : L? Lr, with r < p.3.2 Quantization o f a H am iltonian field theory 43Outline of the proof o f Theorem 3.5. Cut the potential off in “field space” and define Vn12 as'v(q) i f | V ( 9) | < n0 otherwiseVn(q)Then Hn = Ho + AVn is self-adjoint on @{Hq) by the Kato-Rellich theorem (see Theorem X.12 of [48]). Clearly, Vnty —► VW as a function in L^Q, d^)-space for all p G [1, oo). Because Hn is self-adjoint and bounded from below, one can define its exponential e~tHn. The first step is then to obtain various bounds for e~tHn.Step 1: We have the following bounds:1. For any t > 0,sup ||e”tVn||i < oo ,nand is uniformly bounded in t in any compact subinterval o f [0, oo).2. I fp < q then for each t there is a constant Ct, depending on p, q and t but independent of n, so that for all xp E Lq it isl l e - ^ l l p < Ct \WU ■For fixed p and q} Ct is uniformly bounded for t in a compact subinterval of [0, oo).3. There is a constant E , independent of n, so that||e - ‘" " ^ | |2 < e ^ lM I, . (3.31)W hat we want to emphasize here is tha t the last bound shows that Hn = H0 + Vn IS bounded below. This is important and together with Duhamel’s formula, see below, allows us to conclude that the same will hold in the n —> oo limit, which is the content of the next step.Step 2: Let ip € L2(Q,di/). Then T(t)xp := limn_>oo e~tHnip exists and T(t) is a strongly continuous semigroup of self-adjoint operators satisfying ||T(£)|| < eEt. Farther, there is a unique self-adjoint operator H satisfying H > —E so that T(t) = e~tH.12One should not confuse Vn with VA: They are both “cutoff potentials”, but Vn is identically zero ifV > n while its Fourier representation is not restricted. On the other hand VA will, in general, never vanish but its Fourier modes are restricted.3.2 Q uantization o f a H am iltonian field theory 44The proof of this step relies on Duhamel’s formulat= e~tHmip + J e~(t- u)Hn(Vm - V„)e~uHmipdu ,0which holds because both sides applied to a vector in @{H0) solve the same first order differential equation. Specifically, this formula is used to show that for vectors ip G L2(Q, dis), the sequence e~tHnip is Cauchy in L2. We can hence take the limit and meaningfully define T(t)ip := limn e~tHnip. The resulting T(t) is then a strongly continuous semigroup and we denote by H its (symmetric) generator. Finally, since H is bounded below, which also follows from (3.31), it is self-adjoint.Step 3: Ho + V is essentially self-adjoint on @ = @{H0) n @(V) and for rj G Hrj = H0ti + V tj.This last step establishes the domain & of essential self-adjointness for H0 + V. In addition, it also shows that on 0 everything is “as it should be” , that is, H is indeed the sum of Ho and V. □Clearly, the above theorem is not yet sufficient to claim that we have in fact constructed a quantum field theory on the cylinder, for in the above theorem Ho is nothing but the generator of a hypercontractive semigroup and not the “free Hamiltonian” . Likewise, V is just some operator that is assumed to satisfy some hypothesis, and not necessarily the “interaction potential” . To complete the construction we are thus left with the task to show that the free Hamiltonian (3.23) and the interaction potential(3.25) do satisfy the requirements of the above theorem. It is a result of the work of Glimm and Jaffe that this is indeed the case. In particular, for the interaction potential(3.25) considered as a function on Q space, we have a stronger result than e~xv G L l . We in fact have (Theorem 2.1.4 of [35])Theorem 3.6. As a function on Q, we have e~xv G IP for all p < oo.Remark: The proof relies on the decompositionV = V A - (V - V A) = V A - V A ,and on the separate bounds on each of the two quantities, which are given in the following two lemmas. For the full proof of this theorem and of the subsequent lemmas we refer the reader to the literature, but we also state the two lemmas as we will need them in Section §4.2.1. The first one is (Lemma 2.1.5 of [35])3.2 Q uantization o f a H am iltonian field theory 45L em m a 3.7. Let P(£), the positive polynomial used to construct the interaction potential, be of degree p. Then-C>(logp/2 A) < Ka .Proof The proof is based on the commutation relations of the creation and annihilation operators, while additionally noting that since the potential is cut off, it is necessarily bounded. The bound then follows from a simple calculation using the expansion (3.30), and the note that follows it in particular. □The next result is (Lemma 2.1.6 of [35])L em m a 3.8. Let P (f) have degree p and let e > 0. Regarding V A as a function on Qand with a constant K independent of A and p we haveJ l^ A(g)l2i dg < j \ p(K A~1/2+€)2 i .Proof The proof is based on the fact tha t the integral in the above estimate is thesquare of the Q-space norm of (VA)j Qo, th a t is we haveJ \ V A(q)\V dg = \\(V*yno\\2 .We also note that V A is a sum of (generalized Wick) monomials (3.27) whose kernels satisfy the bound 11B A11 < 0 { A-1/2+c) by Theorem 3.3. And since (VA)J’fio is a state with at most pj particles it is possible, using Theorem 3.4, to obtain an estimate involving the norm of V^~lQ0. An induction then gives the result. □Finally, that H0 is the generator of a hypercontractive semigroup is the content of the following (Proposition 2.2.3 of [35])T heo rem 3.9. e~tH° is a contraction on L 1(Q,du) for t > 0 , and for sufficiently large t it is a contraction from L2(Q,di/) to L4(Q,di/).Remark: It might seem that this theorem does not prove that e~tHo is a hypercontractive semigroup, as per the definition given above. The two are, however, equivalent, and it should be noted that in [51] the term “hypercontractive” was introduced for a semigroup having the properties from the above Theorem 3.9.At this point we have verified that the free Hamiltonian Ho and the potential V satisfy the the assumptions of Theorem 3.5. In other words, we have in fact constructed3.2 Q uantization o f a H am iltonian field theory 46the P(ip) 2 model on the circle. Before concluding, however, we state three more results that we will use in what follows. The first one is (Theorem 2.2.5 of [35]) a stronger version of Theorem 3.9.Theorem 3.10. For some M > 0, e~tH° is a contraction from D* to Lq, ifI < (1 + M t)JP QandThe second one, is the well known fact that e~BH is a trace class operator, with /? the inverse temperature. Since we will occasionally appeal to this result, we state it as aTheorem 3.11. We haveT r (e -^ ) < e ^ ' 1 < oo , for any > 0. The same applies toexp( ~ 0 H - jj,ka*kak) ,feEZwhere /z* > 0.Remark: A careful look at the estimates in the proof shows that the constant K is of order L, so the free energy goes as /3_1L, as one expects.Proof. We give the proof without the “chemical potentials” /Zk for simplicity. The general case is the same because the /Zk can be absorbed into the in the free Hamiltonian, and this only makes things better.The Golden-Thompson inequality [52, 53] states that TY(ei4+5) < Tr(eAeB) for any hermitian matrices A, B. The inequality can be proved e.g. using standard properties of the trace and the Lie-Trotter product formulalim (e^/neB/n)n = eA+B .n —►«>It is possible to apply this kind of reasoning also in the infinite dimensional context to A = Ho, B = XV. Indeed, an appropriate version of the Lie-Trotter formula (see e.g. [48]) then holds, because H is essentially self-adjoint on the domain @{H0) D 0 (V ) ,3.2 Q uantization o f a H am iltonian field theory 47see below. An appropriate version of the Golden-Thompson inequality then holds (see [54, 55]), which together with the operator inequality Tr(XY) < ||X||Tr|Y|, and denoting by | | . || Ptq the norm of an operator Lq —> I f , gives:T t e - fiH < Tr(e~PH°e~Pxv) < l l e - ^ e " ^ 2̂ 0!! • Tr(e- ^ /2)H°)< eK0 ||e~xfiv\\(2+2Kp)/(K0) ||e_(^/2)Ho||2+2 ^ 2< e K 0 ~ 1 ^e - [2X( l +K 0) / K]V ^Kf i / ( 2+ 2Kf i ) L2+2Kt for some constant K > 0 when m > 0, which can in principle be seen from the well known explicit formula in Q-space, given by(e -* * * )(9) = y J I (1 - e - ^ r 1/2 exp ( - ,q i eZwhere q-k = qk for A; > 0. We are also using that e~&v is a multiplication operator in Q-space whose ZAnorms are all finite for 1 < p < 0 0 (which follows by writing V = V A + V A, using Theorem 3.8 and the fact that V A is semi-boimded from below), and we have used the inequality T r ( e ~ ^ ^ H°) < eK&~x. (This latter inequality is obvious for P —> 00. For ft -» 0 it follows from, say, equation (1.2) of [56], while for finite (3 it holds because both functions are monotonically decreasing.) □Later we will also need to appeal to the so called “Rosen bounds” (see [57] for the original work and Theorem 3.1.3 of [35] for the specific version written below) or “Higher order estimates”, and to a result by Glimm and Jaife [39], so we state the results here as aTheorem 3.12 (Rosen bounds). Let i be a positive integer and p the degree of the polynomial P. For e > 0, i > 3 and j = j( i,p ,e ) sufficiently large, there is a constant such thatH%-eN i+€~3 < K ( H + 0 ( \ ) L • I)j , (3.32)and the constant K does not depend on X or j . I f e > 2, we may take j = i. For p < 4 and some K ' we have [39]H i < K ' ( H + 0 ( \ ) L ■ I ) 2 . (3.33)Proof We merely give an argument for equation (3.33). In [39] One can find the proof3.2 Q uantization o f a H am iltonian field theory 48of the following statement, valid for (p4 theory,- [ H l ' \ [HlJ \ « ]] < eH2 + b(e, X) ,for any positive number e and suitably large b. We can use this result as followsH 2 = H ll2HQH ln < H ^ H H ^ 2 = - [Hx0' 2, [//01/2, H}} + H0H + HH0 << - [Hi12, [Hl/2, /f]] + (H0 + H f < eHl + bl + (Ho + H)2 << tH l + b l + (2H0 + XV)2 = tH l + bl + 4H l/2 < tH 2 + (Hxn + 0(X)L ■ I)2 ,where i f a/2 = Hq + X /2V . Noting now that the above result holds for arbitrary A, we could use it with A —> 2A and hence immediately obtain equation (3.33). □We will use these inequalities in many places below and we also note that it implies that H is bounded below as a special case. For simplicity we will absorb the additive constant 0{X)L • I into the Hamiltonian H —this does not affect the definition of the time evolution at (A) = ettHAe~*tH of an observable A , but it has the advantage of turning H into a non-negative operator. The fact that m > 0 implies [35] that in the thermodynamic limit L —> oo, i.e. in QFT on 2-dimensional Minkowski space, H has a state of lowest energy, the ‘Vacuum state” , followed by a mass gap, corresponding to physical particles in the sense of scattering theory. However, we note that the value of this mass gap is not m, and hence this parameter must therefore not be confused with the physical mass. Finally, as for the free theory, consistency with the Wightman axioms should be checked. We again omit the proof and note that the above construction indeed results in 8m interesting field theory according to both the Wightman and Haag-Kastler axioms.Later in Section §4.2.1 it will be important for us that the inequality (3.32) equally applies to the time evolved free Hamiltonian and time evolved number operator because H is a constant of the motion. In this way, the higher order estimates will allow us to transfer information on the number operator densities at the initial time to later times.Chapter 4 N on-perturbative pre-Boltzm ann equationIn this chapter we take the first step in the problem of deriving the Boltzmann equation from the Heisenberg equation. The result we present in this chapter is also one of the main results of this work, i.e. what we call the “pre-Boltzmann equation” (4.46). Although it may appear to be very different from the Heisenberg equation it is, in fact, equivalent to it. To show this, we first decompose the Heisenberg equation by appropriately modifying the so called “projection operator” technique [18] to fit our needs. The essential idea behind the method is to project an observable onto a subspace of the algebra of observables spanned by a set of “relevant” observables. The price we have to pay for such a projection is that the resulting equation is not local in time anymore. It is, however, a much better starting point for certain approximations than the original Heisenberg equation. Next we introduce the main observables of this work, i.e. the number operators, in the context of a hermitian scalar quantum field with polynomial (self-)interaction on the cylinder and prove various estimates satisfied by the particle number operators. Finally, we derive the pre-Boltzmann equation for our model, and we give two alternative formulations of it.Before proceeding we want to remark on the author’s contribution to this chapter. To establish the non-perturbative validity of the pre-Boltzmann equation (4.35), two ingredients are needed: A formal derivation and the rigorous proof of the validity of the formal results. In what follows the two are tightly intertwined, but this distinction essentially corresponds to the work done by the author (the formal derivation) and his research advisor (the rigorous proof of such results). In particular, Theorems 4.4, 4.5, 4.6, and the proof of Theorem 4.7 are the work of the author’s research advisor, see also [58]. They are included in this work for completeness.494.1 General framework 504.1 General frameworkHere we set up the projection operator method [10,18], for the case in which the Hamiltonian is time independent. After this has been accomplished, it will be straightforward to generalize the results to the case of a time dependent Hamiltonian. Such a generalization is useful for the application of the method to situations when there is a time dependent background field. This could be the case for charged particles in a varying electric field or, for cosmologically relevant applications, quantum fields propagating on a curved background like e.g. Robert son-Walker spaces. In this work, however, we will be only interested in the time independent case.The projection method is well known in the literature and our contribution is to adapt it to our needs and to generalize it to the time dependent case. The basic framework is very general and we will explain it, for the sake of simplicity, in a finite dimensional setting. This has the definite advantage that all the steps are completely well defined. In the infinite dimensional case the calculations are formally identical, but the convergence of the various series below cannot be taken for granted.Let H € Mn(C) be a Hamiltonian, self-adjoint with respect to the standard scalar product (• | •) on 7i = Cn. Given an observable G £ Mn(C), we define its time evolution asa , : M„(C) -* M„(C) , a t (G) = eitHGe~itH = G(t) .The time evolution clearly satisfies the group law a t+8 = ott ° a s, i.e. it is an automorphism, and it is also a homomorphism of the algebra Mn(C) for each t, that is a t(AB) = a t(A)at(B). In the context of the Boltzmann equation, we have a density matrix state p, i.e. a self-adjoint, positive semi-definite operator satisfying Trp = 1, and a family of observables Gj, j = 1 , . . . , N , and we want to study the time evolution of their expectation values,9j(t) := Ti(pG j(t)) .The observables Gj , j = 1 , . . . , N tha t will be of interest for us later are number operators, with j corresponding to the mode number, and H will be the Hamiltonian of the P( s.The family of maps Vt, t € R, is referred to as a family of “projections” onto the space of time-t observables because V? = Vt, and we observe, however, that they are not orthogonal. We neither require the maps Vt to be algebra homomorphisms. The projections will serve us to break up the time evolution of an observable into a part “paraHel” to Vt, and a part “parallel” to the complementary projection Qt := id — Vt, where id : Mn(C) —> Mn(C) is the identity. Later, Vt will be chosen in such a way that the latter part becomes small in a suitable sense, and this will then be treated as a perturbation.4.1 General framework 524.1.1 D ecom p osition o f th e equations o f m otionTo start, and to simplify our notation, we note that the Heisenberg evolution equation for a matrix X € Mn(C) that comes from H can be written simply as? - a t ( X ) = i 6 o a t( X) , 6(X) = [H,X] = H X - X H . (4.1)atWe now decompose~ a t = i a t o 6 = i V t o a t o 6 + i Q t o a t o 6 = i a t o p t o 6 + i a t o Qt o 8 , (4.2)dtwhere we have found it convenient to introduce the “Schrodinger picture” operators V t = a_t o V t o a t . The goal is now to replace a t o Q t o8 with an expression that involves only terms with the map V t . To this end one notes the following differential equationd r n dt : ° Qt\ = OLt o i 8 o Qt + a t o — Qt =dt L J ^ ^ dt (4.3)= OLt ° Vt o i 8 o Qt + on o Qt o i 8 o Qt + a t o — Qt .a tThis is an inhomogeneous linear differential equation which we can integrate straightforwardly [cf. appendix B], and the result isott o Qt — Qo ° Yo,t + ! ota o°,t j «tV a o i 8 o Q a + — Qa aso Y8yt ds , (4.4)owhere Y 8yt: M„(C) -► Mn(C) is the cocycle (meaning that Ytl>t2oYt2>t3 = Ytl>t3) defined to be the solution to the differential equation^ Y ,t ( X ) = Yat o i S o Q t ( X ) , Y , , . (X) = X , (4.5)dtfor all X £ Mn(C). To solve the above equation we look for a solution of the form Y a t = Q_4oY8i(OQt, where Ys>t is the “Heisenberg-picture cocycle” . Combining equations (4.1) and (4.5) we immediately get the differential equation for Ys>t^ Y « (J f ) = -Y ,,( o i d o V t{X) , Y,, , (X) = X ,4.1 General framework 53the solution of which is given by the summation formula00 rY . A X ) = J 2 (-*)* / 6 o V a i o - - - o 5 o V„k(X) dka . (4.6)k=0 s C which reproduces the expectation values of the observables Gj(t) in a given state p. That is to say, wt ought to be a normalized (wt( I ) = 1) and positive (wt(X *X) > 0 for all X e M n (C)) linear functional that in addition satisfies the following property:= 9j(t) , (4.8)for all j . The above condition does not specify the state wt uniquely. Indeed, dropping the dependence on t for notational simplicity, let A = olg{Gj, j = 1, . . . , N } be the abelian *-algebra generated by the observables. A joint spectral decomposition Gj = E q= i f j ( a )p(a )i with p(a ) rank-1 projections, allows us to identify A with a subalgebra of the abelian *-algebra of functions F un ({ l,. . . ,n} —* C), and we can also identify Tr(p •) with a positive linear functional on this algebra. It follows by standard arguments then, that there is a non-negative function m : { 1 , . . . , n} —► R of total weight one suchthat gj = 53a = i/j(a )m (a )* We could, e.g. define our state (and reintroducing the tdependence) byn« ,,(* ) = T r [x £ > ( a W a ) ] . (4.9)a=lHowever, this is generally neither the unique, nor the most useful solution to equation(4.8). Instead we consider the unique solution that maximizes the “entropy” , i.e. we pickwt(X ) = T r(ptX) , T rpf = 1 , p* > 0 ,in such a way that the functionalSVt] = -T r[p t l°g Pt\is maximized and such that (4.8) holds for all j = 1 , . . . , N. A standard argument involving Lagrange multipliers, see e.g. Section V.1.3 of [25], shows that, in the generic case1, this maximizer must be of the formNpt = ^ (7) exp( “ X J ^ W