On symmetries in (anti)causal (non)abelian quantum theories
Causality is implemented in quantum theories by analytical continuation of Green functions or propagators according to (anti)causal boundary conditions to be imposed. In a symbolic way this is done by adding to the positive real mass of e.g. a causal Klein-Gordon field an infinitesimal negative imaginary part yielding the so-called (causal) Feynman propagator. An imaginary part of opposite sign would yield the anticausal counterpart of the Klein-Gordon field, while the propagator of an (acausal) field with a purely real valued mass has to be understood in terms of a principle value prescription.
The purpose of our research work to be presented (see e.g. hep-ph/0211460, nucl-th/0212008, and references therein) is to identify and outline non-trivial mathematical modifications in the symmetry structure and standard quantization of (acausal) Hermitian quantum theories induced by (anti)causal boundary conditions, i.e. by the (anti)causal analytical continuation of the underlying Green functions. The crucial point we want to make is that an (anti)causal neutral scalar field should not be considered to be Hermitian, as one (anti)causal neutral scalar field consists of a non-Hermitian linear combination of two (Hermitian) acausal fields. Consequently, the underlying symmetry structure of (anti)causal quantum theories is much richer than of acausal Hermitian quantum theories discussed in standard textbooks. The enrichment of Noether symmetries leads to various kinds of new conserved Noether currents, while some traditional Noether currents usually believed to be conserved are shown to be not conserved.
After recalling a formalism of N. Nakanishi (1972) for the Lorentz-covariant quantization of (anti)causal Klein-Gordon fields, in which creation operators are not obtained from annihilation operators by Hermitian conjugation and in which Green functions are discussed in the context of - what is now called - (tempered) Ultradistributions (see e.g. C.G. Bollini et al. (1999)), we will extend the (renormalizable) formalism to (anti)causal fields of spin 1/2 and 1. The construction of respective spinors and polarization vectors will require the introduction of the concept of Lorentz-transformations for ``complex mass'' fields. The transition to charged fields and the consistent introduction of gauge-invariance in (anti)causal quantum theories will yield a new antiparticle concept in spite of reproducing exactly all results of perturbative QED. The non-Abelian extension of the formalism is straight forward and will be sketched.
In summary, we will not only come to a reinterpretation and enrichment of the symmetry behaviour of quantum theories under e.g. C-, P-, T- and Lorentz-transformations and their products, yet also to a necessary revision of the traditional probability and unitarity concept of Hermitian quantum theories. Implications and applications of unitarity and enriched Noether symmetries are discussed or indicated.