Pseudo-real quantum fields

TL;DR

This paper introduces the concept of pseudo-reality for quantum fields, unifying different approaches to pseudo-Hermitian theories and addressing longstanding issues in gauge theories and quantum gravity coupling.

Contribution

It develops the pseudo-reality framework for quantum fields, providing new conditions and insights for pseudo-Hermitian theories, especially in gauge and gravitational contexts.

Findings

Pseudo-reality conditions for bosonic fields up to quadratic order.

Analysis of a non-Hermitian scalar field theory with mass mixing.

Coupling of pseudo-Hermitian fields to a U(1) gauge field.

Abstract

We introduce the concept of pseudo-reality for complex numbers. We show that this concept, applied to quantum fields, provides a unifying framework for two distinct approaches to pseudo-Hermitian quantum field theories. The first approach stems from analytically continuing Hermitian theories into the complex plane, while the second is based on constructing them from first principles. The pseudo-reality condition for bosonic fields resolves a long-standing problem with the formulation of gauge theories involving pseudo-Hermitian currents, sheds new light on the resolution of the so-called Hermiticity Puzzle, and may allow a consistent minimal coupling of pseudo-Hermitian quantum field theories to gravity. We focus on the $i\phi^3$ cubic scalar theory, obtaining the relevant pseudo-reality conditions up to quadratic order in the coupling; a theory of two complex scalar fields with non-Hermitian mass mixing; and the latter's coupling to a $U(1)$ gauge field. The general principle of pseudo-reality, however, is expected to contribute to the ongoing development of the first-principles construction of pseudo-Hermitian quantum field theories, including their formulation in curved spacetimes.