The Charge Quantum Numbers of Gauge Invariant Quasi-free Endomorphisms

The representations of a group of gauge automorphisms of the canonical commutation or anticommutation relations which appear on the Hilbert spaces of isometries H_\rho implementing quasi-free endomorphisms \rho on Fock space are studied. Such a representation, which characterizes the "charge" of \rho in local quantum field theory, is determined by the Fock space structure of H_\rho itself: Together with a "basic" representation of the group, all higher symmetric or antisymmetric tensor powers thereof also appear. Hence \rho is reducible unless it is an automorphism. It is further shown by the example of the massless Dirac field in two dimensions that localization and implementability of quasi-free endomorphisms are compatible with each other.Comment: 15 pages, no figure

Scattering matrix in external field problems

We discuss several aspects of second quantized scattering operators $\hat S$ for fermions in external time dependent fields. We derive our results on a general, abstract level having in mind as a main application potentials of the Yang--Mills type and in various dimensions. We present a new and powerful method for proving existence of $\hat S$ which is also applicable to other situations like external gravitational fields. We also give two complementary derivations of the change of phase of the scattering matrix under generalized gauge transformations which can be used whenever our method of proving existence of $\hat S$ applies. The first is based on a causality argument i.e.\ $\hat S$ (including phase) is determined from a time evolution, and the second exploits the geometry of certain infinite-dimensional group extensions associated with the second quantization of 1-particle operators. As a special case we obtain a Hamiltonian derivation of the the axial Fermion-Yang-Mills anomaly and the Schwinger terms related to it via the descent equations, which is on the same footing and traces them back to a common root.Comment: AmsTex file (uses amstex.tex and amsppt.sty) 22 ouput page

Unitary evolution in Gowdy cosmology

Recent results on the non-unitary character of quantum time evolution in the family of Gowdy T**3 spacetimes bring the question of whether one should renounce in cosmology to the most sacred principle of unitary evolution. In this work we show that the answer is in the negative. We put forward a full nonperturbative canonical quantization of the polarized Gowdy T**3 model that implements the dynamics while preserving unitarity. We discuss possible implications of this result.Comment: 5 pages, no figures. V2 discussion expanded, references added. Final version to appear in PR

QED in external fields from the spin representation

Systematic use of the infinite-dimensional spin representation simplifies and rigorizes several questions in Quantum Field Theory. This representation permutes ``Gaussian'' elements in the fermion Fock space, and is necessarily projective: we compute its cocycle at the group level, and obtain Schwinger terms and anomalies from infinitesimal versions of this cocycle. Quantization, in this framework, depends on the choice of the ``right'' complex structure on the space of solutions of the Dirac equation. We show how the spin representation allows one to compute exactly the S-matrix for fermions in an external field; the cocycle yields a causality condition needed to determine the phase.Comment: 32 pages, Plain TeX, UCR-FM-01-9

Neutral Particles and Super Schwinger Terms

Z_2-graded Schwinger terms for neutral particles in 1 and 3 space dimensions are considered.Comment: 13 page

Slowly decaying classical fields, unitarity, and gauge invariance

In classical external gauge fields that fall off less fast than the inverse of the evolution parameter (time) of the system the implementability of a unitary perturbative scattering operator ($S$-matrix) is not guaranteed, although the field goes to zero. The importance of this point is exposed for the counter-example of low-dimensionally expanding systems. The issues of gauge invariance and of the interpretation of the evolution at intermediate times are also intricately linked to that point.Comment: 8 pages, no figure

Kakutani Dichotomy on Free States

Two quasi-free states on a CAR or CCR algebra are shown to generate quasi-equivalent representations unless they are disjoint.Comment: 12 page

On the ultraviolet behaviour of quantum fields over noncommutative manifolds

By exploiting the relation between Fredholm modules and the Segal-Shale-Stinespring version of canonical quantization, and taking as starting point the first-quantized fields described by Connes' axioms for noncommutative spin geometries, a Hamiltonian framework for fermion quantum fields over noncommutative manifolds is introduced. We analyze the ultraviolet behaviour of second-quantized fields over noncommutative 3-tori, and discuss what behaviour should be expected on other noncommutative spin manifolds.Comment: 10 pages, RevTeX version, a few references adde