G^(*)) •1 Generic here means that the maximizer is strictly inside the convex set of all positive linear functionals on Mn(C).4.1 General framework 55We now assume that the functions fij{t) have been chosen in this way, i.e. we are in the “generic case” for all t. In our applications below, the nature of the observables Gj implies that the functionals wt always exist, and we will simply assume the same here.We can now use the reference states wt to construct the maps Vt. First we note the following relationa _ a Tr _d ftM W‘ dni{t) Z(t) ~d ^ t ) Tr [e~ S* W([X] d ,M Z(t)Z(t) Z{t)= - w ^ G ^ X ] + w ([X]Si(<) = - ^ [ A G ^ t ) * ] ,where AGi(t) := G{(t) — gi(t)I. If X = Gj(t), we define the “correlation matrix” byc n(t) ■= - = V t [AG,(«)AGJ(<)] . (4.10)This matrix is positive semi-definite and we assume it to be also invertible. We denote the inverse as C%*(t) and remark that it can also be written as<4ji»with gj(t) = wt [Gj(t)] as before. The projection operator is now defined asNPt{X) = « ,,[* ] / + £ c i j ( t ) wt [AG,(t)X] AGj(<) =N(4.12)This projector is known in the literature as the “Kawasaki-Gunton projector” [10, 59]. We collect a number of immediate properties of the above projector in the followingL em m a 4.1. For any X € Mn(C) and any t € IR, we haveQ-t o ^{oL-t o V t o a t(X )) e span {G j(t) - gj{t)I \ j = 1 , . . . , N } . dtMoreover, for any Y € span{G i(£),. . . , we have Vt o 8(Y) = 0, and we alsohave wt[X] = T r[pVt(X)] for any X E M„(C).Proof. To prove the first statement, we introduce Vt — ot-t o Vt ° dt above, and we4.1 General framework 56also introduce wt = wt o a t . The first statement is then seen to be equivalent to the statement that £ V t{X ) G span{Gj — 9 j{ t)I | j = 1 ,. . . ,N } . Now we have, using the “summation convention” :Vt(X ) = wt(X ) + C*(t) Wt[X(Gi - <*(t)/)](G, - 9j(t)I) .When taking the ^-derivative of this expression, we recall equations (4.10) and (4.11) from which it immediately follows thatj t wt(X) = wt{X(Gi - 9i(t)I)} | r M m = mt[X(Gi - 9i(t)I)] 9j( t ) .This term cancels precisely the derivative of the second term in ^ V t when the derivative hits gj(t) in that term. The remaining terms are given by a linear combination of Gj — 9 j ( t ) I , as claimed.In order to prove the second statement, we just follow the definitions and use the cyclicity of the trace as well as the fact that each Gj(t) commutes with the density matrix p = Z(t)~l exp [ — A4j{t)G j(t)]. The last statement is again a straightforward consequence of the definitions. □As we will see momentarily, the properties stated in Lemma 4.1 lead to significant simplifications to equation (4.7). In fact, the last statement of the lemma says that we can evaluate the expectation value of any projected observable in the arbitrary (but fixed) density matrix state Tr(p •) by computing the expectation value of the nonprojected observable in the state wt for any fixed time t.Let us now derive the desired integro-differential equation for the 9 j(t). To obtain a particularly simple form, we shall make the initial state assumption that p is such thatw0(X ) = Tr(pX) , for any X G Mn(C); (4.13)in other words that p = Z ~ l exp [ — f°r some pj G IR. The physical meaning ofthis hypothesis will be explained below. W ith this assumption in place, we proceed as follows: We take the expectation value in our density matrix state Tr(p •) for X = Gj in equation(4.7). Then the first term on the RHS is seen to disappear using the last two statements of the previous lemma. The second term on the RHS is seen to disappear when acted upon by Tr(p •) using the last statement in the lemma, and the assumption(4.13) on the initial condition. For the first term under the integral in (4.7) we use4.1 General framework 57Lemma 4.1 repeatedly to writeT r[pV8 o 6 o Qs o Ys>t o a t o S(Gj)] = w a[6oQ 3 o Y st o <5(<3j(i))] == wa[SoY atto6{Gj{t))] .Finally, the last term under the integral disappears when acted upon by Tr(p •), because it is in the span of Gj(s) — gj(s)I , again by the preceding lemma, and this is annihilated by Tr(p •). Thus we arrive at the following:T h eo rem 4 .2 (Robertson equation). Let G j , j = 1, . . . , N be a set of hemnitian, mutually commuting complex n x n matrices, let p be a density matrix (self-adjoint, positive definite matrix of unit trace) of the general form Z ~ l exp(— ^ PjGj). Let gj(t) = Tr(pGj(t)), where Gj(t) = e~xtHGjeltH is the time evolved observable with respect to a self-adjoint Hamiltonian H, and let Y tjS : M„(C) —> Mn(C) be the cocycles defined as above in equation (4.6). Then the equationt= J ws[i6° Ys,t oz<5(Gj(t))j ds (4.14)oholds, where £(X) = H X — X H . For an arbitrary density matrix state p, i.e. if we do not make the initial state assumption (4.13), the equation takes the formt= (P ~ w o) [Yo,t ° W )] + J ws [iS o YSit o *£((%(*))] ds . (4.15)oHere, we are using the shorthand (p — wo)(X) = Tr(pA') — Wq{X) for any X .The equation in the above theorem is known in the literature as “Robertson equation” , see [60] for the original work and [10] for a review of its use. The initial state assumption (4.13) on the density matrix state p is made for convenience. It can be dropped at the expense of another term in the Robertson equation which encodes the corresponding initial state, see eq. (4.15). This term describes the way in which the influence of the chosen initial state persists to later times. In the model studied below, Gj will be the number operators at the initial time, with j a mode number. In that case, the initial condition can be viewed as saying that the initial state is quasifree, or in a sense, as uncorrelated as possible. This is a physically reasonable assumption, since what one wants to study is not the effect of strong correlations persisting from the initial state to later times, but the process of approach to equilibrium.4.1 General framework 58Another heuristic argument supporting the “initial state assumption” is the following. Suppose the only information we have about a state is the knowledge of the expectation values of a restricted set of observables, as is generally the case when measuring the state of the system experimentally, since it is impossible to measure all the (infinite) correlations between various observables. Then we would like to know what is the state that satisfies the “maximal ignorance hypothesis” , i.e. the state that only contains information about the restricted set of observables and at the same time maximized the entropy for these observables. Such a state is a “minimum bias” state and is by definition given precisely by wt. And by assuming that our initial state is wo we axe stating that what we are really interested in is the time evolution of our selected observables and not in their possible correlations. This can be either because the initial state really was quasifree, or because we can neglect the effect of the initial correlations in the dynamics we are interested in. Clearly, physically it will not always be possible to make this assumption, but for us it is meaningful to do so.4.1 .3 G eneralization for a tim e dependent H am iltonianOur derivation is somewhat different from the one in, say, [10], and it has the advantage that it can be transferred, relatively straightforwardly, to the case when the dynamics is given by a time dependent Hamiltonian H (t) which is smooth in t2. Let us explain briefly the changes that have to be made to the statement and proof of Theorem 4.2 in that situation. The time evolution of a matrix € Mn(C), with t0 some initial time, is now: ^fn(C) —► Mn(C) , a tyto(Xto) := E t ^ X ^ E ^ ,where= in J H (rn) * ' ' H (ti) dri • • • drn . (4.16)n = 0 t o < T i < . . . < T n < tThe Heisenberg equation in this case is^ < W * t o ) = iSt ° a t,«,(*<„) , St(X) := [ f f ( t) ,X ] , (4.17)and the square brackets [ , ] denote the commutator of two matrices, exactly as in the time independent case, see equation (4.1). Here a tyS is again an algebra automorphism and a cocycle, i.e. we have ottySoa3yU = a*tU. Note that now, unlike in the time independent2This situation is of interest because we would eventually like to apply the formalism to field theory in curved (time dependent) backgrounds, even though this will not be done in this work.4.1 General framework 59case, the time evolution automorphism a tyS depends on two parameters t and s rather than one. Additionally, the derivation St now depends on the time parameter t. The main consequence of these two facts is that a tyS and St do not commute in this case, i.e. a tyto oSt ^ 8 t o a tyto.If we now introduce the “Schrodinger picture” derivations 8t>to := at0,t ° $t° &t,t0 in addition to the “Schrodinger picture” projectors Vt,t0 = ctto,t0'Ptoa t,t0, the decomposition of the equation of motion now proceeds much like in the time independent case. We proceed to decompose the Heisenberg equation of motion (4.17) asq________________________— a tyto = iSt o a t,to = ott,to © iSt,to = ott,t0 ° Pt,to ° *^,to + a t,t0 © Qtyto ° ^t,to • (4.18)In complete analogy with (4.4) we now havet° Qt,to = Qto,to ° yto ,t~!■ J a s,to oW)VUtoso that equation (4.18) now becomes'P8,to ° * $s,to ° Qs,to + ^ Qs,to O YSit ds ,—P t ° © O!t,to + Qto ° n ,,t ° ° a t,t0 +tJ ^Vs © iS8 O Q8 - aS'to^iottO'S o V s o OLa M ) O a to ,s o YSyt © i8t o a tyto dstoTo obtain the above equation we have written Y3yt = a ^ s © Vs,t © at,t0, with YSft being now given by00 rY,A X) = J 2 (-*)* / ^ ° ^ ° - 0 ^ ° v °^ x ) Ak(J • (4-19)t = n* u 8 Mn(C) be the cocycles defined as above in equation (4.19).4.2 Num ber operator densities for the P(ip)2 m odel 60Then the equationt= { p - w o) [Y0,t° i£ t(G jW )] + J ws [z<5s oY s,t oz<$t (Gj(t))] dstoholds. Here, we are rising the shorthand (p — ojq)(X) = Tr(pX) — wo(X) for any X .4.2 Number operator densities for the P { y >)2 modelIn this section we introduce the number operator densities for the P( @(H) is the time-t (interacting quantum) field of the P( @(H) , ap(t) := i L 3/2 f dx[up(£)1?*),9(t>x )) : = iL J / ( 4>x ) •0In other words, we have(np(i)e"*px,nk(<)e_,kx) = ptelHt a* e~lHt , (4.22)and a* = a*{t = 0) in the notation used in Chapter 3. Because i f is a constant of the motion (ettHHe~xtH = if), we may write the creation and annihilation operator expression for H [cf. equation (3.28)] in terms of a#(t) rather than a*(t = 0) = ajjf for any t. This will be extremely useful below in Section §4.2.1 when we present the estimates of the number densities at some time t. Note however, that we cannot separately write Ho or V in terms of a*(t) at arbitrary times t because H0 and V are not, separately, conserved ([if, H0] ± 0 ^ [H, V]). Using (4.20) we can also express the field at time t in terms of creation and annihilation operators at time t. After a standard computation, which is easily performed in momentum space, we obtain■ (423)The commutation relations for a^t) and a^ t) are the same as the time-0 ones, see equation (3.8). We now define the “number operator densities”4N „ (t) : 2 (H ) ->■ 2 (H ) , N J t) := , (4.24)which are the quantities of main interest for this work. It also follows from (4.22) that the time-evolved number densities are given byATk(t) = eUHN ke - itH = ± ak(t)'ak(t) ,4This definition is motivated by our desire to consider the L -» oo limit, i.e. the thermodynamic limit.4.2 Num ber operator densities for the P(ip) 2 m odel 62where the second equality follows by a simple and straightforward calculation noting e.g. that ^aic(t) = i[H, UkM] — ^kflk- Using the relation (3.8) we immediately find 7Vp(t)] = 05. Finally, by splitting the Hamiltonian as H = H0(t) + XV(t) it is easy to see that- i[lf, WPW] = e > 0 we have the bounds|n*(t, A) - n*(s, A) | < K \ \ t - s |a ^ 1_e ,and|nt (t,A)| < Kuj-k i+l , urith a constant depending on e and L.Remark. The proof shows that the constants are of the order of Tr(p/P) where j is some positive number depending on e. This quantity is not uniformly bounded in L (in fact typically it is oc LP), hence the estimates are not guaranteed to be preserved in5The number operators hence form a set of mutually commuting operators, which was a key assumption to obtain the Robertson equation (4.14). The second fundamental ingredient for the application of the framework of Section §4.1 is that the time derivative of the “relevant” observables be given in terms of a commutator. By direct computation of dN^t) /d t and i[H, N^t)) we see that they are indeed the same, as one would expect. We do not write the resulting expression as it gives no insight whatsoever for the purposes of this work, but merely note that the equation of motion—the non-linear Klein-Gordon equation—is essential to establish such a result.4,2 Num ber operator densities for the P( 0 there is a constant K so thati i * w i i = « ) > # ] ( ' + w « r p / 2 u < w + e •Proof. We calculate that [N^t), H] = X[N^(t), V(t)] is a finite sum of operators W of the formw ( t ) = wk i/2 y : *(k-q ^ ^ e* ? ; ; E 'eyqj) ■qi»-"*qn€Z lli^ j^q* iex levwhere n < p, and where X U Y = { 1 , . . . , n}. By Theorem 3.4, we have- 1/2 q i M ( E ^ q < ,E « 6yq«)1/2Ui^j u llk|/where in the last hne we have used Holder’s inequality and that logx < K x 6 for large x > 0 and S > 0. □4.2 .2 Perturbative expansion estim atesLater we will also consider a perturbative expansion of the quantities rik(t, A) in the coupling constant A. Such perturbation expansions are known not to converge, but it is still of interest to know to what extent they can be trusted as asymptotic series. Unfortunately, we have been unable to get reliable estimates on quantities like the remainder term in the perturbative expansion of np(t) up to a given order. But it is possible to get, without too much difficulty, estimates on related quantities, e.g. if we let the time parameter t be imaginary. The same type of arguments also provide estimates on the error in the perturbative expansion of n^(t, A) for small t, essentially because the function in question is analytic in t for suitable states p.Despite the fact that our estimates on the error term in the perturbative expansion are not satisfactory for the main purpose of this work, the development of the Boltzmann equation, we will nevertheless present our results here as they provide at least heuristic support of the use of the perturbative expansion. Also, it could be seen as a general illustration of our method, which should be applicable in other contexts. The reader only interested in the main line of the argument can skip this section. We first describe the states that we consider, which are the density matrix states of the form (j3 > 0)p = e“ W V e - W 24.2 Num ber operator densities for the P(y>)2 m odel 66We are allowing both p and a = a* to depend on A, and we postulate thatdndAn< K nfor some constant and all n = 0,1,2, — Under this assumption, n^(t) can be continued analytically to complex t as long as Im< < p/2. Our result in particular covers the case a = / , i.e. p = e~&H. Our result is now the following:T h eo rem 4.6. Let p\ be a density matrix for each A > 0 satisfying the hypothesis above for some ft > 0, and let n*(t, A) = Tr(pxeitH^ Nke~ttH^ ) . Let r^ (t) be the remainder in be the Taylor expansion of nk{it) up to order N . Then for t < P/2, we have the estimate, „ (XK)N(Np)\I ^ “ N \u tProof We write n^(it, A) as a Taylor series to N -th order. The remainder in this series is given by the Schlomilch formula1\ NrN( \ , t ) = j ( \ — s)N(d " n k) ( i t , s \ )d s .Thus, we have to estimate the 7V-th A-derivativeA) = ~ T r .When we carry out these derivatives, they get distributed over the factors inside the trace. When derivatives hit <7, then we use our assumption that this is estimated by the factor K raised to the number of derivatives. When j derivatives hit one of the exponential factors, we use the iterated Duhamel formulaB — ^ c - s o H _ f e - ( s o - s i ) H y e - ( 8 i - s 2) H y m m m e - ( * j - i - 8 j ) H y e - S j H3 dAJ J80>3i>...>8j>0We will use this for so = P/ 2 — t > 0. Using the inequalities Tr(AB) < ||i4||Tr|B| and |TV(i4B)| < (rIV|i4|2)1/2(T r|5 |2)1/2, it is straightforward to see that the desired estimate will follow if we can show thatk I J - 1/27 UVUsing that TVk < lĵ H q < K u ^ H by the first order estimates H0(t) < K H (see [57]),4.2 Num ber operator densities for the P(ai>...