Non-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 Poincaré 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 Poincaré group. By doing so, we establish a more solid theoretical foundation for the emerging field of non-Hermitian quantum field theory. Published by the American Physical Society 2024This 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 Poincaré 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 Poincaré group. By doing so, we establish a more solid theoretical foundation for the emerging field of non-Hermitian quantum field theory. Published by the American Physical Society 2024

arXiv:2307.16805Physical review. D/Physical review. D.Mar 11, 202411 citations

Bound States of Non-Hermitian Quantum Field Theories

Carl M. Bender, Böttcher, S, Jones, H F +2 more

The spectrum of the Hermitian Hamiltonian ${1\\over2}p^2+{1\\over2}m^2x^2+gx^4$ ( $g>0$ ), which describes the quantum anharmonic oscillator, is real and positive. The non-Hermitian quantum-mechanical Hamiltonian $H={1\\over2}p^2+{1 \\over2}m^2x^2-gx^4$ , where the coupling constant $g$ is real and positive, is ${\\cal PT}$ -symmetric. As a consequence, the spectrum of $H$ is known to be real and positive as well. Here, it is shown that there is a significant difference between these two theories: When $g$ is sufficiently small, the latter Hamiltonian exhibits a two-particle bound state while the former does not. The bound state persists in the corresponding non-Hermitian ${\\cal PT}$ -symmetric $-g\\phi^4$ quantum field theory for all dimensions $0\\leq D<3$ but is not present in the conventional Hermitian $g\\phi^4$ field theory.The spectrum of the Hermitian Hamiltonian ${1\\over2}p^2+{1\\over2}m^2x^2+gx^4$ ($g>0$), which describes the quantum anharmonic oscillator, is real and positive. The non-Hermitian quantum-mechanical Hamiltonian $H={1\\over2}p^2+{1 \\over2}m^2x^2-gx^4$, where the coupling constant $g$ is real and positive, is ${\\cal PT}$-symmetric. As a consequence, the spectrum of $H$ is known to be real and positive as well. Here, it is shown that there is a significant difference between these two theories: When $g$ is sufficiently small, the latter Hamiltonian exhibits a two-particle bound state while the former does not. The bound state persists in the corresponding non-Hermitian ${\\cal PT}$-symmetric $-g\\phi^4$ quantum field theory for all dimensions $0\\leq D<3$ but is not present in the conventional Hermitian $g\\phi^4$ field theory.

Jan 1, 200123 citations

Variational Solution of Non-Hermitian Quantum Field Theory in Two Dimensions

Paul Romatschke

This work describes a variational technique to solve non-Hermitian quantum systems, in particular those with negative coupling quartic self-interaction. Differences of applying variational methods from standard quantum mechanics to non-Hermitian quantum mechanics -- including potential pitfalls -- are highlighted. A successful implementation of the variational approach for the non-Hermitian case is presented. The technique is used to analyze scalar quantum field theory in d=2 with negative quartic self-interaction in the Hamiltonian formulation in lattice discretization.This work describes a variational technique to solve non-Hermitian quantum systems, in particular those with negative coupling quartic self-interaction. Differences of applying variational methods from standard quantum mechanics to non-Hermitian quantum mechanics -- including potential pitfalls -- are highlighted. A successful implementation of the variational approach for the non-Hermitian case is presented. The technique is used to analyze scalar quantum field theory in d=2 with negative quartic self-interaction in the Hamiltonian formulation in lattice discretization.

hep-thCross-listedarXiv:2609.19798Sep 18, 2026

The spectrum of the Hermitian Hamiltonian

{1\\over2}p^2+{1\\over2}m^2x^2+gx^4

(

g>0

), which describes the quantum anharmonic oscillator, is real and positive. The non-Hermitian quantum-mechanical Hamiltonian

H={1\\over2}p^2+{1 \\over2}m^2x^2-gx^4

, where the coupling constant

g

is real and positive, is

{\\cal PT}

-symmetric. As a consequence, the spectrum of

H

is known to be real and positive as well. Here, it is shown that there is a significant difference between these two theories: When

g

is sufficiently small, the latter Hamiltonian exhibits a two-particle bound state while the former does not. The bound state persists in the corresponding non-Hermitian

{\\cal PT}

-symmetric

-g\\phi^4

quantum field theory for all dimensions

0\\leq D<3

but is not present in the conventional Hermitian

g\\phi^4

field theory.