Pseudo-Hermitian QFT: relativistic scattering and symmetry structure

TL;DR

This paper extends pseudo-Hermitian quantum mechanics to relativistic quantum field theory, demonstrating a consistent scattering framework with preserved unitarity, Lorentz invariance, and CPT symmetry, despite non-Hermitian features.

Contribution

It introduces a relativistic pseudo-Hermitian QFT framework with distinct in/out metrics and a novel symmetry formalism, ensuring probability conservation and Lorentz invariance.

Findings

Scattering matrix remains Lorentz invariant and unitary in the pseudo-Hermitian framework.

The CPT theorem holds within the pseudo-Hermitian QFT.

A perturbative Dyson series expansion is developed for the pseudo-Hermitian scattering matrix.

Abstract

Unitarity is a cornerstone of quantum theory, ensuring the conservation of probability and information. Although non-Hermitian Hamiltonians are typically associated with open or dissipative systems, pseudo-Hermitian quantum mechanics shows that real spectra and unitary evolution can still emerge through a suitably defined inner product. Motivated by this insight, we extend the pseudo-Hermitian framework to relativistic quantum field theory and construct a consistent formulation of scattering processes. A novel structural feature of this theory is the presence of distinct metric operators for the in and out sectors, connected through a nontrivial metric projector that guarantees global probability conservation under pseudo-unitary time evolution. We further develop a general symmetry formalism, showing that each symmetry generally corresponds to two pseudo-unitary operators associated with the in and out metrics, respectively. Within this framework, the scattering matrix admits a perturbative expansion through the Dyson series and remains Lorentz invariant and unitary, remarkably in complete agreement with the conventional Hermitian case. The fundamental CPT theorem is also shown to hold. Our results provide a rigorous foundation for interacting pseudo-Hermitian quantum field theories and open new directions for exploring their possible physical implications beyond the standard Hermitian paradigm.