PCT , a fundamental symmetry of quantum field theory, is derived from the assumption of Lorentz invariance and positivity of the spectrum.What happens if we assume only Lorentz invariance and PCT symmetry?Hamiltonians having this property need not be Hermitian but, except when PCT is spontaneously broken, the energy levels of such Hamiltonians are all real and positive!In this talk I examine quantum mechanical and quantum field theoretic systems whose Hamiltonians are non-Hermitian but obey PCT symmetry.These systems have weird and remarkable properties.Examples of such Hamiltonians are H = p 2 + ix 3 and H = p 2x 4 .Hamiltonians such as these are complex deformations of conventional Hermitian Hamiltonians.Thus, in this talk I study the analytic continuation of conventional quantum mechanics into the complex plane.PCT , a fundamental symmetry of quantum field theory, is derived from the assumption of Lorentz invariance and positivity of the spectrum.What happens if we assume only Lorentz invariance and PCT symmetry?Hamiltonians having this property need not be Hermitian but, except when PCT is spontaneously broken, the energy levels of such Hamiltonians are all real and positive!In this talk I examine quantum mechanical and quantum field theoretic systems whose Hamiltonians are non-Hermitian but obey PCT symmetry.These systems have weird and remarkable properties.Examples of such Hamiltonians are H = p 2 + ix 3 and H = p 2x 4 .Hamiltonians such as these are complex deformations of conventional Hermitian Hamiltonians.Thus, in this talk I study the analytic continuation of conventional quantum mechanics into the complex plane.

European Physical Society Conference on High Energy Physics (EPS-HEP)Dec 1, 20010 citations

Failure of the mean-field Hartree approximation for a bosonic many-body system with non-Hermitian Hamiltonian

Matias Ginzburg, Simone Rademacher, Giacomo De Palma

Mean-field Hartree theory is a central tool for reducing many-body dynamics to an effective one-particle evolution. This approximation has been employed when the Hamiltonian governing the many-body dynamics is not Hermitian. Indeed, non-Hermitian Hamiltonians model particle gain/loss or the evolution of open quantum systems between consecutive quantum jumps. Furthermore, the validity of the Hartree approximation for generic non-Hermitian Hamiltonians lies at the basis of a quantum algorithm for nonlinear differential equations. In this work, we show that this approximation can fail. First, we consider the most general bosonic mean-field Hamiltonian with two-body interactions and we prove that, for non-Hermitian Hamiltonian, the Hartree equation does not capture the correct second time derivative of the one-particle marginal state at $t=0$ in the thermodynamic limit. Then, we analytically solve a model of $N$ bosonic qubits with a purely anti-Hermitian Hamiltonian, compute the limit for $N\to\infty$ of the one-particle marginal state and show that it does not agree with the solution of the non-Hermitian Hartree equation. We further show that there exists an initial condition such that the exact one-particle marginal state undergoes a finite-time transition to a mixed state, a phenomenon that is completely absent in the case of Hermitian Hamiltonians. Finally, we simulate a non-Hermitian Hamiltonian with real spectrum and show that, also in this case, there are initial conditions for which the Hartree equation does not reproduce the correct one-particle marginal state in the limit $N\to\infty$ . Our findings challenge the validity of the mean-field Hartree approximation for non-Hermitian Hamiltonians, and call for additional conditions for the validity of the mean-field regime to model the dynamics of particle gain and loss and the open-system dynamics in bosonic many-body systems.Mean-field Hartree theory is a central tool for reducing many-body dynamics to an effective one-particle evolution. This approximation has been employed when the Hamiltonian governing the many-body dynamics is not Hermitian. Indeed, non-Hermitian Hamiltonians model particle gain/loss or the evolution of open quantum systems between consecutive quantum jumps. Furthermore, the validity of the Hartree approximation for generic non-Hermitian Hamiltonians lies at the basis of a quantum algorithm for nonlinear differential equations. In this work, we show that this approximation can fail. First, we consider the most general bosonic mean-field Hamiltonian with two-body interactions and we prove that, for non-Hermitian Hamiltonian, the Hartree equation does not capture the correct second time derivative of the one-particle marginal state at $t=0$ in the thermodynamic limit. Then, we analytically solve a model of $N$ bosonic qubits with a purely anti-Hermitian Hamiltonian, compute the limit for $N\to\infty$ of the one-particle marginal state and show that it does not agree with the solution of the non-Hermitian Hartree equation. We further show that there exists an initial condition such that the exact one-particle marginal state undergoes a finite-time transition to a mixed state, a phenomenon that is completely absent in the case of Hermitian Hamiltonians. Finally, we simulate a non-Hermitian Hamiltonian with real spectrum and show that, also in this case, there are initial conditions for which the Hartree equation does not reproduce the correct one-particle marginal state in the limit $N\to\infty$. Our findings challenge the validity of the mean-field Hartree approximation for non-Hermitian Hamiltonians, and call for additional conditions for the validity of the mean-field regime to model the dynamics of particle gain and loss and the open-system dynamics in bosonic many-body systems.

quant-pharXiv:2601.13038QuantumJan 20, 20260 citations

Bound states of non-Hermitian quantum field theories

Carl M. Bender, Stefan Boettcher, H.F Jones +2 more

arXiv:hep-th/0108057Physics Letters ADec 1, 200137 citations

t=0

in the thermodynamic limit. Then, we analytically solve a model of

N

bosonic qubits with a purely anti-Hermitian Hamiltonian, compute the limit for

N\to\infty

of the one-particle marginal state and show that it does not agree with the solution of the non-Hermitian Hartree equation. We further show that there exists an initial condition such that the exact one-particle marginal state undergoes a finite-time transition to a mixed state, a phenomenon that is completely absent in the case of Hermitian Hamiltonians. Finally, we simulate a non-Hermitian Hamiltonian with real spectrum and show that, also in this case, there are initial conditions for which the Hartree equation does not reproduce the correct one-particle marginal state in the limit

N\to\infty

. Our findings challenge the validity of the mean-field Hartree approximation for non-Hermitian Hamiltonians, and call for additional conditions for the validity of the mean-field regime to model the dynamics of particle gain and loss and the open-system dynamics in bosonic many-body systems.