Pseudo-real quantum fields

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.