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This paper explores quantum field theories with pseudo-HermitianHamiltonians, where PT-symmetric Hamiltonians serve as a special case. Inspecific regimes, these pseudo-Hermitian Hamiltonians have real eigenspectra,orthogonal eigenstates, and unitary time evolution. So far, mostpseudo-Hermitian quantum field theories have been constructed using analyticcontinuation or by adding non-Hermitian terms to otherwise HermitianHamiltonians. However, in this paper, we take a different approach. Weconstruct pseudo-Hermitian scalar and fermionic quantum field theories fromfirst principles, by extending the Poincar\'e algebra to include non-Hermitiangenerators. This allows us to develop consistent pseudo-Hermitian quantum fieldtheories, with Lagrangian densities transforming appropriately under the properPoincar\'e group. By doing so, we establish a more solid theoretical foundationfor the emerging field of non-Hermitian quantum field theory.

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