>«j>0 'In order to continue, it is convenient to work in the Q-space representation. We introduce'XV|e-(«o-*i)JSrVe-(ai-«a)»Vr . .. Ve~siH H 1/2|2 dJs < K j (jp)\ . (4.26)the Holder space L9(Q, di/), with 1 < q < oo, as the Banach space with the norm \\ty\\q = ( f |^ |9 noting that H = L2. In these spaces we have the usual Holderinequality which states that |(4>, \P)| < ||4>||p||^ ||9, when 1 = - + J. This implies in particular that \\ f̂\\p < ||4>||9 if q > p , since is a probability measure of total weight 1. If T : U -> Lq is a linear operator, then we let ||T ||9jT. be the operator norm; the ordinary operator norm on H = L2 is a special case of this. The Holder norm of V can be estimated as follows for any j , see Theorem 3.8In the last step, we use Theorems 3.4 and 3.3, ||V(iV + I) p|| < K, as well as the fact that the V-*- 1f2o contains at most p(j — 1) “particles” . An induction then showswhere in the last step we have used Stirling’s formula. Thus, as a multiplication operator, V is an operator Lr —> Lq for q = r — l / ( K j ) with norm (K j)p. The second ingredient is Theorem 3.10, i.e. the fact that e~sH is a contraction from U to Lq as long as J < (1 + K s)K Therefore (by duality),These facts are now put together to estimate the integrand of equation (4.26).We first note that of the j interval lengths so — s i , . . . , Sj-i — Sj, Sj, at least one willin question is the last one, i.e. when Sj > r / j . Case (2) covers the rest. Both cases are< K ( 1 + (j - .be greater than or equal to so/j. We consider two cases. Case (1) occurs if the interval4.2 Num ber operator densities for the P( so/j. We first estimatee~>,HH l/2 < K s -1/2 _which leads to Tr < K s~ lrTrjQ ~ ( s k - i - S k ) H yk=l‘- s‘+l)wV)||xA’*Ve-(,i->i+l)wy*ywhere we are using the shorthand notations*—1 j —1X = TTe- (**-*‘+‘)JfV • Y = n .k=0 f c = t + lThe trace term on the right side is now estimated using that s* — si+1 > S o / j . In order to tame the factor of V under the trace, we write V = V ( I + N)~P(I + N )p, and we use the Rosen inequality with e = 3 to estimate (I + N )p < K H P, as well as the identity ||V (I + W)“pll < K. This gives- s 0H/(2j)^ ^ g K j / s o ^ j j p G~soH/(2j) || ^ QKj /8owith a constant K. In order to estimate \\XX*Ve~(s'~s'+1')HY*Y\\, we use the mapping properties of the multiplication operator V and the contractions e~rH between the Holder spaces IP. We then get2 j+ l(4.27)k = lbecause each of the 2j + 1 factors of V has norm (K j)p as an operator from U -» Lq, where q = r — l / (K j ) . This decrease in the Holder index for each factor of V is compensated by the increase in the Holder index caused by the 2j + 1 heat kernels e - { s k - 8 k+i )H jn x and Y in (4.27). In the last step we have used Stirling’s formula. Thus, in total we have shown that in case (2), we have/ 1 2\ J/2I TV e- (*°-*1>/,Ve"(*1_*2)wV • • • V e - ‘>HH 1/2 J < s ~ 1/2K j ap)\4.3 The pre-Boltzm ann equation 69and upon integration over the s*, this gives the desired bound eq. (4.26). Case (1) is dealt with similarly. □4.3 The pre-Boltzmann equationThe projection operator technique from Section §4.1.1 is now combined with the results of Section 4.2.1 to derive what we called the “pre-Boltzmann equation”. This is an equation for the expected number densities nk(£) = Tr[p7Vk(t)] and is, in a sense, a preliminary form of the Boltzmann equation, with which it shares some features. The number estimates from Section §4.2.1 are used to establish that the various ingredients (infinite sums) in the pre-Boltzmann equation make sense, even non-perturbatively. Despite some common features, it is also very different from the BE [cf. equation (2.10)] as it is still an exact equation. The Boltzmann equation, on the other hand, only holds when we consider appropriate limits like the thermodynamic limit L —> oo, and the long time limit t —> oo. In this chapter we will not yet take these limits.We first need to define the reference states wt [cf. equation (4.9)] for our model and the set of observables {Nk} where k € Z. In accordance with our constructions in Section §4.1.2, we let wt be the density matrix statewt(X) = Tr(ptX ) , p, = Z(t)~ l e x p (— 5 > k(t)iVk(t)) ,kezwhere X is, say, a bounded operator, Z(t) = Tr[exp(— MpW^pW)] and where the quantities /xk are defined through the formulaM i ) = _ l o g i w T T > 0 ' (4 2 8 )We now first discuss the rationale behind this definition and then argue that it is also mathematically meaningful. Recalling the discussion in Section §4.1.2, we anticipate that the above choice for the //k(£) is precisely the one we need to make in order to satisfy the requirement wt(Nk(t)) = Tr(p N k(t)) = nk(t), which is a crucial ingredient in the construction of the Kawasaki-Gunton projector (4.12). Indeed, it is easy to see thatnk(t) = I V [ ^ k(i)] =Tr[ptNk (t)} = A ^ t)] == ^ m o g Z ( t )Z(t) dfik(t)4.3 The pre-Boltzm ann equation 70Formally, the “partition function” Z(t) is given by (see e.g. Section 18.2 of [15])Z{t) = J J ( i - e‘ w'(,)V 1 , (4.29)kezfrom which it straightforwardly follows that Tr[ptiVk(2)] = nk(2) if the /xk are given by equation (4.28).To see that equation (4.28) makes sense, let us assume first nk(2) > 0 for all k € Z. As shown in Theorem 4.4 in Section §4.2.1, when the initial density matrix state p € (%)is such that also pH* € S i(H ) for all j , then we have nk(2) < K u ^ 4+e. It then easily follows that the partition function is convergent:Z(t) = exp^— £ l o g ( l — e-Mk̂ )^ < exp(k < exp^Ff 5 ^ ^ k 4+c) < 00 • kez kez kezThus, pt is indeed a trace class operator for any 2 € IR, and the state wt is well defined. The situation is the same when some nk(2) = 0, essentially because this means that P>k(t) = +oo, and this only improves the convergence properties. Thus, wt is a well- defined state if the initial state of the system p is such that p W has a finite trace for all j > 0.A different way to characterize the state wt is to say that it is the unique quasifree state with respect to the time 2-observables ak (2), whose 2-point function iswt(ak(t)*ap(t)) = L rip(2) <5(k, p) , (4.30)and whose n-point functions are zero for an odd number of creation/annihilation operators, and factorize into 2-point functions for an even number. More precisely, for X , Y C Z we havewt (n\ i e X j e Y J / :X -> y bijectivej€XIt is important to realize that for fixed 2, this factorization formula for wt will not hold for the creation and annihilation operators ak(s) at another time s ^ 2 unless the model is free, A = 0. The above factorization formula also demonstrates once again that the state is well-defined also when one or more n k(2)’s happen to be equal to zero, and we have, as desiredrik(2) = wt(Nk(t)) , for all 2 G IR, k € Z. (4.32)4.3 The pre-Boltzm ann equation 71The correlation matrix [cf. equation (4.10)] is found to be diagonal,CpkM = wt[(AWt) ~ np(t)I)(Nk(t) - nk(t)I)] = -^nk(<)(nk(t) + l)<5(k,p) .We now define our projector according to the general recipe laid out in Section §4.1.1. To be on the safe side, we first consider only a subset {N k : k € Z, |k| < A} of observables where A < oo, and we put pk = +oo for |k| > A in wt and denote the correspondingly changed state as w f. This change has the effect that equation (4.32) is valid only for |k| < A, and that eq. (4.30) returns zero for |k| > A. Also, A is not a cutoff of the theory, but merely a restriction on the set of momenta k in nk(t) that we monitor.The projector as in equation (4.12) is then:V?{A) = Wf (A) I + £ ^ {l Nr Z ~ ( 7 {Z )A) (Np{t) -WSK np(<)(np(t) + 1} / ._ d w A(A) ( ^= w?(A) I + £ ^ 4 ^ (Np(t) - np(t)I) ■|p|To obtain the formula in the second line, we used the analogue of equation (4.11) for Gj —> TVp. In writing that expression, we have also anticipated that, in the expressions below, A will be a power series in ajjf (£), and wt{A) can then be written as a corresponding power series in np(t) by equation (4.31). The operator d /d n k(t) then acts as the usual partial derivative operator on such an expression. In particular, it is clear from the last line that the np(f)’s in the denominator in the first line will always cancel, and so the case np(t) = 0 will never cause any problems.We also need to choose our initial conditions, i.e. the quantum state p that we would like to investigate. As we have just explained, in order for the states wt to be well defined, we require pH* to have finite trace for sufficiently large non-negative j . Furthermore, we would like to have an initial state so tha t the Robertson equation is valid without a “memory term” , cf. Section §4.1.1, Theorem 4.2. Thus, we would ideally like to choose as our state as p = po, whereA) = |e x p ( - £ M k A T k) . (4.34)kezfor some /xk. In other words, we would like to choose our state to be quasifree (with respect to the time-0 creation/annihilation operators a%(0)!), and we would also like our initial state to be translation invariant6. In the finite dimensional context, we were free to6That is, invariant under the 1-parameter group generated by the momentum operator P =4.3 The pre-Boltzm ann equation 72make this assumption. Unfortunately, in the present model with infinitely many degrees of freedom, a technical difficulty arises because we also need the initial state p to satisfy the condition that p W be trace class for sufficiently large j . This condition is needed not only in order to guarantee that the wt are well defined for all times t G IR, but it turns out to be essential also in order to give sense to the other ingredients in the Robertson equation in the present infinite dimensional context, see below. Unfortunately, there seems to be a conflict between demanding that pHj £ ^ i('H ), and that p be quasifree,i.e. equal to (4.34) for some p*. The reason for this conflict seems to be the presence of the non-trivial interaction XV in the Hamiltonian. The problem disappears if we only interpret the pre-Boltzmann equation in a perturbative sense (see Section §5.3), but here we wish to have formulae that hold non-perturbatively. So we are forced to introduce e.g. a damping factor into equation (4.34), such as taking7p = Z~x e x p (— /3H - Y ^ * N k)kezwhere (3 is arbitrarily small but positive [cf. Theorem 3.11]. This has the effect of creating a memory term in the Robertson equation, whose form is given by equation (4.15), with Gj —> Np. The memory term will clearly be of order /3. We will not bother much about the memory term, since this depends on the precise choice of the initial state. Also, when we pass to the perturbative expansion and consider various scaling limits such as the long-time/dilute-medium or long-time/weakly-interacting-system limits in the next chapter, we can take the initial state as quasifree, and in that case the memory term vanishes.4.3.1 DerivationW ith our definition of the projection operators and everything else in place, we can now formally appeal to the result obtained above in Theorem 4.2, see equation (4.15). This suggests:Theorem 4.7 (“Pre-Boltzmann equation” ). Let \p\ < A, where A < oo, and let the state p satisfy the “initial condition” (4.34). Then, i f Np(t) are the number operators for the (p4 theory, we have the following equation for the expected number densities np(t) =T , M L)aka*'7In order to see that this state satisfies H’ p € , one can e.g. use the Rosen estimates, Theorem 3.12,noting that on the right side we can replace H by H + pkNk.4.3 The pre-Boltzm ann equation 73Tr(pNp(t)), and dropping the reference to A on wt :t“j^npM = “memory term” — A2 j d s w s{[V (s),[V (t) ,N p(t)]}) +0oo *+ £ ( - i A r 2 £ I d s j w3{[V(s),[V(a1) ,N kl(cri)]}) (4.35)n=1 l*il,-,l*»l H B t y ) ! ! is a bounded operator with exponentially bounded norm on the closure H B (H )H of B(H) under | | . ||i.The lemma proves that the domains in the composition Ytl>t2 o Yt2tt3 match up, and the series formula (4.6) for the evolution cocycle then also shows that the cocycle condition holds,Y*,* o Y*2,t3(A) = Y tl,t3(A) for all A € H B(H )H .Prom this, it is now simple to demonstrate that it satisfies the desired differential equation.Proof of Lemma 4-8: Let A be a bounded operator. Prom the series expression for Y*>s, we can estimate, dropping the superscript “A” on the projectors and wt for notational simplicity:OO *ll»W*)lli < £ J \ \ S o V n o . . . 8 o V Ck( A ) \ h ^k = 0 8 < 0 \ < . . . < 0 k < ts f > ‘ / ( „ 7)*=0 ><»!< <^<< 'Pi' IP*ISA* - i |iu„,((W Pl(<7() - n Pl(ai)l) [V V l+ l) , Â Pi+1(cri+ i) ] ) |1 = 1np,(ai)Our aim is to show that each term under the sum on the right side can be bounded by ||;4||i(|£ — s|AlfA)fe/fc!, which will imply that the series converges absolutely for any t, and the inequality (4.36).In order to estimate the integrand, we use another lemma, which is at the heart of our analysis, and which makes crucial use of the estimates n^(t) < K uĵ 4+€ that were derived above in Theorem 4.4.Lemma 4.9. For all k G Z we have(4.38)4.3 The pre-Boltzm ann equation 75as well asM P M , [ m JV*(t)]])| < (4.39)uniformly in t and s. Furthermore, if A is a bounded operator, we also have|to,(JV,(«)j4)| < KtJpn p(s) || A ||i . (4.40)A proof of this lemma is given below. When we now use the estimates (4.38) and (4.40) from the lemma on the terms under the last integral in (4.37), we note that the dangerous factors of nlfs )~ 1 precisely cancel out with the corresponding factor in the estimate, and the subsequent factors of cjp and its inverse also cancel. In formulae, we havell« o 7 V ,o ...« o * V k(i4)||I <<(*a)*piii % in*.*p>.)]Hi n np,~ia‘i “p[ f p‘IP.I. ...,|P*I a k> . . .> a i> soo< J 2 {Klt ll^lli = e ^ '^ 'l l A l l ! ,k=0so the convergence of the series (4.37) follows for all t for any A with ||/L4i2|| < oo. This proves Lemma 4.8. □To complete the proof of the theorem, the only thing left to do is to check that the combination i/>s[5oYt)So<$(./Vp(£))] is well defined. We can estimate ws[<5o Yt>so ' * * > > «}• This shows convergence, and completes the demonstrationof the theorem up to Lemma 4.9. □Proof of Lemma ^.9: The proof of all estimates is rather similar; we show the first estimate (4.38). We havew .([JV k W , V ( t ) ] J V p « ) = w .([N k(t), V(s)]Np(s))= w ,(N k(t)V(s)NP(s)) - w ,{Np(s)V(s)Nk(t)) .The terms on the right side are estimated in exactly the same manner. We demonstrate the argument for one of them. We have, using the Cauchy-Schwarz inequality together with the fact that ws(N^(t)V(s)Np(s)) = ws(Np(s)l/2Nk(t)V(s)Np(s)^ 2):w.(AlWV(*)Arp(*))| < (ws(Np( s y '2Nk(t)2Np(Sy ' 2) )• (w»(iVp(s)1/2V(s)2iVp(s)1/2) )1/21/2Using the inequality (\&, (j4 + 2?)24>) < 2(4>, (A2 + B 2)\I>) for hermitian A, B , together with equation (3.33), Hq < K H 2, we can estimateNk(t)2 < K u£ 2H$ < K u f H 2 < 2K ^ 2(H0{s)2 + A V (s)2) ,so we findws(Nk(t)V(s)Np(s))\2 < K u ? ■ w, ( n p(s )(H0(s )2 + A2V(s)2)) ws [N p{s )V {s f) . We continue our estimation by using Theorems 3.4 and 3.3, which imply that V(s)2 < ||V (a)(/ + iV(s) ) - ^ 2||2 (I + N ( s ) f < K ( I + N(s))p .This then allows us to estimatew,(Nk(t)V(s)Np(s))\ < K u ^ L > , ( N p(s)(I + N(s))py + + w ,{H l(s)N p(s)) w.{Np(s)(I + A T ) ) ’ .Now, we recall that the total number operator is N = Jlkez Nk, and the free Hamiltonian(4.41)4.3 The pre-Boltzm ann equation 77is Ho = similarly for the quantities at time s. Thus, in view of theinequality just given, we have reduced the problem to that of estimating quantities of the form Wg(N\il (s) . . . N^r(s)). For this, we need a simple combinatorial formula. To derive this formula, let X be any finite subset of IN, and for each i G X, let a* G N. Then we have, using elementary Fock-space algebra:,*ex / iex v iex ' Zi=oT T — —i i Z ( t ) 9 * 1*exTrj^expiex 1Ci=on at?* z ( t ) n i _ e-<*p,+?. n iiexniexiex 1 _ e-MPi(s)dg*x 1 — e-/W s)+&M P i,*ex d**e~^kn&=01&=o »ex &=o= n np /s)^«i(nPi(s))i e xIn the above calculation we are using expression (4.29) for the partition function Z(t) and that np(t) = e_Mp̂ / ( l — e_/ip), see equation (4.28) and the explanation after it. The ij>n{z) are the degree (n — 1) polynomials defined iterativelyJxpi(x) = 1 , and tpn+i(x) = — [x(l + x)^n(x)] .This formula implies (dropping the reference to “s” in nPi(s) on the right side to lighten the notation):wf ( W k , w - ^ w ) = ^ E EXiU...UXn={l>-”>P} ^ 42)n « * . • * > - n *(Pn» • • • nPntp\xn\(nPn) .jeXi jex„With the help of this formula, we can now easily estimate quantities like e.g. w8(N(s)p). In such an expression we have a p-fold iterated sum over expressions of the form (4.42). The key point is now that, after taking into account the Kronecker delta’s, we are left with iterated sums each of which is accompanied by at least one factor of n^(s). Because we have the estimate rik(s) < Kcc£4+c from Theorem 4.4, such a sum will converge. If we have e.g. an expression of the form w s ( H q ( s ) 2N ( s ) p ), we can make a similar argument.4.3 The pre-Boltzm ann equation 78Now, after taking into account the Kronecker delta’s, we are left with iterated sums which, at worst contain a factor of (from the squared free Hamiltonian), and at least one factor of fik(s). Again, because we have the estimate n^(s) < K lĵ 4+€ from Theorem 4.4, such a sum will converge. By making simple arguments of this kind, we thus easily arrive at the basic estimates:w, (Np(s)H0(s)2) < K n v (s)u>l , ws (Np(s)(I + N(s))p) < K n p(s) , (4.43)which allows us to continue the estimation of (4.41) asw»(Wk(t)V(«)JVp(s))| ^ + = •ffnp(«Vk1{ wp + £̂" }a .from which the estimate \ws(N^(t)V(s)Np(s))\ < K tip(s)ujpuĵ 1 follows since for K big enough we have K ujp > K ' because ujp > 0. We find that this arguement leads to the same estimate for \w3(Np(s)V(s)Nk(t))\, which concludes the proof of the first inequality (4.38). The second inequality (4.39) is dealt with in a very similar fashion. For the third inequality (4.40), we can argue e.g. by saying thatw,{Nk(s)A) = ws(Nk(s)l/2A N k(s)l/2)< Wil + H r ' A V + H y 'W w s i N k l s y V t f N ^ s ) 1' 2) .We continue the estimation byw.(Nk(s)1/2H 2Nk(s)V2) = w ,(N k(s)H 2) < 2ws (Nk(s)H0(s)2) + 2X2ws(Nk(s)V(s)2)< 2ws(Nk(s)H0(s)2) + 2 K \ 2w,{Nk(s)(I + N ( s ) f ) .The expressions on the right side have already been estimated in equation (4.43), and hence the desired inequality (4.40) follows. □4.3.2 A lternative form o f th e pre-B oltzm ann equationIt is convenient for later purposes to write the pre-Boltzmann equation (4.35) in a way which makes more manifest the dependence of the integrands on the number densities rip(s). For this, it is convenient, to introduce the “collision kernels” B A(E, p, s) by the formulaB A(E ,p ,s ) := ^ f d t e - iE^ w A( [ V ( s l N p(t)}) . (4.44)R4.3 The pre-Boltzm ann equation 79From now on we will drop the reference to A, for simplicity of notation. The terminology for these kernels will become clear later, where we will relate them to scattering cross sections. These kernels are distributions in E that are defined for any s € IR and p € Z. We claim that the pre-Boltzmann equation can be written entirely in terms of these kernels. We will demonstrate this now for the collision term on the right side of the pre-Boltzmann equation. Similar arguments can also be applied to the memory term. However, this will later be set to zero anyway by an appropriate choice of initial state, so we will not discuss this here.The statement is clear for the first term on the right side of the pre-Boltzmann equar tion, since the factor of E in front of B (E , p, s) can be converted to a ^-derivative in the integrand, which in turn yields the resulting first term on the right side of the pre-Boltzmann equation in view of ^ATp(t) = i[H, Np(t)\ = i\[V(t), Np(t)], using [H0(t), Np(t)] = 0. To cast the remaining terms in a similar form we note the following chain of equalitiesAwr([K (t) ,^ p (* )] )= « r ([« » W + AV(i),JVp(t)]) = wr ({H(t),Np(t)}) == w T ( [ H ( T ) , N p (t)}) = A™t ([V (t),JVp(<)]) == Aiot ([K(t), Np(t)}) - A w rflK M , Afp(r)]) = (445)This equality puts the terms appearing in the second half of the pre-Boltzmann equation into a form similar to the first, and so we can again express them through the collision kernel B (E ,p , s). Above we use [H0(t), N p(t)] = 0 in the first and fourth equality, and energy conservation, i.e. time independence of the Hamiltonian, to go from the first line to the second line. Combining equation (4.45), the definition of wt , and that of the collision factor, equation (4.44) then leads to the following proposition:P ro p o sitio n 4.10. The pre-Boltzmann equation can he expressed in terms of the colli4.3 The pre-Boltzm ann equation 80d_ d tsion kernels B (E , p, s) ast■np(t) = “memory term” — J ds J dE etE^~8̂ B(E ,p ,s) +o Roo t- J 2 [ d s f dE f dnr d V f dnE ei£?(n- s)B(E,fei,s) • (4-46)n=1 0 R Aa»(s,t) Rn |fcil.-»|fcn|x := , px x := Y iE v 0* ■ iex (g l)iex iexWe also recall the notation p = (o;p, p) to denote on-shell momenta.5.1 Collision kernel evaluationOur aim in this section is to derive an alternative expression for the collision kernels B (E ,p , s) in terms of (local) “Moller operators” , or local 5-matrices. We define the local Moller operators as5.1 Collision kernel evaluation 84which are unitary operators on H. Here, H0(s) is the free Hamiltonian defined at time s by Ho(s) = etaHHq e~l8li. It can be written alternatively in terms of the time-s creation and annihilation operators ajf(s), see equation (4.20). The true 5-matrix of the theory (which exists for this model, see [61]) is given in terms of these local Moller operators byS = s- lim S (0 ,t)S (—1,0) = s- lim S ( —t,t)t-+oo t—► ooon a suitable domain of vectors in H, in the thermodynamic limit L —► oo. We also note the following propertiest^ 5 (s , t)il> = —iX (s) 5(s, s) = 1 , (5.2)atfor suitable vectors € H. The terminology “local Moller operators” arises from the fact that they are equal to the 5-matrix of a theory wherein the interaction is switched on in the time interval [5, t] only, i.e. they are “local in time” .5.1.1 Step 1Using now the above properties and the fact that e1̂ -3 1̂1 Np(s) = Np(t) weeasily see thatA w8 (s), iVp(<)] ̂ = - id tW a(Np(t)) = - id tW s (et{t~s)H Np(s)e~x{t~s)H>j =— -id tW 8(S*(s,t)Np(s)S(s ,t)) = - i d t u;s (5*(s,t)[iVp(s),5(s,£)]) .With this result we rewrite the collision kernels B (E ,p , s) as£ ( £ ,p ,s ) J dt e- iE(t~8) ws (s*{s,t)[Np(s),S(s,t)ty . (5.3)RWe can summarize the above results in the followingP ro p o sitio n 5.1. The expected number densities np(t) satisfy the integro-differential equation (4.46), where the collision kernels B (E ,p ,s ) are given by equation (5.3).Before we proceed with our derivation of the BE, we pause to note some properties of the local 5-matrices just introduced. Equation (5.2) implies the following differential equation: Q—5(s, t)il> = -iX V 0(t) 5 (s, t)tl>, 5 (s ,s) = 1 , (5.4)5.1 Collision kernel evaluation 85where we have introduced the “interaction picture” potentialV0(t) := ei(t- s)Ho(5) V(s) e . ( 5 . 