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arXiv:0901.1889

New Treatment of Ghost states and Unitarity problems in the PT-Symmetric $-\phi^4$ Scalar Field Theory

Abouzeid M. Shalaby

Published Jan 13, 2009• 0 citations
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Abstract

In this work, a new ansatz is introduced to make the calculations of the metric operator in Pseudo-Hermitian field theory simpler. The idea is to assume that the metric operator is not only a functional of the field operators $\phi$ and its conjugate field $\pi$ but also on the field gradient $\nabla\phi$. Rather than the locality of the metric operator obtained, the ansatz enables one to calculate the metric operator just once for all dimensions of the space-time. We calculated the metric operator of the $i\phi^{3}$ scalar field theory up to first order in the coupling. The higher orders can be conjectured from their corresponding operators in the quantum mechanical case available in the literature. We assert that, the calculations existing in literature for the metric operator in field theory are cumbersome and are done case by case concerning the dimension of space-time in which the theory is investigated. Moreover, while the resulted metric operator in this work is local, the existing calculations for the metric operator leads to a non-local one. Indeed, we expect that the new results introduced in this work will greatly lead to the progress of the studies in Pseudo-Hermitian field theories where there exist a lack of such kind of studies in the literature. In fact, with the aid of this work a rigorous study of a $\mathcal{PT}$-symmetric Higgs mechanism can be reached.

Related papers

Comment on “New ansatz for metric operator calculation in pseudo-Hermitian field theory”

Carl M. Bender, Gregorio Benincasa, Hugh F Jones

In a recent Brief Report by Shalaby, a new first-order perturbative calculation of the metric operator for an $i{\ensuremath{\phi}}^{3}$ scalar field theory is given. It is claimed that the incorporation of derivative terms in the ansatz for the metric operator results in a local solution, in contrast to the nonlocal solution previously obtained by Bender, Brody, and Jones. Unfortunately, Shalaby's calculation is not valid because of sign errors.

arXiv:0906.4293•Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Dec 17, 2009•3 citations

Reply to "Comment on 'New ansatz for metric operator calculation in pseudo-Hermitian field theory '", arXiv:0906.4293

Abouzeid M. Shalaby

In this report, we reply to a recent comment by Carl M. Bender, Gregorio Benincasa and Hugh F. Jones on our work 'New ansatz for metric operator calculation in pseudo-Hermitian field theory (Phys. Rev. D. 79, 107702 (2009)). In fact, they figured out that there exist sign errors in our work which leaded to the conclusion of the invalidity of the ansatz introduced in our work. Here, we show that, the ansatz is valid in d+1 space-time dimensions, which by itself is a new and very important result. The importance of the work comes from the fact that it is the first time to have a metric operator for a quantum field theory which is local in the fields as well as valid in 3+1 space-time dimensions. Moreover, it is composed of the operators in the Hamiltonian itself which makes the Feynmann diagram calculations for the physical amplitudes go the same way as in conventional theories.

arXiv:0908.1802•Aug 12, 2009•0 citations

Differential realization of pseudo-Hermiticity: A quantum mechanical analog of Einstein’s field equation

Ali Mostafazadeh

For a given pseudo-Hermitian Hamiltonian of the standard form: H=p2∕2m+v(x), we reduce the problem of finding the most general (pseudo-)metric operator η satisfying H†=ηHη−1 to the solution of a differential equation. If the configuration space is R, this is a Klein-Gordon equation with a nonconstant mass term. We obtain a general series solution of this equation that involves a pair of arbitrary functions. These characterize the arbitrariness in the choice of η. We apply our general results to calculate η for the PT-symmetric square well, an imaginary scattering potential, and a class of imaginary delta-function potentials. For the first two systems, our method reproduces the known results in a straightforward and extremely efficient manner. For all these systems we obtain the most general η up to second-order terms in the coupling constants.

arXiv:quant-ph/0603023•Journal of Mathematical Physics•Jul 1, 2006•24 citations

Beyond PT-symmetry: towards a symmetry-metric relation for time-dependent non-Hermitian Hamiltonians

