Poincare symmetries and representations in pseudo-Hermitian quantum field theory

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 Poincare ' algebra to include non -Hermitian generators. This allows us to develop consistent pseudoHermitian quantum field theories, with Lagrangian densities that transform appropriately under the proper Poincare ' group. By doing so, we establish a more solid theoretical foundation for the emerging field of nonHermitian quantum field theory.