5)The latter can be rewritten in terms of “time-s free fields”, denoted x), as follows.They are defined by demanding that, at time s we have y?o(s,x) = y?(s,x), and bydefiningx ) : • (5.8)n=2 {as one can easily see by running through the definitions. Here, the normal ordering prescription indicated by double dots is defined by expanding ipo out into creation and annihilation operators at time s, see equation (4.20), and then moving the annihilators to the right::¥>S(«,x): = Y ; n ^ ( ^ ) W n ^ M ) ^ ) . (5-9)Pl,...,pnez»ex jeYwhere X , Y C Z and \X U Y\ = n. Note that this prescription depends on the time s chosen. If we combine this with the differential equation (5.4) given for the local 5- matrix, then we can express it in terms of a formal power series in the free field (po- This5.1 Collision kernel evaluation 86is obtained by integrating the differential equation (5.4) in the sense of formal power series in A, and the solution isS (M ) = d r i...d r n Vb(r„). . . Vofa) . (5.10)n=® S —oo), and is then called “Haag series” [64, 65]. But a similar expansion is also valid for finite s, see e.g. [66, 67, 68]; it is given below in equation (5.22).5.1.2 Step 2To further proceed in the derivation we will express the local Moller operators in terms of time-s creation and annihilation operators. We claim that the following formula holdsS(s,t) = Z l + Z2irLY^ £ M {s,t{ X p ^ Y q) 5 (p x ,qy )(5.12)TT TT aP,(s)11 ( O . T j . i . U /2 112Lw^)V* (2 1 ^ )1 /2 ’je Y v iexwhere X , Y C N with \X U Y\ > 1, the overall factor Z accounts for the correct normalization, and we refer the reader to the remarks preceeding (5.1) for an explanation of the above notation. This formula can also be obtained in perturbation theory by using equation (5.9) in the expression for the interaction picture potential (5.8) and substituting the result in the formal power series expression of the local 5-matrix equation (5.10). This will give us at the same time a perturbative formula for the quantities M . We want to emphasize, however, that in order to get the above equation, we actually do not have to make a perturbative expansion of the local 5-matrix since it is valid non-perturbatively. And the reason for this is that on Fock space we can expand every operator as a sum of (products of) creation and annihilation operators. We have5.1 Collision kernel evaluation 87collected some details about this latter point in Appendix C, and refer the interested reader to it as well as to Section 6 of [69] where an expansion like ours was studied. The “momentum conservation” delta in the above expression can be understood by noting that the local Moller operators commute with the momentum operator on %.For completeness, we now give the perturbative expression for the local scattering amplitudes M here in terms of (standard) position space Feynman diagrams. One can expand the time ordered products equation (5.10) into normal ordered products of the free field • Yq) = f ^ ( - iX ) n J 2 CgL" [ d2x' ■ ■ ■d21" •n=0 G:|V(G)|=nsS(*) :o= £ (n - 2 j) \ j \ 2> M ’ : Vo~V (x ) ■> > (5-16)where Sm2 = ]imy^ x{ws[ipo(x)ip0(y)] - ((fio(x)(po(y))0}, (cf. equation (5.43)) and [n/2] denotes the integer part of n/2. The advantage of working with Wick-reorderd products is clear: Their expectation value in the state ws vanishes identically.As for the A4, the quantities M. have a perturbative expansion. To obtain this expansion, we may Wick-reorder the interaction potential (5.8); we obtain the following expression27TM t) = L ^ 2 bn f d* :Vo(*>x )n=l iwhere we have dropped a term proportional to 1 because it is irrelevant for what follows. We note, however, that in the above expression we now also have terms linear and quadratic in the fields 0, by Theorem 4.4. As an aside, we can now combine equations (5.15) and (5.11) to getM(s+a,t+a)(px -> qy) = M (s,t{px -> Qy ) exp{ia(ux - uy) } • (5.19)Before proceeding to the evaluation of B (E , p, s) we need one more result, and we state it here as aP ro p o sitio n 5.2. Let X , Y , X \ Y ' C IN, with \X U Y\ > 1 and \X' U Y'\ > 1 . I f \X\ = \X ’\ and \Y\ = |Y'|, we havew>(. n ap(,(s ) n % ( « ) :i i a« /s ) i i ap<(s) :« ) =\*'€X' * r e v jeY iex ) (5 20)= ( l I nft(S)I'l5(Pi’Pi;)+ (*< ^*J))(lI( 1 +" % ( * ) ) >' i e x ' \ e Y 'and zero otherwise.Proof. The proof relies on Wick’s theorem for the factorization of the above expression in products of expectation values of “two point functions” , i.e. expressions containing a creation and an annihilation operator only. A crucial observation here is that because the5.1 Collision kernel evaluation 91products of creation and annihilation operators above are normal ordered with respect to the state w3, the only non-vanishing contributions will arise when “contracting” one creation (annihilation) operator from the first normal ordered product with an annihilar tion (creation) operator from the second one. Let us prove, then, that when \X\ ^ \X'\ or \Y\ 7̂ \Y'\ the above expectation value vanishes.Suppose, for the sake of the argument, that \X\ = \X'\ but \Y\ > \Y'\. Applying Wick’s theorem we will then be able to “fully contract” all the annihilation operators corresponding to the sets X, X ' and Y ' but not Y . The above expectation value will hence be proportional toSimilarly, if |Y'| > |Y|, the expectation value (5.20) will then be proportional to a (vanishing) term like the above one where the creation operators are replaced by annihilation operators. It is also clear that this argument equally applies to the case when |X | ^ |X '| but \Y\ = \Y% We are hence left with the case |X | ^ \X'\ and \Y\ ^ |Y'|.For definiteness, let us assume first tha t |X | > \X'\ and |Y| > \Y'\. Applying Wick’s theorem we will be able to fully contract all the creation and annihilation operators corresponding to the sets X ' and Y ’ but not X and Y , the result being thus proportional to. |y|-|Y'l \x\-\x’\ vw*(: n aq/s) n apj(s):»)=0'^ j=i i=i 'which vanishes because of the normal ordering prescription. If, on the other hand,> |X '| and \Y\ < \Y'\ then the result will be proportional toAY'\-\Y\ \X\-\X'\ v”'»( n aq;(s) n )=° •^ j=i i=i . *An identical argument applies to the case |X | < \X'\ and \Y\ ^ \Y'\, thus showing that if |X | ^ |X '| or \Y\ / \Y’\ the expectation value in (5.20) is indeed zero. Finally, when |X | = |X '| and |Y| = \Y'\, a direct application of Wick’s theorem leads to the above result, and where the single contractions have been evaluated as w(a^a^) = np<$(p, k)and ty(apa j) = (1 + np)£(p, k). □5.1 Collision kernel evaluation 92C o m p u ta tio n of B (E , p, s): After this preparatory work has been done we are ready to express B (E , p, s) in terms of the dressed local scattering amplitudes. We haveE 2B (E ,p ,s ) - J / d , e -iE'‘- \ ( 5 ( s ,t)*[iVp(s ) ,5 (S,()]) == fr / dt e"iE<‘"s) X X L S (P x ,< Iy ) L 8 (p'x , , ^ y , )X , Y P1,...,P |X |€Zpi,...,pJx / |eZ ’ Q i.- .q m e z q/ v >q/y/|GZ• -+ -+ n ) •w,ap',(s) „ Oq',(s) ° q / s ) T T O p,(s)n (21J', )V2 .n (2ijq;,)v2 =• [n^ ’ n (2L̂)i/2 n= ^ / d< e - iE(‘- s) ] T X XI i<5(Px.qy)<5(Px'.qy')x,y Pl,...,P|X|ez p ',...,Pjx / |GZ ’ q i ’" ’q m GZ q'x,...,q' ,,GZy ;)M (J,t)(x p -+ y,) L [X > (p ,q ;) - X * ( p -p*)]jey iexw. ■pr a p ' / S) t t a q '/ ( s ) . .T T a q/ * ) T T O p /5 )11 /97V, , M/2 11 /9 7 V , , M /2 s 11 /97V, M /2 11/ 6* ( 2 ^ ) 1/2 ( 2 ^ q ; J 1/2 “ (2Lo7qi )1/2 11 (2L(Jp.)1/2#227T x,y Pi,-,P|x|€ZP',...,pjx,|ez q i ,...,qm e z q/iv ..)qjy / |6 zMX ̂ (p .q j) - X ^ ^ P ’P^]j e Y i e xws t t a p ' / 5 ) T T . .T T ° q / S ) T T a P,(S)11 tor,., . M/2 11 (O T ,. , . M /2 ’ 11 (or,., m /2 11J A (2^ ) 1/2 A i ( 2 ^ q '- ) 1/2 " h r (2L^ )1/2 “ (2Lv )1/2In the above chain of equalities the first one amounts to using the expansion (5.15) for both S and S* while neglecting the terms proportional to the unity as their contributions vanish. In fact, there axe two such contributions arising from the expansions of S and S*. The one associated to S can be dropped as the unity commutes with Np. The contributions arising from the unity in the expansion of P<)]J iex P,ez jeY iexieY (5.21)• -> Yq) | L S (p x ,q Y) n Pi(s) [l + nqj(s)l ,i e x jeYafter performing the sum over the p', and q'-, and noting that for Kronecker deltas in a finite volume we have <5(k,p)6(k,p) = <5(k, p). Finally, the factors \X\\ and |F |! arise from the permutations in equation (5.20). This is the expression we were looking for, and we want to emphasize that it is an expression which is valid non-perturbatively. As we have stated above, the expansion of the 5-m atrix (5.15) in terms of creation and annihilation operators is valid non-perturbatively, and this is all we needed to arrive to the above result. That such an expansion can be understood also perturbatively or the fact that we have presented a perturbative expansion of the local scattering amplitudes (5.18) should not conceal the non-perturbative validity of equation (5.21). The next step will be to consider the infinite volume (thermodynamic) limit L —> oo and the scaling (weak coupling and/or low density) limit. Before that, however, we give an alternative perturbative expansion for B(E, p, s) in terms of retarded products.5.2 The collision kernel in terms of retarded productsWe now give an equivalent alternative way to expand J5(FJ,p, s) in a formal power series in A. This derivation relies on an alternative expansion of the interacting field ip in terms of the free field ipo. The relevant formula is well known in the literature (“Haag’s series”), see e.g. [64, 66, 67, 72]. It is, in our notation(p(t, x) = Ifidt, x) + Y f Hn x); v0(n) ® ® Vo(r„)) d"r , (5 22)n=1 [»,(]»which is valid for t > s. Here, the notation 7Zn means a “retarded product”. Mathematically, it is convenient to take the view that it is a bilinear map Tin • F x (nJr)5.2 The collision kernel in term s o f retarded products 94End('H), where T is the linear space of classical local expressions of a fictitious classical field po of the form A = / W [p^x), dpt(x) , . . . , d rp jiAi) with n factors; it is given by a sum of multiple commutators multiplied by step functions, see e.g. [67]. However, if the arguments of the retarded products contain derivatives, then this simple formula becomes ill-defined, essentially because one then has to perform renormalization2. In this case, the retarded products may be thought of as defined by a combinatorial formula in terms of time-ordered and anti time-ordered products T and T respectively, (see e.g. [66, 67]), which is*=1 X W = {l,...,n} jex i e YThis then leaves one with the product of defining the ordinary time ordered products, see e.g. [63, 66, 67, 70, 73]. It follows from these constructions that the time ordered/retarded products have an expression in terms of a f (s), the creation/annihilation operators at time s.The retarded products owe their name to their support properties: If B, Ai are2When carrying out this renormalization, it turns out to be of considerable advantage to consider the arguments of the retarded products to be classical expressions, and this is why we proceed here in this way.5.2 The collision kernel in term s o f retarded products 95polynomials in ipo and its partial derivatives, viewed again as “classical expressions”, then we have (see [74])s u p p f tn ^ B ^ J L ^ ^ C {(xi , . . . , x n,x)\x i G s u p p ^ fl J “ (suppB),x e suppB, for al i i = 1 , . . . , n J ,where where J ±(5) denotes the causal future/past of a set S C IR x S 1, and where we define the support of an expression A = f W[ip0(x), d(po(x) , . . . , &r(po(x)]f{x) d2x to be equal to the support of the testfunction / .We can now start with our task of expanding expressions on the right side of the pre-Boltzmann equation in perturbation theory. This is accomplished essentially by inserting a perturbative formula for the (interacting model) number operator Np(t) into eq. (4.44). In order to do this in an efficient way, we proceed as follows. First, we note that, by the Glaser-Lehmann-Zimmermann (GLZ)-formula (see [67, 65]), we have3:P^), *M*)1 == £ (fa + m+M / [^n(Vo(s); (g ) V „ fo )),* » (# ,(* ) ; Then, multiplying both sides with a step function and using the support properties of the retarded products, we get:9(t - .)[V M , Np{t)] = j r ( Np(t); V0(s) ® (g ) V0{ c , ) ) d V .n=o n - \ i= i )We now use this expression in our expression (4.44) for the collision kernel B (B ,p , s). In this formula, we may multiply the integrand by a step function 9(t — s), because the opposite step function $(s — t) would give a contribution to B(E, p, s) that is analytic for ImE > 0, and which would for this reason vanish when substituted back into the3 Note that N^t) when expressed in terms of the free field ipo is not a local expression in T . However, it is still local in time, and this is sufficient in order for the retarded (or time ordered) product to make sense in two spacetime dimensions.5.3 Weakly interacting and dilute system s 96pre-Boltzmann equation. The (unproved) assumption here is clearly that when we deform the intergration contour in the upper half plane, the contribution arising through this deformation vanishes in the limit where the integration (semi-circular) domain is extended to an infinite radius. Therefore, assuming that these contributions are irrelevant, our collision term becomesThis is our second expression for the collision factor. The expectation values of the retarded products in the states ws can be evaluated in terms of Feynman integrals with “propagators” (5.17) using a version of Wick’s theorem [68, 66, 67], because the retarded products are expressible in terms of ajjf(s), and because the states ws are quasifree, cf. (4.31). The propagators can be evaluated in terms of the factors n^s), but we will not show this here. In higher dimensions d > 0 (or for any model that is not super-renormalizable), the fully renormalized retarded product must be understood in the above expression.In summary, our pre-Boltzmann equation (4.46) together with the perturbative expression for B(E, p, s) gives us a closed set of integro-differential equations for the unknown quantities n,/£). These equations are not particularly simple, but we will see in the next section that they form a good starting point for a further expansions, namely a simultaneous expansion essentially of the inverse observation time 1 /£, and the coupling constant A (or typical initial density np(0)).5.3 Weakly interacting and dilute systemsWe would now like to take the long-time limit and the thermodynamic limit (L —> oo) of the number densities {np}. In order to have any reasonable hope that such a limit might exist, it is clearly necessary on physical grounds to take simultaneously (at least) one of the two further limits:(a) The low density limit (i.e. the initial densities np(0) —> 0) and/or(b) The weak coupling limit (i.e. A —> 0) .Both limits have been studied previously. For instance, the weak coupling limit has beenstudied in [6] and in [9] in the context of a lattice Fermi gas with 2 —> 2 interactions5.3 Weakly interacting and dilute system s 97within the framework of (non relativistic) quantum mechanics. The low density limit, on the other hand, has been analyzed in, e.g. [12] for a quantum particle interacting with a random potential. For weakly interacting particles the common feature is the emergence of the scattering amplitude in the Born approximation whereas for dilute gases a full scattering amplitude emerges.The idea in both cases is to introduce a new small parameter, e, into the problem to “control” the limits. The physical meaning of this parameter is not that of an additional coupling constant, but instead it characterizes the initial density matrix state p(e) of the system (or the strength of the interactions for weakly interacting particles), as well as the time over which we observe its evolution. Roughly speaking, the time duration over which we wish to observe the system is of order e-1, whereas the initial densities (coupling constants) are of order ea for suitable a > 0. The idea is then to consider an expansion of the observable quantities in the new small parameter e. There are two related issues associated with this limit:1. Does the limit of {np(t)} exist?2. What equation does it satisfy?In what follows we will study what limiting equation do the (suitably rescaled) quantities {np(t)} satisfy in either (or both) the low density as well as the weak coupling limit. In doing so, we will assume that the answer to the question 1. above is yes, and we will arrive at different answers for dilute and weakly interacting gases. Both answers have some features in common and the BE as stated in the Introduction will arise only when both limits are considered simultaneously. These results will be obtained using the pre-Boltzmann equation and the perturbative formulae for the collision factor B(e, p, s) derived in sections §5.1.2 and §5.1.2. Our derivation is in part formal because we do not control the perturbation series and we will interchange several limits and integrals without proper mathematical justification, but we believe a more rigorous derivation could be given.5.3.1 T he therm odynam ic lim itBefore proceeding to explicitly consider the scaling details of the weak coupling and low density limits we reconsider the pre-Boltzmann equation by taking the thermodynamic limit L —> oo. As already mentioned, we are going to discuss this limit only in a heuristic fashion as we believe that a rigorous treatment would be considerably more involved while leading to the same conclusion.5.3 Weakly interacting and dilute system s 98Since the quantities n ^ t) were defined as densities [cf. (4.24)], we expect that they will possess a well defined thermodynamic limit. Physically this means that the particle densities is finite. Assuming this to be the case, the pre-Boltzmann equation is expected to continue to hold in said limit, at the very least in the sense of formal power seriesin A. To obtain the collision factors, one must then only make the following (standard)replacementsnp(£) , p € Z — —> np(t) , p