On the duality condition for quantum fields

TLDR

A general quantum field theory is considered in which the fields are assumed to be operator‑valued tempered distributions, potentially including any number of boson and fermion fields. The authors state and prove a theorem linking complex Lorentz transformations to the TCP transformation, and use it to examine duality conditions under physically reasonable field assumptions. They extend the algebras of field operators to analyze duality conditions in the context of the TCP‑related theorem. They find that local internal symmetries commute with both the Poincaré group and the TCP transformation.

Abstract

A general quantum field theory is considered in which the fields are assumed to be operator-valued tempered distributions. The system of fields may include any number of boson fields and fermion fields. A theorem which relates certain complex Lorentz transformations to the TCP transformation is stated and proved. With reference to this theorem, duality conditions are considered, and it is shown that such conditions hold under various physically reasonable assumptions about the fields. Extensions of the algebras of field operators are discussed with reference to the duality conditions. Local internal symmetries are discussed, and it is shown that these commute with the Poincaré group and with the TCP transformation.