Mathematical aspects of fermions in quantum field theory

Typical quantum field theoretic treatments of neutrino oscillations have been confined to Fock states; we extend these oscillation formulae to incorporate general quasifree states and nonunitary mixing relations. In regards to quark confinement, we highlight some of the mathematical structure that underlies 't Hooft's duality formula. In particular, we consider a general (nonunitary) orthogonal mixing transformation by relating the creators of flavour states to those of the mass eigenstates via a Bogoliubov transformation. We calculate the correlation function that relates the creation of a #lambda#-flavour particle and the detection (at a later time) of a #mu#-flavour particle. The resulting formula contains oscillation terms whose arguments depend on both the difference and sum of the particle energies. In dealing with general quasifree states, we use a standard technique that relates the cyclic representation of a quasifree state with two tensored Fock representations. After determining the action of second-quantised operators on this doubled Fock space, we calculate the general oscillation formula. We make this formula explicit for the class of states which are U(1) gauge invariant and have characteristic operators that are functions of the Hamiltonian; this formula has the aforementioned nonstandard oscillation terms. Using independent methods (the KMS condition), we determine the oscillation formula for thermal states; the results agree with our more general formula. Finally, we use a particular quasifree state to calculate the oscillation formula between light and heavy neutrinos. We review 't Hooft's duality formula which plays a central role in his explanation of quark confinement. We introduce the notion of induced projective representations and define partial Fourier transforms which allow us to relate representations induced by different subgroups. Using this framework, we construct a particular induced representation of the group Z"3 x Z"3 whose structure resembles that of the quantised magnetic and electric flux lines in 't Hooft's model. Partial Fourier transforms of this representation allow us to perform our own sort of duality transformations. (author)SIGLEAvailable from British Library Document Supply Centre- DSC:DN056639 / BLDSC - British Library Document Supply CentreGBUnited Kingdo