G IRLd/dn p , p € Z — 8/5np , p € IR2irL J dx ( • ) , x € [0,2n\ L~>°°> J dx ( • ) , x € IR0 Rp e Z / * / ( p > . p G l R(p, x) = —Pot + px , p € Z L~*°°> px = - p 0t + px , p € IRL<5(p,k) , p ,k e Z L~̂ °°> 5(p —k ) , p , ke l Ro;p = y /p2/L + to2 , p € Z L̂ °°> cj(p) = \ /p 2 + m2 » P € IRIn the second line 8/drip is the functional derivative with respect to the particle number densities np(t) in the infinite volume case [cf. Section§3.1, equation (3.1)], and the other modifications are self-explanatory and intuitive. After the thermodynamic limit, our QFT is defined on R̂ 1,1̂ rather than R x S. The pre-Boltzmann equation now readstiip(t) = J As J dE ei£(T- s)B (£ ,p , s) +0 RT+ / d a f d n . . . d Tn J dki . . . d k n J d E eiB(r' - s)B (E , k u s)n=1 0 S < T i < ~ < T n < T R Rn Ti+1n5S 3 / ,*- 1 Ti Rand we still expect this equation to be an exact equation.As a preparation for later, we scale the time t and introduce a rescaled “time” T by writingt = T / e ,5.3 W eakly interacting and dilute system s 99which in addition to a change of variables s -> s/e and E -¥ eE in the integral on the first line below and analogous changes for the remaining terms, we then use to rewrite the above equation asid rn p(T /t) = ^ f d s f d E eiE(T- s)B (e£ , p, +0 RT+ £ y f Jds J A n . . . drn J d k ^ .d k n J d E eiE^ B ( e E , k lt J )Ti+1n * s 3 f ) ■* 1 e n R(5.23)We note that so far we have only performed a trivial change of variables and so the above formula (presumably) still holds exactly. In this form, it is also formally clear that one needs to consider some additional scaling if the result is to be finite in the e —t 0 limit. Finally, we will also consider the A —► oo limit in order to have a closed set of equations for all the momenta.In order to write the rescaled collision factor we use the property (5.19) to shift the squared local scattering amplitude\M(S4 X P -> Y„)\2 = Mjv,t-4XV -*■ - f Yq)e -« “x-**> == \M{0,t- a)(Xv -»• y ?) |2 ,and rewrite the rescaled collision kernel (5.21) in the thermodynamic limit asbUe,pA\ = £ |X | ! | y | ! [ d n XpUyq [ dt e g e - £t I M (0,t/({X P -> Yq) I* ••EX t K ,j & R R (5.24)• Hpx - qy)[X^(p _ (̂p _ p*)1 II nv.ish) II f1 + ni/s/e)) >jeY i e x i e x jeYand where we have performed another change of variables t — s —>t and introduceddk dkd n XpuK, == I I d n p.d n % ’ dIIk = 2 w(k) = 2 (k2 + m2)1/2 ’i e xjey5.3 W eakly interacting and dilute system s 100the Lorentz invariant measure on the mass hyperboloids IHmIHm := {(a;, p) € R2| cu = ± \ / p 2 + m 2} .5.3.2 T he B oltzm ann equation for d ilu te gasesTo introduce the long time-dilute medium limit we have to additionally rescale the particle number densities n ^s ) and in addition introduce the physical time T. We writet -> T /e , n ^ t) = ea^,(T,e) , if>(T) := limi^(T,e) , (5.25)for a suitable a. The idea of the above replacements is to take e -» 0+ while keeping T fixed (so that t —► oo) and at the same time keeping the initial density i/p(0, e)—and hence the initial state (4.13)—fixed (so that n^O) —► 0). We claim that the limiting quantities, if they exist, satisfy an equation which is similar to the Boltzmann equation, and which reduces to the Boltzmann equation if in addition we assume that the collision time is long (e.g. when A becomes small). Note also that the last equation/definition above is clearly an assumption in the sense that the full mathematical demonstration of this claim would require us to control the limit e —► 0 of ^ (T , e), and for this we would have to look at the full non perturbative dynamics of the model. In principle this ought to be possible using methods similar to those described previously (cf. Section §4.2.1), but we do not believe that such an analysis would necessarily offer more insight into the nature of the limit than a formal derivation. We will give such a formal derivation here by starting simply with the assumption th a t the limits in (5.25) exist in a suitably strong sense. We then use the pre-Boltzmann equation to see what equation does this limit satisfy. Above we also have an as of yet unspecified parameter a , which we will keep arbitrary and will eventually argue tha t if the e —> 0 limit of the pre-Boltzmann equation is to be finite, than we will be forced to consider a = 1. By additionally assuming that we can exchange a limit with an integral, we will arrive at equation (5.28).We would now like to take the limit as e —> 0+ of the rescaled collision factore -a~lB{eE, p ,J ) ,where the e-a_1 factor comes from the LHS of the pre-Boltzmann equation after the rescalings in (5.25) have been used. Later we will argue that, if we choose a = 1, then the limit exist, and is in fact independent of E.To state what the limit is, we define the Boltzmann collision factor in the low density5.3 Weakly interacting and dilute system s 101limit BJjp, T) to be the expressionB U p , r ) - - 2 £ | y | ! [ d n (P l, q i , . . . , q |y|)(2^)2s ^ L + £i -jeY $ V jeY '|x (p> £, ->• ?1, • • • , 2|y|) |% (7 ’)H.a( r )++ £ l W I / dn (P i.P 2. ‘b .- - .q m )(2 ^ )25(2)(£ 1 + P 2 - E - E 2 j )i&r i ' j=» Jm (Ei ’P2 - >£-22> -" > V |) | H.,(t )h>2(7') .(5.26)where M ( X P —> Yq) now denotes the full, not dressed scattering matrix element of the theory (as opposed to the local dressed matrix elements in the pre-Boltzmann equation), with the energy-momentum conservation delta’s taken out. Then we claim that we have, in the sense of distributional boundary values:lim B.V. t T ^ B f e g . p , - ) = , (5.27)c-+0+ Im£<0,Im£->-0 \ € / td(p)plus a contribution that is the boundary value of an analytic function for ImE > 0, but that will not contribute to the expression eq. (5.23). Indeed, for that contribution, we can deform the contour of the d isin tegration in eq. (5.23) to the trivial contour within the half plane ImE > 0, as the exponent elE T̂~3̂ provides a damping there (note that s < T), and the same applies to the other d^-integrals in eq. (5.23). Substituting the limit (5.27) into the limit of pre-Boltzmann equation (5.23) then delivers the final resultoou(p)drh,(T) = Bv(p,T) + » - ! ) • / ds J d r i ... drnn- 1 o s 1) “rescattering terms” in (5.28) are suppressed by powers of A. Thus, the leading contribution will arise from the 2 —► 2 scattering processes and a single collision factor. The corresponding leading approximation of equation (5.28) for small A is (in d. dimensions):u>(p)3rit,(r) = ^ / d n P2d n qid n q2 (2^)d6(i)(p+p1- q 1- g 2) •(5.29)• |M p>Pj 2 A ) f (MhCHxJT) - H f ix c r ) ) ,where the matrix element is now denoting the B om approximation. This is indeed the relativistic5 version of the familiar Boltzmann equation. Finally, another previously mentioned feature of the above equation is now clear, namely that in two dimensions the right hand side of equation (5.29) vanishes. This is because the momentum conservation delta enforces {p,Pi} = {Qi,^}*In order to get a somewhat better qualitative understanding when the “rescattering terms” can be neglected, let us introduce the ZA-norm of a function f p on the mass hyperboloid Hm as H/Hz,1 = / \fp\d n p. Then the ZA-norm of the n-th rescattering term5Note the relativistic kinematical factors implicit in dll, as well as in the expression uj(p)&r, which is equal to p^dp for the homogeneous state that we consider, because the î ,’s are independent of the spatial coordinate.5.3 W eakly interacting and dilute system s 103in eq. (5.28) as a function of p is immediately estimated bysup||B „(T )||t i j dBT ||B „ (n ) | |i i \\SBv(t{)/ ■ ■ • ||5Bi,(TB)/6 !/||t ,_>ti0 < ri< ...< rn 0. Therefore, in terms of the physical time £, the condition that T To would mean, for finite but very small e, that et 0+, the limit, if it exists, will still be analytic for Imp® < 0 and Im q? > 0, because in the integral over the time coordinates in equation (5.13) we can safely continue the frequency arguments in the exponentials to the indicated domain. In other words, we expect that there exists a function T x p->Yq analytic in this domain and such that the following relation holds in5.3 Weakly interacting and dilute system s 104the sense of distributionslim M { o m (Xp -> Yq) = B. V.€“♦0+ gJ-K*/(q3)ImpP<0, Im g9>0Here “B. V.” means distributional boundary value of an analytic function [75]. The existence of this limit follows from the work of Epstein and Glaser [76], to arbitrary orders in perturbation theory6. One would also expect this to be true non-perturbatively, but we have not been able to see it. In the following, we will denote with M (0>00){XP —> Yq) this boundary value. It corresponds to the matrix elements of the (full) scattering matrix with an interaction switched on at t = 0. The relation with the full matrix element is easily seen from the following formal calculationm (x t yt) := AVoo,+oo)(*p -> Y,) == — ^.A4(+oo,+oo) { X p ̂ ^ g ) *A4(—qo,+oo) (-^ p ̂ Y q j' j == - J 6a §-gM is,+0o)(X p -+ Yq) =R= - / d a £ m (o,+0o) (X p -► =" , (5-31)= Hux - 0Jy)M{0t+ao){Xp -*■ Yq) I d• eftvr-ux) =R= 2 ttS ( u x — w y ) i { y } y — v x ) M ( o , + o o )(^p -> ^g) == 2irS(wx - ojy) B.v. i(p°x — )}) ■ImpP<0, ImgP>0Here we have again used equation (5.19), whereas in the last line we have employed the definition given in (5.30). The non trivial assumption here is, of course, that the boundary value of (px — 0y )^ x p->y, can indeed be restricted to u x = cjy, which is certainly not obvious (see [63] for related work). Thus, up to a standard energy conservation delta function, the full matrix element is equal to the scattering matrix element with interaction switched on at time t = 0, multiplied by the energy. Finally, we also note6Here, it is essential that one takes the parameter m in the free Hamiltonian to be the true physical mass of the theory. In other words, it is assumed that the mass has been renormalized. It should also be noted that in [76] the adiabatic limit is defined in terms of some sort of averaging procedure in momentum space around the mass hyperboloids, rather than a boundary value prescription as above. However, the two are seen to be equivalent.FXp—yYq ({p?>p<)}> {($>% )}) • (5-30)5.3 W eakly interacting and dilute system s 105the distributional equality8 = 0= i(uy — Wx ) M q,oo (Xp —> Yq)B.V. i{jPx - £ ) ^ y f({ (r f ,p ,)} ,{ (# « £ )} )lmp?<0, lmg?>0(5.32)With this preparation in place we axe ready to take the e —► 0+ Hmit of the local scattering element. Using the property (5.19), we can write the relevant integral [cf. (5.24)] asg J d t e-« e E \ M i0̂ X p -+ y?)|2 = -i£ j At e~*« e | | A <( 0 -> y,)|2 =R R= Jdt e~'Et €§t\M(°A-dXp y,)|2L =RRR= / * e " i £ t ^ ) § - s M (s^ X p - + y , ) } >=Q =Roo= 4 / d< e_* -+ -+ y,)}<=0+00+ J d t e-*Et R e { x * 0i!)(Xp -+ Yq) ^ M (, ^ X p -> y ,) } ^ .— OONow, the integral in the last line above is analytic in Imi£ > 0 and will hence not contribute when we evaluate its contribution to the collision factor in the pre-Boltzmann equation as we could then deform the contour of the dE (respectively dEi) integrations into the lower half plane and get zero. We hence need only consider the integral on the next to last line above, i.e. with integration domain [0, oo). In the e —► 0 limit, the integrand becomes independent of t except for the exponential exp(-iE t) , and we can trivially perform the dt integration which yields simply a factor of (E — iO)”1. Using5.3 W eakly interacting and dilute system s 106expressions (5.31) and (5.32) to obtainei r / dt e~iEt\M (o,dx p -*■ y»)l2 =|(u>X — CJy)A^(o,oo)(^p —► Yq) Re < B.V.Qj ) E x — Eylmp?<0, lmg?>0(5.33)= 4ir2S(uix ~ u y ) \M (X p -*• y, ) |2 ,where A i(X p —> Yq) is now the full scattering matrix element of the theory, with the energy-momentum delta’s taken out To obtain this expression we also used the well known distributional equality= r f ( x ) ,where & denotes the Cauchy principal value. Equation (5.33) now no longer depends on E and we will use it in order to evaluate the limit of the collision factor (5.24).R em ark . Strictly speaking, the results we have obtained in this section are only valid for the “standard” scattering amplitude M. and not to the dressed one M.. Because of that, the above analysis only justifies the use of equation (5.33) in the low density limit where the dressed amplitude gets replaced by the standard scattering amplitude. We will, however, use equation (5.33) formally also for the dressed amplitudes and proceed to “derive” the Boltzmann collision factor for weakly interacting systems, see Section §5.3.2. Albeit mathematically not justified, the main reason for us to do so, is the fact that in the latter case we obtain some physically interesting results.T h e collision fac to rTo obtain the desired equation (5.26) we now use the results in the previous section and the scaling assumptions (5.25) to (formally) writelim e~a~1El(eE,p, = lim V |X | ! |y | ! fa(|x|- 1)“ 1 / dIIXpUy, (2tt)2<5(2)(p - q ) ■e-*0+ \ C / e—»0+ Jj% Y R• \m {x p -> y, ) | 2 [ s(p ~ - Y , s (p ~ p<)] II r) + • • •j e Y iex iex5.3 W eakly interacting and dilute system s 107where the dots mean tha t we are omitting terms that axe of higher order in e and will hence vanish when we take the limit. Heuristically speaking, they come from the fact that in the low density limit 1 + &f>(T) « 1. From the above expression we now immediately see that the sum over all sets Y c IN is unrestricted by the limit. On the other hand, for the limit to be finite, it must be that a(\X\ — 1) — 1 = 0, from which the condition a = l/(|Af| — 1) follows. By noting then that \X\ > 2, we deduce that a < 1. However, if we pick a < 1 then all the terms in the above sum such that \X\ < 1 + 1 /a will contain an inverse power of e and will diverge in the e —> 0 limit. (And there will always be at least one such term as 1 + l / a > 2 when a < 1.) It follows that we will obtain a finite limit only for a = 1. Taking now the limit in the above equation we getBm e - o - ^ e S . p . J ) = 2 ^ | y | ! [ d n {PliPj}uyq (2 qy ) ■j e Y JR■ |M ( X P y ,)|2 [ £ <5(p - « * , ) - £ *(p - m )l n t i n ■j e Y i e x ie xTo obtain the desired result, equation (5.26), we now simply integrate against the remaining deltas <5(p—q7) and 6 (p —p*), and note that we are working with a P T -invariant model so that M ( X P —> Yq) = M (Y q —> X p).5.3.3 T he B oltzm ann eq u ation for w eakly interacting system sWe will now consider the so called A2t scaling limit of equation (5.24), i.e. the “weak coupling” limit. This scaling limit corresponds to the physical situation of a weakly interacting system which we observe over a very long time. The way to consider this limit is to rescale the expansion parameter A to zero in such a way that A2£ = T remains finite in the limit. We also note at this point that the number densities np(t) depend on A and we denote them by n£(t). Additionally, we need to make the assumption that they possess a finite limit. That is to say, in the weak coupling limit we assumeA -> y/e A , t - > T / e , hm n£(t) = f ^ T ) , (5.34)The full mathematical proof of the existence of the limit would require us to look at the full non-perturbative dynamics of the model. While this ought to be possible in principle, we have not done it. We will, again partially formally, derive an equation that the quantities f p(T) must satisfy if the limit (5.34) exists in a sufficiently strong sense.First, let us suppose that the adiabatic limit A i := limA4(s>t) [cf. equation (5.15)]exists in the sense that we can take the limit s —> — oo and t —* oo to obtain the full5.3 W eakly interacting and dilute system s 108(not local) dressed scattering amplitude. This very assumption actually turns out to be wrong for any particle distribution np(t), and we will explain in much more detail what has to be done to make things work (see Section §5.4). However, for pedagogical purposes, let us ignore this fact for the moment and let us pretend (incorrectly) that M. exits. Since M. — 0(A), we may then also formally calculate its “Born approximation” as __A -> 0 AThen, let us formally define B \(p, T), the Boltzmann collision factor in the weak coupling limit, asB*(P,T) := y E l*l! ly l! \x\ / dn^uyq(27r)2 5(2)(p + qxl- gY) iex, je Y £l*Wyl=P (5.35){(n w)i1+ + w ) - /pm n +m ) } - ̂ jeY i e x 1 i e x 1 jeY *where p in the sum above is the degree of our interaction polynomial [cf. equation (5.8)]. We now claim that formally the quantities f p{T) satisfy the following equationo° rw(p) d r f J J ) = Ba(p, T) + E C - 1)" J d n . . . dr„n _1 s(px - qY) Y , <*(P - Qj) Pi) ̂P '-jeY i e x• n m v n(1+f ^ n )) + { / dn^ <5(2)(py - gx yie x jeY J ̂£■ [x^p-c|j)-5i s (p - pi)l n m t ) n i 1+̂ /r))}=' - jex ieY J ieY j e x >= { f d n XpUy, 8m{px - qY) \ Y 5(P ~ q*) - Y 5(P “ P*>1 ' ̂£ L je Y i e x J• I I U T ) n ( 1 + f 4 T ) ) ) - { [ dnXpUy, &(2)(PX - g Y ) •ie x jeY > ̂ i■ [e5(P-^)-e ̂ p - pi)1 n /p/r) n(i+w )}='-jeY i e x J i e x jeY '= f d n XpUy, <5(2)(px - £y) f Y 5(p _ qi) “ X ^ P “ Pi)] ’£ L j s Y i e x J• {n w n n (i+w ) - n ̂ ik^ w ) } ■' i e x je Y jeY i e x JLet us explain what we did above. Prom the sum over all possible X , Y C N in (5.37), we pick two terms corresponding to specific X , Y and X \ Y \ and such that X r = Y and Y f = X . If we now introduce the schematic notation (|X | —> |y |) to denote the summand in (5.37) corresponding to the sets X and y , we extract the terms (|A | —> |y|) and (|A71 —> |y'|) = (|y | —► |A |) and we sum them. That is, above we have computed (|A | —► \Y\) + (\Y\ -> |A |). The chain of equalities is then obtained by noting that in the first equality we have replaced the dummy integration variables p* «-> q; in the5.3 Weakly interacting and dilute system s 110second integral and then rearrange the various terms. As a last step we now integrate over the momenta corresponding to the various delta functions to obtainwhere X 1 = X \ {1}. Using the above result we can finally write B \(p ,T ) as in (5.35).Before proceeding with the discussion we now want to inspect #a(p> T) in more detail. Specifically, we want to comment on the sum in equation (5.35). First of all we note that we can physically interpret the above sum as the sum over all the \X\ \Y\ scatteringprocesses, i.e. a sum over all processes in which the interaction amongst \X\ initial particles results in \Y\ final particles. Such a process occurs with an amplitude that in the Born approximation is given by iX. In the above form the collision factor emphasizes the fact that a process and its inverse ought to be considered on the same footing. In the form of equation (5.37), on the other hand, the change in the number of particles with momentum p is due to two unrelated processes: A first process in which there is a particle with momentum p amongst the \X\ incoming ones, and another process where it is amongst the \Y\ outgoing ones. Clearly, the expansion (5.35) is physically more meaningful.Given now such a (physical) interpretation and by recalling the expansion of the local 5-matrix (5.15) (where |X |, \Y\ > 1), one is led to think that the above sum includes all the possible processes, with arbitrary \X\ and \Y\. We will now argue that this is not