Poincar\'{e} symmetries and representations in pseudo-Hermitian quantum field theory

TL;DR

This paper develops a foundational approach to pseudo-Hermitian quantum field theories by extending the Poincaré algebra, enabling consistent construction of scalar and fermionic theories with real spectra and unitary evolution.

Contribution

It introduces a first-principles method to construct pseudo-Hermitian quantum field theories through algebraic extension, moving beyond analytic continuation methods.

Findings

Constructed pseudo-Hermitian scalar and fermionic QFTs from first principles.

Extended Poincaré algebra to include non-Hermitian generators.

Established a consistent theoretical framework for non-Hermitian QFTs.

Abstract

This paper explores quantum field theories with pseudo-Hermitian Hamiltonians, where PT-symmetric Hamiltonians serve as a special case. In specific regimes, these pseudo-Hermitian Hamiltonians have real eigenspectra, orthogonal eigenstates, and unitary time evolution. So far, most pseudo-Hermitian quantum field theories have been constructed using analytic continuation or by adding non-Hermitian terms to otherwise Hermitian Hamiltonians. However, in this paper, we take a different approach. We construct pseudo-Hermitian scalar and fermionic quantum field theories from first principles by extending the Poincar\'e algebra to include non-Hermitian generators. This allows us to develop consistent pseudo-Hermitian quantum field theories, with Lagrangian densities that transform appropriately under the proper Poincar\'e group. By doing so, we establish a more solid theoretical foundation for the emerging field of non-Hermitian quantum field theory.