Energy inequalities in integrable quantum field theory

Negative energy densities are an abundant and necessary feature of quantum field theory (QFT) and may lead to surprising measurable effects. Some of these stand in contrast to classical physics, so that the accumulation of negative energy, also in quantum field theory, must be subject to some constraints. One class of such constraints is commonly referred to as quantum energy inequalities (QEI). These are lower bounds on the averaged stress-energy tensor which have been established quite generically in quantum field theory, however, mostly excluding models with self-interaction. A rich but mathematically tractable class of interacting models are those subject to integrability. In this thesis, we give an overview of the construction of integrable models via the inverse scattering approach, extending previous results on the char- acterization of local observables to models with more than one particle species and inner degrees of freedom. We apply these results to the stress-energy tensor, leading to a characterization of the stress-energy tensor at one-particle level. In models with simple interaction, where the S-matrix is independent of the particles’ momenta, this suffices to con- struct the full stress-energy tensor and provide a state-independent QEI. In models with generic interaction, we obtain QEIs at the one-particle level and find that they substantially constrain the choice of reasonable stress-energy tensors, in some cases fixing it uniquely.:Acknowledgements 4 Contents 5 1 Introduction 7 2 Constructive aspects of integrable quantum field theories 13 2.1 General notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2 Particle spectrum and one-particle space . . . . . . . . . . . . . . . . 15 2.3 The scattering function . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.4 Full state space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.5 Asymptotic completeness; closing the circle . . . . . . . . . . . . . . . 29 2.6 Connection to algebraic quantum field theory . . . . . . . . . . . . . 33 3 Locality and the form factor series 37 3.1 Locality and the form factor series . . . . . . . . . . . . . . . . . . . 38 3.2 Local commutativity theorem for one- and two-particle form factors . 44 3.3 Transformation properties of the form factors . . . . . . . . . . . . . 58 3.3.1 Form factors of invariant operators and derivatives . . . . . . 62 4 Structure of form factors and the minimal solution 64 4.1 Classification of two-particle form factors . . . . . . . . . . . . . . . . 64 4.2 Existence of the minimal solutions and asymptotic growth . . . . . . 68 4.3 Computing a characteristic function . . . . . . . . . . . . . . . . . . . 74 5 The stress-energy tensor 77 5.1 The stress-energy tensor from first principles . . . . . . . . . . . . . . 77 5.2 The stress-energy tensor at one-particle level . . . . . . . . . . . . . . 83 5.3 Characterization at one-particle level . . . . . . . . . . . . . . . . . . 88 6 State-independent QEI for constant scattering functions 94 6.1 Candidate for the stress-energy tensor . . . . . . . . . . . . . . . . . 94 6.2 A generic estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 6.3 Derivation of the QEI . . . . . . . . . . . . . . . . . . . . . . . . . . 99 6.4 Discussion of the QEI . . . . . . . . . . . . . . . . . . . . . . . . . . 105 6.5 Supplementary computations . . . . . . . . . . . . . . . . . . . . . . 108 7 QEIs at one-particle level for generic scattering functions 110 7.1 Derivation of the QEI at one-particle level . . . . . . . . . . . . . . . 111 7.2 Extending the scope of the QEI result . . . . . . . . . . . . . . . . . 117 7.3 A general recipe to obtain QEIs at one-particle level . . . . . . . . . 119 8 Examples 123 8.1 Models with one scalar particle type without bound states . . . . . . 123 8.2 Generalized Bullough-Dodd model . . . . . . . . . . . . . . . . . . . 125 8.3 Federbush model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 8.4 O(n)-nonlinear sigma model . . . . . . . . . . . . . . . . . . . . . . . 130 9 Conclusion, discussion, and outlook 134 A Constructive aspects of integrable quantum field theory 137 A.1 Representation theory of the Poincaré group in 1+1d . . . . . . . . . 137 A.2 Discrete symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 A.3 S-function and ZF operators in a basis . . . . . . . . . . . . . . . . . 143 A.4 Improper rapidity eigenstates . . . . . . . . . . . . . . . . . . . . . . 145 A.5 Bound states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 A.6 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 B Literature survey: Form factor conventions 158 C Stress-energy tensor 159 C.1 Stress-energy tensors for the free scalar field . . . . . . . . . . . . . . 159 C.2 A weaker notion for the density property . . . . . . . . . . . . . . . . 163 C.3 Stress-energy tensor at one-particle level generating the boosts . . . . 164 Bibliography 16