so, and that in fact the above sum reduces to the sum over all “physical” processes. To do so, we split the above sum into two classes of terms: (a) (unphysical) processes that violate momentum conservation, (b) everything else, i.e. physical processes.For processes (a), we note that the momentum conservation delta will be of the form S ^ (k ) with k ^ 0 and hence vanishing. And it will be of this form precisely because momentum is not conserved—we cannot arrange the sum of the (on-shell) 2-momenta to vanish by respecting at the same time some elementary physical assumptions. An example of such a process is a 1 —> 3 process, that is the “decay of one particle into three/ d n x ,uy, *<*>(2+px l- gY) f ( n f 4 T ) ) (X + (1 + o n ) +p je Y iGX*- / p ( T ) n / P. m n ( i + /q jm ) l -i e x 1 jeY J[<*nYiuxJm (2 + PY - i x ) \ ( U f 4 T ) ) ( l + U T ) ) l [ ( l + / Pj(T))-£ L jex ieY1- U ( T ) Y [ U T ) Y [ ( l + f qj(T))]iev1 j e x -I5.3 Weakly interacting and dilute system s 111particles with same mass as the decaying one” and it is an easy exercise of relativistic kinematics to show that such a process violates momentum conservation7.The terms of type (b) are the physically allowed processes and these will, in general, contribute to the change in the particle number. However, we additionally point out that as one might intuitively imagine, the “mere propagation” of particles—a process where \X\ = \Y\ = 1—is also to be excluded from the above sum. The reason for this is that the momentum conservation delta will enforce equality of p and q1 which will in turn enforce that the term in curly brackets will vanish. We also note that a specific choice for the interaction polynomial might impose additional constraints to the above sum. For instance, if the interaction polynomial contains only even powers in the field, then all the processes with \X\ + |y | odd will vanish as the corresponding scattering amplitudes would vanish identically.Last, but certainly not least, we also want to point out a subtle point about the form of collision factor (5.35). That we have obtained a sum over all (physical) collision processes in the AH limit is a model dependent feature that in fact further restricts the sum so that \X\ + \Y\ = p, where p is the highest degree in our interaction polynomial. This is easily understood by noting that at tree level, i.e. in the Born approximation, the sum of incoming and outgoing particles can be at most p. However, when we later briefly extend our analysis beyond the weak coupling limit in Section §5.4, we will obtain corrections (in the coupling constant) to equation (5.36). And one aspect of these corrections will be that the scattering amplitude will emerge in its full perturbative expansion. Because of this, the only change to the Boltzmann collision factor (5.35) will come from the fact that the sum will now have to be extended to all physically allowed processes. And this will hold independently of the specific model, the reason being that in quantum field theory an arbitrary number of particles can be created, albeit this will only occur at correspondingly high orders in perturbation theory.5.3.4 Physical in terpretationHere we give a physical interpretation of the results we obtained in Sections §5.3.2 and §5.3.3. In general we can say that in both cases we obtain an equation that is different from the BE as we introduced in Chapter 2. The main difference lies in the fact that our equation is non-Markovian whereas the BE is local in time. Such a non locality manifests itself in the form of additional (infinite) terms on the right hand side of our equation.7It should be noted that this is a model dependent feature: Had we started with a theory with more than one field and with different masses, then if the sum of the masses of the three outgoing particles was less than the mass of the decaying one, such a process would be allowed.5.3 W eakly interacting and dilute system s 112It is only by truncating (by hand) our equation that we could recover the Boltzmann equation. Alternatively, we obtain the BE by considering both the low density and weak coupling limits at the same time. The physical interpretation of these additional terms is attributed to particle rescattering, i.e. multiple scatterings between particles. Intuitively one would expect no particle rescattering at least in the low density limit, but the fact that we axe really considering a long time limit in addition to any other limit clears why particle rescattering will inevitable be present. In fact, even if the density of the particles is very low, in the limit of infinite time every particle will undergo multiple (actually infinite) collisions.Another difference is given by the presence, in our results [cf. equations (5.26) and (5.35)] of a sum over infinitely many collision processes whereas in the BE (2.10) there only appears a single collision process. This difference is easily understood by recalling that the BE gives an answer to the question: “How does the particle number (locally) change in a sufficiently dilute or weakly interacting system?” . A pragmatic, physical answer would then involve the identification of (typically) one specific process that is the principal physical reason for such a change and, eventually, arrive at the BE (2.10). In principle, however, it is intuitively clear that the particle number will not change only because of one specific process, but because of all the possible processes a particle can be involved in. And our derivation is “mathematical” rather than physical, thus answering the previous question “in principle” rather than “in practice”. We consider it a remarkable feature of our derivation, and of quantum field theory, that this latter intuitive feature emerges naturally. But let us now make some remarks which are specific to the two cases separately.In the low density limit, the collision factor B u(p, T) involves a sum over all 2 —► n scattering processes. A heuristic “explanation” of this fact can be given by arguing that in a dilute system the likelihood of more than two particles colliding is negligible and hence only two particle scattering will contribute. The two distinct terms in (5.26) are then understood as follows. Keeping in mind that we are interested in the change of the number of particles with specific momentum p, as is clear from the LHS of (5.28), we observe that a particle with momentum p can be either in the set of outgoing particles or it could be one of the two incoming (colliding) particles. In the first case, the contribution will be positive while in the second case it will be negative, as intuitively clear and as the signs in (5.26) reveal. For weakly interacting systems, on the other hand, there is no reason why only two particles should scatter at any time. And this heuristic argument is “confirmed” by our (formal) “derivation” .By the above heuristic discussion on the physical interpretation of some aspects of5.4 Beyond the weak coupling lim it 113Dilute systems Weakly interacting systems2 —> n processes vs. All (physical) processes Pull scattering matrix vs. Born approximationVacuum amplitudes vs. Dressed amplitudes*Table 5.1: Comparison of the results in the low density and weak coupling limits.*: This is only obtained by going beyond the weak coupling limit, see Section §5. .̂equations (5.26) and (5.35), we can say that a characterizing feature of the collision factor B u(p,T) is the emergence of 2 -* n scattering processes “only” . An analogue characterization can be given for the collision factor 2?a(p> T) in the weak coupling limit as well. In this latter case, what we observe is that the scattering amplitude emerges in its Bom approximation. And this too can be given a physical interpretation: If the interactions between particles is very weak, then higher order corrections will be completely negligible. This ought to be compared with the emergence of the full scattering matrix in the low density limit.There is one more difference between the weak coupling and the low density limits that is worth pointing out. In Section §5.4 we will outline one possible way to extend our results beyond the weak coupling limit. Although very formal and not complete, our results do exhibit one particularly interesting feature, namely that if one wants to consider corrections (in the coupling constant) to the weak coupling limit equation, than the full scattering amplitude emerges again. The point here is, however, that it is not the “vacuum” scattering amplitude like the one we have in the low density limit but the “dressed” one. This last feature can be physically understood as follows. For a very dilute system, the single collisions will not “feel the effect” of the surrounding particles. On the other hand, a weakly interacting system might be very dense so that each collision will “feel” the surrounding “bath of particles” . And this feature is encoded in the dressed propagator. We summarize these conclusions in Table 5.1.5.4 Beyond the weak coupling limitIn our considerations, an important role was played by the “dressed local amplitudes” These formally represent a scattering amplitude for particles in a medium characterized by occupation numbers n ^s ) , and an interaction that is switched on during the time interval It was important to obtain the limit of this as t -¥ oo, togetherwith either A 0 (weak coupling) or np(0) -> 0 (low density). If one wants to go beyond these limits, one is confronted with the problem of making sense of the limit5.4 Beyond the weak coupling lim it 114M = lim^oo without the weak coupling/low density assumption. This limit,however, simply does not exist, unless all np(s) = 0, which is the situation covered in the existence proof of the adiabatic limit by Epstein and Glaser, cf. [76], and which we appealed to above. To show that the dressed amplitudes A4(s,t) do not possess such a limit is much simpler and in this section we will see why this is so.The physical origin of this problem appears to be that in a medium characterized by a non-trivial distribution of occupation numbers (i.e. with a finite density of particles), the notion of a “free particle” is different from that in the vacuum. Because of this, one may expect that the energy of a free particle £(p) in a medium will, in general, not be the vacuum energy, £(p) ^ cj(p), see e.g. [77]. Consequently, one should not expand the theory around the free theory given by the free Hamiltonian Ho as in equation (3.23), but instead use a different Hamiltonian that is based on the dispersion relation E = £(p) for the underlying free dynamics. This has been observed in the case of thermal equilibrium states e.g. by Landsman in [78], where he analyzed why “naive” thermal perturbation theory, i.e. perturbation theory in terms of the vacuum free Hamiltonian, leads to a non vanishing self-energy of the thermal propagator on the mass shell. It is also related to the well-known fact that the Feynman propagator for finite temperature equilibrium states, i.e. KMS states [79], is not Lorentz invariant, see e.g. [80, 81, 82]. Seen in this fight then it is not surprising that the adiabatic limit of M (a>t) does not exist—we (implicitly) attempted a perturbative expansion of the local scattering amplitudes in terms of an unphysical notion of free particle in a medium, and consequently a “wrong” free Hamiltonian.Unfortunately, these considerations do not tell us what we should take for £(p), i.e. what is the correct notion of free particle in a medium characterized by occupation numbers n^s). We will, however, explain how a “self-consistent” determination of this quantity is possible in the next subsection. The basic idea is that £(p) can be (self-consistenly) obtained by a “renormalization condition” on the self-energy in perturbation theory around a free theory characterized by the dispersion law E = £(p). This renormalization condition is similar in nature to that appearing in ordinary perturbation theory in the vacuum. There, the dispersion relation is of course E = cj(p) = >/p2 + m2, and the renormalization condition concerns the value of the constant m2. This is the mass parameter that has to be used in the free Hamiltonian, and the renormalization condition is (cf. [31, 32, 83, 84])5.4 Beyond the weak coupling lim it 115In the above equation II is the “perturbative self-energy” , which is formally defined as the sum of all “one particle irreducible” diagrams with two external legs, see Section §5.4.1. The meaning of the above condition is that the parameter m 2 in the free Hamiltonian must be equal to the mass of the interacting theory. This observation will play an important role in what follows.Before presenting the our self-consistent approach to the problem, however, we want to comment on the rigour of this approach, or better, on its lack of rigour. What follows crucially relies on the assumption that in a generic state we will have a functional (dispersion) relation between a particle’s momentum p and its energy £(p). As (physically) reasonable as it seems, however, this assumption hides a non-trivial consequence that necessarily invalidates the results in this section. This hidden subtlety is as follows. In [85] some consequences of the KMS condition are studied on the properties of (quasi- )particles for which it is assumed that a functional relation between their momentum and energy exists. One of the conclusions they obtain is that such an assumption leads to a unit S-matrix, which means that there is no scattering between these (quasi-)particles. In other words, the assumption of the existence of a dispersion relation implies that the model is free. Clearly, then, such an assumption cannot be made for an interacting model like ours. One might argue that we are not considering a KMS state, but as remarked in the introductory chapter to this work, KMS states are relatively “simple” (equilibrium) states. We, on the other hand, axe considering a generic non-equilibrium state and hence cannot expect any “simplifications” with respect to the KMS case.Nevertheless, there are compelling reasons for us to present our results. The first and most important one is that our “derivation” of the BE in the weak coupling limit was not justified. We are clearly referring to the fact tha t we assumed that we could consider the limit of a divergent quantity, namely of the full (not local) dressed scattering amplitudes. Something ought to be done to “remove” the divergencies and the results in this section are a step in that direction. They also provide a strong argument in support of the claim that a fully rigorous derivation would require some kind of mass renormalization. Clearly, the problem then remains as to how should such a renormalization be done. In the light of our results, a reasonable attem pt appears to be to assume the existence of a “dispersion correlation” (sharply peaked around a dispersion relation) and hence circumvent the results from [85]. On physical grounds we expect such a “dispersion correlation” to exist—physicsits do measure momenta and energies of particles in various states after all. We have not pursued this idea at all.Another reason for presenting our results is that they offer some practical novelties when it comes to solve a problem using the Boltzmann equation. Specifically, we “show”5.4 Beyond the weak coupling lim it 116that the scattering amplitudes should be computed with a modified propagator and we also find how to (self-consistently) compute it. This, in turn, might lead to improved quantitative physical predictions. Finally, in Section 5.4.2 we also briefly argue that it is necessary to justify the use of the BE beyond the Born approximation. And in order to do so, it will again be necessary to tackle the divergencies of the (dressed) scattering amplitudes.It appears that we are facing a formidable problem and we hope that what we did will shed some fight into its possible solution.5.4.1 Self consistent m ass renorm alizationAs we have said, the ansatz we make is that the energy of the particles in a medium is 3 p ) 4 u/(p). This “simple” assumption has a number of non-trivial consequences, as it will become clear in what follows. In particular, it will allow us to (formally) avoid certain divergencies in the perturbative expansion of Ai(s,t), and we will obtain a prescription to compute £(p) once we push our argument to the end. Before proceeding, we note that we will make “one step back” so we can make “two steps forward” in that we will go back to the finite volume case so we could use the general framework set up previously to compute the Feynman propagator in the present case and then obtain its thermodynamic limit by (formally) letting L —> oo, as done previously. (In the finite volume case we will be denoting the energy of the particles by Sp rather than £(p), which we will use to denote its thermodynamic limit.) And as we also want to give an explicit example of the kind of equations tha t one has to solve in order to find £(p), we will be specifically working with a quartic interaction, that is, we will have V(t) = A/4! <̂4(t, x).If we assume that the energy is £p, then the appropriate choice for the free dynamicsisw = t E ^ s) W - (5-4°)pezWe can now immediately compute the interaction picture fields y?o(^x ) according to (5.6). Recalling the expression of the interaction field in terms of the creation and annihilation operators (4.23) we immediately get»(•■»)- i j s E { ( 5 ^ j i 7 5 « **> + "■' } •where “h.c.” stands for hermitian conjugate. This field satisfies an equation akin to the5.4 Beyond the weak coupling lim it 117Klein-Gordon equation+ £{d, s)] tp0(t, x) = 0 , [d? + + u ;s(aIap)ei]0( - t ) } =" S I E + "*>+ [(1 + Tip(s)) == ^ + e-* ^ > 0 (- i) ] + rip(s) [ e ^ + e - ^ j =- si I +"■<’> *«<■> - *■>} ■R P€Zwhere we are now denoting p = (E , p), p = (£p(s), p) and we have used n ^s) = n_p(s), which is valid for a spatially homogenous state and for simplicity we assume this to be the case. Treating the general case would not add anything to our conceptual understanding of the problem while complicating some of the expressions below. Provided now that the thermodynamic limit of the above expression exists, the momentum space representation of the Feynman propagator that will be part of our Feynman rules can be straightforwardly read from the last line.Our next task is to determine the interaction potential as required for the perturba- tive expansion. Clearly,V(s) := H - H0(s) ,and what we want now is to have an expression of V (s) in terms of the interacting field5.4 Beyond the weak coupling lim it 118(p(s, x). If we now define5 & ) , g%s) := £Jis)up ,we have thatffd(s) = j £ N J s) £ J s) pezLpezso that the interaction potential is LV (s ) = 2 { t1 ~ : d*vXs< P)d,f(s , p ) : +pez2n (5.41)+ [wp “ ffp(s ) ] : P M S> P ) '} + ^ J d x . :tp4( s , x ) : ,which we can use to compute the “interaction picture” potential. Before that, however, we want to comment on the above expression, and on the “smallness” of the potential specifically. First of all, we note tha t by introducing £p(s) into the Hamiltonian we have not modified it, as can be seen immediately by computing Ho(s)+V(s). This observation gives us some clues about £p(s). In fact, in the A -* 0 limit, the Hamiltonian reduces to the standard free Hamiltonian N pljp/L . We hence expect that£p(s) = a;p + 0 ( A) , (5.42)as in this case H^s) immediately reduces to what one would expect and V(s) vanishes when A —> 0. This also tells us tha t V(s) = (9(A), and can hence be indeed treated as a small perturbation to the free dynamics thus justifying the use of perturbation theory. At this point, however, (5.42) has not in any way been established yet, so we complete our ansatz with this additional condition.Recalling equation (5.5) for the interaction picture potential, we see that in addition5.4 Beyond the weak coupling lim it 119to the interaction picture fields we need to computejr0(f, p) := ds0̂ ’p ) : +2 7T+ [WP _ $ s) ] : P)>Po{t,p ) :} + 4 / d x : \ -2 x 1 ,4 ; 2u>„ I " + Jpez ” p27tLv r f d E Y . ^ r n ^ s( £ ^ - E 2 ) * - i M -Ti 1 p€2 “ pwhere we have used rip = n_p again, we now use Wick re-ordering formula (A.l) for the quartic term in the potential and get: oo and that such a limit is well defined and finite up to the considerations that we present here. In the thermodynamic limit, the momentum space representation of the Feynman propagator istial results in having additional terms proportional to the unity. Such terms would end up contributing a phase factor to the local scattering amplitudes S which is cancelled in the product S*S. Dropping all the c-number terms we hence rewrite the interaction potential asp) dtWo(t, p) :s +our local 5 matrix, the usual 4-point interaction vertex and an additional (2-point) mass “counter term”. It is the latter term that is required in order to avoid certain(5.45)andSm2(E, p) = n^ s ) S(£ \p , s) - E 2) .Consider now the contribution to Ad (2 <-► 2) coming from Figure 5.1. Omitting some irrelevant numerical and phase factors, and recalling that p = (£p(s), p) is the on-shell5.4 Beyond the weak coupling lim it 121Figure 5.1: Divergent third order contribution to A4(2 2). The dashed lines do notcontribute and denote “external amputated legs”.(c)(a) (b)Figure 5.2: Examples of 1PI diagrams with two (a) and four (b) (amputated) external legs. The diagram in figure (c) is not 1PI.momentum corresponding to the modified mass shell £p(s), we evaluate it asf d2xi d2x2 A i d2y2 d2x d2y d2z .R• 6{2)(x - Xi)Si2)(y - x 2)5{2\ y - yi)6{2)(y - y2)A3(x - z)A(y - z) = = J d2x d2y d2z Ap(x — z)A p (y — z)e1̂ == <5(2)(p1 + p 2 - 91 - ^ 2)AF(£ (p i,s ) ,P i) f d2x A%(x)etx^ h &>R2which is divergent as can be seen by considering (5.45): Formally we haveA ^ p ^ . p O a - y_ + rlp(-)<5(A p1, S) - A p i , 5)) •It is now clear that the divergence comes from an “on-shell Feynman propagator”. Similar divergencies occur in other diagrams. However, if we sum up all diagrams, then we show that there exists a choice of £(p, s) such that these divergencies cancel. In fact, this £(p, s) is defined by equation (5.47).To do this let us introduce some terminology. We call a diagram “one particle irreducible” (or 1PI) if it is connected and cannot be disconnected by cutting through any one internal propagator, see Figure 5.2, and we will denote by V)(p; s) (Vf (p; s)) a generic 1PI diagram with two (four) external legs. Let us then introduce the 2-point vertex function, V2(p;s) = V2(E ,p \s ) , as the sum of all 1PI diagrams with two (amputated)5.4 B eyond th e w eak cou p lin g lim it 122++(b)+ +• i p i y + i f i . . i p i . +(C)Figure 5.3: Graphical representation of (a) the 2-point vertex function V2 (p), (b) the 4-point vertex function V4 (pi,p2? <72) and (c) of the re-summed 2-point vertex functionv2(p).external legs. Similarly, we define the 4-point vertex function, V4 (pi,p2 ,9i, tf2; s), as the sum of all IPI diagrams with 4 (amputated) external legs. Dropping all the dependencies on the time s in what follows, we schematically writev fo ) = y v i{P), w •Finally we also define the “re-summed 2-point vertex function” V2 (p) by the geometric seriesVip) ■■= v2(p) + V2(p)AF(p)V2(p) fn= 0v2(p)1 - V2(p )AAll these objects can be graphically represented as in Figure 5.3.Having identified an on-shell propagator as the origin of the divergence, we immediately see that all the terms of the formV?(£1.p2.2i-£2) ^ ( £ 1)V2 (£i) ’will be divergent as well. Not only, it is also clear that there will be similar divergent contributions for each of the external four momenta p 2, ^ and q2. This also clears the statement that the problem is not specific to M .(2 «->• 2) or to some IPI graph with two external legs in particular: It is a generic feature of M{\X\ |T |).To have a more compact expression of the above mentioned divergent contributions5.4 B eyond th e w eak cou p lin g lim it 123Figure 5.4: Schematic graphical representation of the on-shell propagator divergences in Ad( 2 h 2 ) .to A d (2 <-> 2) we perform a formal resummation of both all the IPI diagrams with two and with four external legs and writeH V4(£1.P2.2i>22) ^ ( £ i ) V2(Pi) = V (̂ p , ) ^ ( E i ) , (5.46)hjwhere the sum is over all IPI diagrams with four and two external legs. And again there will be one such term for each of the four momenta in V4. Next we note that (5.46) does not account for all the divergences. Another set of divergent contributions will come from terms containing two on-shell propagators, which after a formal resummation can be written asv^ v E r i v i 2 ^ (E 0 V^ E l)^(P1)v^Ei)We note that the above term formally represents infinite divergent terms, each of which has two on-shell propagators. In general, there will be similar terms containing n on-shellpropagators with n = 1 ,2 ,__ We again formally re-sum all these terms and obtain avery compact expression for the divergent contributions to Ad (2 «->• 2) that arise because of on-shell propagators and are “attached” to one of the four external momenta. We have OO nv*(EvE2’l v i 2) ' 5 2 f a E i ) v*(E1)) ̂ ( £ 1,£2,21.22)^ (£ 1) v2(2i) •n = lConsidering now the contributions coming from all the four momenta we get the final expression representing all the divergent contributions to Ad (2 2) arising because ofthe quartic interaction vertex. It isV,4(p1,p2,2 i,2 2)[A(Pi)V2(f>i) + A(p2)V2(p2) + A (?1)V2(£1) + A(£2)V€(£2) ,and can be graphically represented as in Figure 5.4. We again note that the above expression is a formal object in th a t it represents a divergent quantity, but we will5.4 Beyond the weak coupling lim it 124use it to better understand the overall structure of > 1(2 H 2) and eventually derive an equation for £(p, s) which, once solved, will lead to the cancellation of all such divergences order by order in the coupling constant. We also observe that the above four contributions are all formally identical and from now we only concentrate on one.Up to this point we have only been concerned with the contribution to the scattering amplitude due to the quartic term in the interaction polynomial and we now turn our attention to the remaining mass (counter) term. To this term will correspond a vertex with two propagators attached to it. The resulting matrix elements will then be constructed out of the Feynman propagators A f (E , p) in equation (5.45), but the rationale behind the construction of the diagrams will be the same as before. For instance, to the diagram in Figure 5.1 will correspond an identical one with the “bubble” replaced by a single “mass vertex”, whose contribution in momentum space can be immediately read off the thermodynamic limit of (5.44) to beS M \ E ,p) = E 2* + w2(p) - P 2 . 2 i > 2 2 ) ^ ( E i ) ( ^ ( £ i ) + S M 3 ( E . j ) = V i ^ 2 i ' E 2 ' l v S . 2 ) ^ i ) n ( P i ) ■ Formally, the situation is now identical to the standard vacuum case and we can again5.4 Beyond the weak coupling limit 125re-sum all the self-energy contributions to obtainv* l£ v E r h ’3 j Y l (A (£i)n (2 i)) = ~ ,from which we see that if II(p) = n(£(p, s), p) = 0 then indeed all the on-shell propagator divergences cancel. And this is precisely our self-consistency/renormalization condition that we use to (mass) renormalize the theory, determine £(p, s) and to complete our Feynman rules. We write it asU (E,p) = 0 . (5.47)E =£(p,a)R em ark . We want to emphasize at this point that we call U(E,p) “self-energy” only because of its formal similarity with the self-energy of the full propagator in the vacuum theory. That is, our “self energy” has the same diagrammatic expression as the per- turbatively computed self energy in the vacuum theory. But we also want to point out that aside from the obvious difference between the two arising because they are to be computed with different propagators, a fundamental difference between the two is that in the vacuum theory the vanishing of the self energy is justified by the known analytic properties of the full non-perturbative propagator, see [83, 84], whereas in our case we are not even willing to conjecture this fact.We now have an equation that will, in principle, allow us to find the modified mass shell £(p, s) to all orders in the coupling constant. A quick inspection of the above condition straightforawrdly gives us the “first order” equation for £(p, s). In fact, to order (9(A) equation (5.47) is simplyn(S(p, s),p) = <5M2(£(p, s), p) = 2w(p) £(p, s) + 2u \p ) + j Sm2 = 0 , which yields£(p, s) = «(p) - ^ 8m2(s) + 0 (X 2) = « (p ) - + 0(X 2) . (5.48)RAn immediate observation about the above result is that our “modified” ansatz, i.e. the assumption that £(p, s) = a;(p) -I- (9(A) is verified—our assumption was indeed “self consistent”. In turn, this means that our split of the Hamiltonian as summarized by equations (5.40) and (5.41) results in an interacting potential of order (9(A), so perturba5.4 Beyond the weak coupling lim it 126tion theory is applicable. Also, the above result is telling us that the energy of a particle is reduced with respect to the vacuum case when it is propagating in a surrounding medium. Stated differently, for a given spatial momentum p, the energy of a particle will be maximal in the vacuum. We find this result particularly interesting.Additional insight into the modified mass shell £(p, s) is given by the truncating equation (5.47) to second order. The resulting equation is^ P ' a ) = u { p ) - 8 ^ p ) J ^ W ) +RA2 f d2k d2o ~ ~ _“ 2aJ(pjJ (2t t)2 (2t t)2 Â Ek ~ Eq~ ^ P’s)’k - q ~ P)ME>”k )M E „ q ) •RWhat is immediately clear from this formula is that it is a highly non trivial equation. A reasonable attempt to solving it is by means of an iteration procedure, the first step of which is to use equation (5.48) on the right hand side of the above equation. At any rate, however, the full determination not only of £(p, s) but of the Feynman propagator(5.45) itself requires the knowledge of n ^ t) , that is of the very quantity we are solving the Boltzmann equation for. It is worth noting though that in principle we have a closed set of equations, and on physical grounds we expect it to have a (unique) solution. The mathematical proof of this statement, however, appears highly non-trivial and has not been addressed at all in this work.5.4.2 A hierarchy o f equations— work in progressIn many physical applications the BE is used by computing the scattering amplitudes beyond the Born approximation, see [5] for an application of the BE to baryogenesis. In this latter case this is not done to increase the accuracy of the prediction as it is a necessary feature to obtain any results at all. In other words, the standard out-of- equilibrium decay mechanism for baryogenesis is invisible at tree level. Because of this, one would like to justify, even formally, the use of the full scattering amplitude in the BE as opposed to the Born approximation “only” . And the results from the previous section show that this ought to be possible as once the existence of the dressed scattering amplitudes has been established, the corrections will come precisely in the form of the full amplitude, as in the low density case. Clearly, however, these will be sub-leading corrections to the Born approximation and here we very briefly present an idea on how one could proceed to establish such a result. Note also that this section should be considered as “work in progress” , which nevertheless offers an interesting perspective on5.4 Beyond the weak coupling lim it 127what we could expect to obtain.Remember that in the weak coupling case we assumed the particle number densities np(t) possess a finite limit f p{T) under the scalings (5.34). Consistently with these assumptions, we could consider an expansion of np(t) in terms of the coupling constant A, i.e. we could writeThis expression then ought to be used together with the pre-Boltzmann equation (4.46)where we are using the same notation as previously, see equation (5.35). Comparing the above expression with equation (5.35) one could note two differences. The main one is the emergence of the full scattering amplitude M. as opposed to its Born approximationinteraction polynomial. This is easy to understand. Since the scattering amplitude will now include higher order contributions, we will have creation of particles also because of the lower degree interaction vertices. Let us be more specific. In the Born approximation,will occur because of a 6th degree interaction vertex. The same interaction vertex will be responsible for the 3 —» 3 and 4 —► 2 processes. By allowing higher order corrections to the scattering amplitudes, on the other hand, these processes can, and will, occur because of the quartic interaction vertex. In particular, the 2 —> 4, 3 —> 3 and 4 —> 2 areone considers the quartic interaction only! Another remark about the formal structure ofnP(t) = f p(T) + A fW (T ) + \ 2fW (T ) + ■■■ .to obtain a hierarchy of equations that could be used to compute sub-leading corrections to equation (5.36). We do not present the results of this calculation but emphasize on the fact that now the Boltzmann collision factor B \(p ,T ) will be different from the corresponding one in equation (5.35). Specifically, we now expect the collision factor tobed n x iuyq(27r)2 8{2){p + qiX. The second one is closely related to this fact and is the fact that the sum over all the scattering processes is now not restricted to \X\ + \Y\ = p, with p the degree of thea process like 2 —>• 4, i.e. the creation of 2 (additional) particles after a collision of 2second order (in A) processes for the Yq) = M (Y q —► X p). Last but certainly not least, we should say that we also expect higher order collision factors £ ^ ( p ,T ) ,# ^ ( p ,T ) , . . . to emerge from this approach, but we have not explored it that fax.It now seems appropriate to present the Feynman rules we expect to be part of the “practical” conclusions of the this (work in progress) analysis. They are written here for reading convenience and are a straightforward adaptation of the usual Feynman rules one could find in quantum field theory textbooks, like e.g. [32]. Without discussing the rules at all, and for the quartic interaction only, we have:1. Draw all possible topologically inequivalent Feynman diagrams for a specific collision process.2. To each external line associate its own 2-momentum corresponding to the specific particle.3. Each vertex connects four lines and momentum conservation is imposed at it.4. To each internal line associate a “dressed propagator”& f ( E , p) = w (p) . ^ W P , s) + io + n^ s) ^ p ’s) ~ E2\5. To each vertex associate a factor (zA).6. A diagram with L loops will have L undetermined momenta k*, i = 1 , . . . , £ . Integrate over each of these momenta with measure d2ki/(2n)2.7. A symmetry factor is associated to each diagram (see [32] for a discussion of symmetry factors).8. The scattering amplitude is given by the sum of all diagrams computed by the previous rules.In the above rules it is also assumed that mass renormalization has been carried over. If this is not so, one should include the “mass counter terms”. The rules for the full polynomial interaction are easily generalized, the only change being that there will now be different vertices corresponding to the various powers of the interaction polynomial. Clearly, each vertex ought to be weighted by the appropriate coupling constant.5.4 Beyond the weak coupling lim it 129As stated previously, these results are just the tip of the iceberg of what one might expect to obtain in pushing the study of the derivation beyond the weak coupling limit. We have not done much about it and think it would certainly be interesting to do so, not only from a conceptual point of view. As the Feynman rules above show, there are some practical novelties such an analysis would offer and it certainly is of interest to see if any other modifications to the “textbook” Boltzmann equation would emerge.Chapter 6 ConclusionsIn this thesis we have investigated various conceptual issues related with the Boltzmann equation (BE). The BE has originally been obtained in a heuristic way as an attempt to better understand non-equilibrium processes, and despite its remarkable mathematical complexity, it has been widely employed in “practical” physical applications. Because of its experimental success it has been generalized in order to account for the quantum mechanical aspects of microscopic phenomena as well as to be able to use it in a (general) relativistic setting. Currently, the BE is employed in a wide range of applications, including early Universe theories like Nucleosynthesis and Baryogenesis1.It is perhaps precisely this experimental success of the BE that prompted us, amongst many others, to try and understand its conceptual status better. The original derivation, in fact, was not free from dubious and non-trivial assumptions. Amongst them, most notably, the so called “Stosszahlansatz” , or assumption of “molecular chaos”. According to this assumption, every collision between particles is statistically un-correlated to any other collision, and this is assumed to hold at every single instant of time. (Mathematically this translates into a factorization of the multi-particle distribution function into a product of single particle distribution functions.) The conceptual problem with such an assumption lies in the fact tha t it was assumed to be valid at every instant of time. Stated otherwise, it is conceptually perfectly acceptable to assume the factoriza- tion property at some initial time to» saY- But whether this condition will be “replicated” at later times depends, in principle, on the underlying dynamics. Intuitively, in a very dense and strongly interacting system where a collision would occur “almost at the same time as the previous one” , one would certainly expect the two collisions to be (strongly) correlated, and the Stosszahlansatz violated. Clearly then the BE will not provide a1 Nucleosynthesis is the theory that aims to account for the abundance of the lightest elements in the Universe, while Baryogenesis is the theory with the goal of explaining the observed matter-anti-matter asymmetry in the Universe.130131valid description of the system. Prom this also follows another major conceptual issue with the BE, i.e. unlike the Heisenberg (or Schrodinger) equation, it is not a fundamental, or “exact” , equation of physics. Not only, its relation the Heisenberg equation (HE) was also not entirely clear, in the sense that it was not clear to what extent could the BE be derived from the HE, and hence be understood as a particular approximation to the latter.These, essentially, were the concerns we have raised and studied in this thesis. Specifically, with this work we have questioned the validity of the Boltzmann equation. We wanted to understand whether the BE can be obtained, rigorously, starting from physically accepted first principles, i.e. Quantum Field Theory on Minkowski space.To attack the problem we have employed the “Projection Operator Method”. This technique allowed us to “decompose” the equations of motion, i.e. Heisenberg’s equation, and obtain what is known as the “Robertson equations” for the number operator densities. This was done for a specific model, the real scalar quantum field with polynomial (self-)interaction on the two dimensional cylinder IR x S. The Robertson equation allowed us to obtain what we called the “pre-Boltzmann equation”. The non-perturbative existence of the latter was proved2, hence establishing the equivalence of the pre-Boltzmann equation with the HE. The latter is hardly any simpler to solve exactly than the original HE, but it is a much better starting point to study some of its limits. In particular we were interested in the long-time limit of the HE.After the pre-Boltzmann equation has been obtained, we proceeded to consider its perturbative expansion. Moreover, as we were interested in understanding the Boltzmann equation on Minkowski space, we also considered the “thermodynamic limit” , i.e. we formally considered the L —> oo limit, with L being the radius of the cylinder. The next step was to consider the long-time limit of the perturbative expansion of the pre-Boltzmann equation, now formally considered on Minkowski space.Previous work on the Boltzmann equation suggested that the long-time limit by itself should not be enough: In order to obtain a meaningful limit, one ought to additionally consider either an increasingly dilute medium or a weakly interacting one. Specifically, for a dilute gas we rescale the time t and the particle number densities f p(t) as t —> T/e and / p —» ea/ p, where the exponent a had to be a = 1 if the result was to be finite. On the other hand, the weak coupling limit was considered by rescaling t -* T/e, and A -» y/eX. Clearly, in both cases the non-trivial assumption was that / p had a meaningful limit. But by making this assumption, we were able to derive the Boltzmann equation in both cases. The two cases exhibited a number of differences, and they2The non-perturbative existence of the pre-Boltzmann equation relies heavily on the estimates for the number densities we presented in Section §4.2.1, and was established by my research advisor.132confirmed the intuitive expectations we had about the two cases. Most notably, in the low density limit only 2-particle scattering processes should be considered and the scattering amplitude naturally appears to all orders in the coupling constant. In the weakly interacting case, on the other hand, the scattering amplitude appears only in the Born approximation, i.e. at “tree level” , while all possible particle scattering processes enter the Boltzmann equation. We should remark however, that in the latter case we obtain (sub-leading) corrections to the scattering amplitude, and these come precisely in the form of the full scattering amplitude.A physically very interesting issue arises in the case of the weak coupling limit, and it is related to the “mass renormalization” of the scattering amplitudes. From a mathematical point of view, the (perturbative) scattering amplitudes are divergent unless some “renormalization” procedure is applied. And the naive use of the usual mass renormalization one encounters in the vacuum case is not sufficient because the scattering amplitudes in this case are not Lorentz invariant. This is not a surprise as we considered a generic state, which in general is not Lorentz invariant. This problem appears to be mathematically extremely challenging and is still open. Our contribution to its solution came by means of a “self consistent” working assumption motivated by the physical understanding of the problem. Very briefly, in a generic state the dispersion relation of a particle, i.e. its energy, will not be its vacuum one. We hence made precisely the assumption that the particles’ energy £(p) was not the vacuum one, i.e. £(p) ^ cj(p). In addition, we considered a different perturbative expansion that depended on £(p). This allowed us to impose a condition, i.e. the vanishing of the “perturbative self-energy” on the (modified) mass shell, that effectively “removes” the divergencies, or better, it ensures there are no divergences in the first place. In addition, the vanishing of the perturbative self energy allows us to determine the particles’ energy, and we find that it depends on that is, it depends on the very quantity one would like to solvethe Boltzmann equation for. This “circularity” is typical of self-consistent working assumptions.We aslo want to remark that the Boltzmann collision factor, i.e. the right hand side of the textbook BE, is not the only term tha t we obtain on the RHS of our equation, even in the long-time limit. We do, in fact, obtain (non-Markovian) correction terms. These are “remnants” from the pre-Boltzmann equation and are associated with the rescattering of particles. Intuitively one could understand the emergence of these terms by thinking that as we consider longer and longer time intervals, particle rescattering will necessarily become a competing effect, albeit smaller. Also, when such multiple collisions are taken into account, it is not too surprising that the resulting terms are non-Markovian, i.e. that133the entire history of the system ought to be considered. Hence, strictly speaking, we do not obtain the Boltzmann equation, but a more general equation, which contains the Boltzmann collision factor as it first, Markovian, term. In this sense, our result is more general than the BE in that it holds for dilute systems and/or weakly interacting ones even when multiple scattering cannot be neglected.Finally, in considering the two previously mentioned limits, we have cleared the conceptual status of the BE, in the sense that we now understand it as a “long-time” approximation of the Heisenberg equation. The role of the Stosszahlansatz is then “hidden” in this long-time limit, which encodes a very specific physical meaning. Together with considerations on the particle density and/or strength of the interaction between the particles, we understand the Stosszahlansatz as an assumption that forces (by hand) the system to “forget about its past” and “neglect multiple scattering”. To impose the Stosszahlansatz would then amount to drop the (non-Markovian) rescattering terms in our scaling limits of the pre-Boltzmann equation. We should say that these are not entirely new results as the scaling limits are known in the literature, but they have not been established in QFT as of yet.Before concluding this thesis we want to point out some deficiencies in our treatment and open issues that certainly justify additional work on the Boltzmann equation. First of all, whereas the pre-Boltzmann equation holds in a non-perturbative sense, the Boltzmann equation as we establish it, is only valid perturbatively. This is clearly not entirely satisfactory and ideally one would like to establish such a result non-perturbatively. Furthermore, the perturbative derivation was not entirely rigorous, even by disregarding the non-convergence of the perturbative expansion of the scattering amplitudes. Some of the manipulations were only formal and although we believe that a rigorous treatment would not alter the end result, it would still be interesting to verify that this is indeed so.The previous remarks on the validity of the perturbative expansion really only apply for the low density scaling of the pre-Boltzmann equation. In the weak coupling limit the derivation is, in fact, much more formal than the corresponding one for the low density limit. Intuitively, the reason is that in a low density limit, the physical state is “similar to the vacuum” precisely because of the extreme diluteness of the particles. On the other hand, a weakly interacting system might not be like the vacuum at all. Mathematically this is reflected in the fact that in the low density limit all the quantities are the vacuum ones, i.e. the Feynman propagator is the vacuum propagator and so are the scattering amplitudes. And (perturbative) quantum field theory has mostly been developed in the vacuum for which many results have been also rigorously justified from134non-perturbative investigations, like the singularity structure of the (full) propagator which justifies the usual mass renormalization prescription. In a generic case, on the other hand, little is known and it should always be dealt with on a case-by-case basis. In our case, these issues enter through the presence of the dressed propagator and the corresponding dressed amplitudes. The latter exhibit the same type of divergence that requires mass renormalization in the vacuum case, but now the well known procedure cannot be applied. Our self-consistent solution, which mimics the vacuum situation, although very reasonable, lacks any sort of justification other than “results justifying the means”. A particularly interesting aspect of our solution is the meaning, if any, of the vanishing of the self energy on the (modified) mass-shell. This would require a deeper, non-perturbative understanding of the full propagator of our theory.We also want to point out here that our solution to the mass renormalization problem represents only a “guidline” of what a rigorous result could be. The main reason for this is that the very assumption of a functional relation between a particle’s momentum and its energy cannot be made for an interacting model. That is, by assuming such a functional relation, we are inevitably led to a free field. This is known to be the case for thermal states, which are relatively “simple” states, and it would certainly be a big surprise if in a generic state the situation would simplify. One possibility then is that the energy and momentum of a particle are strongly correlated with the correlation distribution being narrowly peaked around some dispersion relation £(p). It is, however, clear that to prove such a result a formidable amount of work would be required and we have not even attempted to do so.Another aspect that is worth of additional investigation is the fact that our treatment does not allow for a spatial dependence of the particle number density / p. The reason for this is easily traced to our definition of the number density operator, which is in no sense a local quantity. We know, however, that the textbook Boltzmann equation postulates the existence of a particle number density that depends on space as well as on time and on momenta. So a treatm ent that would allow for a spatially dependent particle number density would clearly be desirable. One such result has been obtained by Buchholtz in [86] where he establishes the validity of the collisionless Boltzmann equation for a massless scalar quantum field within the framework laid down in [87] for the analysis of non-equilibrium states in QFT. So one natural attempt would be to try and employ the local thermal equilibrium states of the latter work together with the projection operator method we have so successfully employed in this work.Finally, another topic that is certainly worth of additional investigations is the extension of our results to a cosmological setting, namely to Robertson-Walker spaces. We135have, in fact, appropriately modified the projection operator method so that it could be applied to a time dependent case, i.e. to the case where the Hamiltonian depends on time. For the case of RW spaces we expect three terms emerging as a result: the Boltzmann collision term as we have obtained it in this work (clearly appropriately modified), a term that would describe Bogoliubov transformation type creation and annihilation of particles, and an additional term to account for the expansion of the Universe. Such a result might then be applied to those cosmological investigations where the Boltzmann equation is a crucial tool, like the Nucleosynthesis and Baryogenesis theories. Given the expected modifications to the usual Boltzmann equation, it would be interesting to see if existing quantitative prediction would be modified by these results.To conclude, we want to note how this work, despite having elucidated the status of the BE within QFT to a large extent, it is far from being a concluding answer to all the questions the BE raises. As we have outlined above, the questions left open with this work alone are many and varied. Needless to say, the BE will clearly continue to attract a lot of intellectual effort. Hugenholtz’s words come to mind once again, and we hope that in the future we can further reinforce the validity of that claim.Appendix AW ick reordering for normal ordered productsIn Chapter 3 we introduced the notion of “normal ordering” . We explained that a normal ordered product of fields is one where all the creation operators have been moved to the left of the annihilation operators. An important point of such a normal ordering prescription, we noted, is that its expectation value in the (free theory) vacuum state is identically zero. In addition, this makes the evaluation of vacuum expectation values products of (free) fields very simple. (For this reason it is more appropriate to talk about “normal ordering with respect to the vacuum state” .) On the other hand, in Chapter 5 we are confronted with the task of evaluating products of normal ordered fields in the states ws introduced in Chapter 4. Because of this, we find it convenient to “normal re-order” (or Wick re-order) said products of fields. For this reason we give here an alternative, but equivalent, formulation of the idea of normal ordering for free fields, in terms of a “subtraction procedure” . It should be noted that this Appendix is not a complete review of the so-called “Wick calculus” , i.e. of the computational rules to deal with normal ordered products of fields. Our main theorem is Theorem A.l below, which we present as we need it in the main body of this work.Following [88] we define the normal ordered prescription for the product of n (free) fields as follows. First we define a “quasi-free” state a; as a state that is completely determined by its 2-point function (i.e. the expectation value of a product of two free fields). More precisely, if Po(x):o = lim \ipa(x)(a:Vo(j/))0] •While the limit of the individual terms on the RHS of the above expression is ill defined, the limit of the difference is well defined. The reason for this is that both terms have “the same singularity structure” .Elaborating the above simple example some more, we see that given two quasi-free states u and xp we can write:ipo(x)ip0(y)'-u = o(y) H- xp [ : o(z):« = ^ o W v + ^ W W - ' u ] 1 -The above formula can clearly be generalized to an arbitrary Wick power and the generic change of covariance formula is the content of the following theorem, the proof of which can be found in [43]:T heorem A .l (Wick reordering or change of “covariance”). Reordering of Wick monomials is given by the formula[n/2] ,:y?g(x):u,= y ; ^ » (A-l)139where Sc(x) = limy^ x{/ip[(p(x)(p(y)\ — w[t = Bt .At this point it is sufficient to multiply the latter equation from the right by Ytjto and formally integrate to obtainH t = H to + f ds B s o Ys>to .JtoFinally, noting that Ht0 = K to, from equation (B.2) we then immediately obtaintds Bg o YSj$ .K t = K t0 o Y k t +to,tLAppendix CExpansion o f operators on Fock spaceIn Chapter 5 we have given an expansion of the local 5-matrix (5.10) in terms of (time-s) creation and annihilation operators, see equations (5.12) and (5.15). Such an expansion could be understood as a perturbative expansion of the local 5 -matrix, but the important point is that it is valid non-perturbatively. Here we argue that any operator on Fock space can be expanded in terms of creation and annihilation operators.For simplicity we will consider here only the case of a single “harmonic oscillator” , that is, we will only consider a single pair of creation and annihilation operators. The generalization of this discussion to the case of relevance in the main body of this work is then fairly straightforward, albeit more involved. Our Hilbert space is spanned by an orthonormal basis {|n)} (i.e. we have (n\m ) = <5(m, n)). The creation operator a* acts on the Hilbert space in the usual waya*\n) = y/n + 1 |n + 1) .The action of its adjoint with respect to the given scalar product, the annihilation operator a, is as usuala\n) = yfn \n — 1) .Our claim is now that given a finite rank operator W on the Hilbert space H, we can write it as follows OO TXT’w = E ^ 7 =f(aT “n - (C.1)V m W nlTo prove this claim, we need to find the quantities Wmfn. Recall that all the vectors in H142143can be constructed by repeatedly acting on the “lowest energy” vector |0) with creation operatorsWe now want to compute the expectation value wPtQ := (p\W q). Using (C.l) we have/ ( O p „ „ , («*)«^ , , = ( p |^ 9 ) = ( ^ r 0 | ^ - ^ - 0 ) == £ ^ f e r a < (a*)P0l(a*)man(a *), 0 ) = (C.2)m ,n=0 v riw.m ,n=0 v r ^To push the computation further we need the following calculation« » ( a * m = \fp \am\p) = \ p - m ) = (a-)— 10) ,which is valid for p > m. For p < ra, on the other hand, we would simply obtain zero. By plugging this result in equation (C.2) we then getp QJp,q ~W„„ = Y Y WmnS(p - m , q - n) . ^ ^P Q / \ 1/2 / x l / 2= E E ( m) Q Wmft6(p-m,q-n)-m = 0 n = 0 x 7 v 7~ e . w U - ( p - 9 ) ; ’m _ ( p _ 5 ) ’m=p—qwhere we have p > m, q > n in addition to p > q, which is required for the last equality.We can now solve the above system of equation for the quantities Wm,n- For instance, for p = q we getw p ,p — ^ 2 ( m ) »m = 0 ' 7which we can use to calculateWo,o = W 0 ,o , witi = W o # + W i }i = > W i t i — W o to — w \ y\ ,144and proceeding like above, we promptly obtainm=0 ' 'We can similarly treat the case p ^ q. For instance, if q = 0 we get WPf0 — wp,oi if Q — 1 we have WPyi = — y/pwp,o and so on. In general, to find WPA we will need to findall the VFpy with (f < q. The upshot of this discussion is that we can indeed write any finite rank operator W on % as in equation (C.l). 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The quantum mechanical boltzmann equation

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The quantum mechanical boltzmann equation

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