Luís Felipe Alves da Silva

A new chapter in quantum mechanics has opened over the past 20 years with the fact that time-independent (TI) non-Hermitian Hamiltonians have a real spectrum and unitary time evolution when they exhibit an unbroken PT -symmetry and satisfy the pseudo-Hermiticity relation.In this Master's thesis, we first propose a method for the derivation of a general continuous antilinear time-dependent (TD) symmetry operator I(t) for non-Hermitian Hamiltonian H(t) and metric operator ρ(t) in a TD scenario.Assuming H(t) to be simultaneously ρ(t)-pseudo-Hermitian and Ξ(t)-anti-pseudo-Hermitian, we also derive the antilinear symmetry I(t) = Ξ -1 (t)ρ(t), which retrieves an important result obtained by Mostafazadeh [J.Math, Phys.43, 3944 (2002)] for the time-independent (TI) scenario.We apply our method for the derivation of the symmetry associated with TD non-Hermitian linear and quadratic Hamiltonians.In the TI scenario, we retrieve the well-known Bender-Berry-Mandilara result for the symmetry operator:].The results here derived allow us to propose a useful symmetry-metric relation for TD non-Hermitian Hamiltonians.From this relation, the TD metric is automatically derived from the TD symmetry of the problem.Our results reinforce the prospects of going beyond PT -symmetric quantum mechanics making the field of pseudo-Hermiticity even more comprehensive and promising.

•Mar 25, 2022•1 citation

Exact Metric Operators as the Ground State functions of the Hermitian Conjugates of a Class of Quasi-Hermitian Hamiltonians

Abouzeid M. Shalaby

We generalized a class of non-Hermitian Hamiltonians which introduced previously by us in such a way in which every member in the class is non-\textit{PT}-symmetric. For every member of the class, the ground state is a constant with zero energy eigen value. Instead of using an infinite set of coupled operator equations to calculate the metric operator we used a simple realization to obtain the class of closed form metric operators corresponding to the class of non-Hermitian and non-\textit{PT}-symmetric Hamiltonians introduced. The trick is that, if $ψ$ is an eigen function of $H$, then $ϕ=ηψ$ is an eigen function of $H^{\dagger}$ with the same eigen value. Thus, knowing any pair $(ψ,ϕ)$ one can deduce the form of the exact metric operator. We note that, the class of Hamiltonians generalized in this work has the form of that of imaginary magnetic field which can be absorbed by the quasi-gauge transformations represented by metric operators. Accordingly, it is expected that the $Q$ operators will disappear for the whole members in the class in the path integral formulation. However, the detailed analysis of this issue will appear in another work.

arXiv:0706.2509•Jun 17, 2007•0 citations

Pseudo-Hermitian QFT: relativistic scattering and symmetry structure

Ruifeng Leng, Cheng-Yang Lee, Siyi Zhou

Unitarity is a cornerstone of quantum theory, ensuring the conservation of probability and information. Although non-Hermitian Hamiltonians are typically associated with open or dissipative systems, pseudo-Hermitian quantum mechanics shows that real spectra and unitary evolution can still emerge through a suitably defined inner product. Motivated by this insight, we extend the pseudo-Hermitian framework to relativistic quantum field theory and construct a consistent formulation of scattering processes. A novel structural feature of this theory is the presence of distinct metric operators for the in and out sectors, connected through a nontrivial metric projector that guarantees global probability conservation under pseudo-unitary time evolution. We further develop a general symmetry formalism, showing that each symmetry generally corresponds to two pseudo-unitary operators associated with the in and out metrics, respectively. Within this framework, the scattering matrix admits a perturbative expansion through the Dyson series and remains Lorentz invariant and unitary, remarkably in complete agreement with the conventional Hermitian case. The fundamental CPT theorem is also shown to hold. Our results provide a rigorous foundation for interacting pseudo-Hermitian quantum field theories and open new directions for exploring their possible physical implications beyond the standard Hermitian paradigm.

arXiv:2510.27404•Oct 31, 2025•0 citations
New Treatment of Ghost states and Unitarity problems in the PT-Symmetric −ϕ4-\phi^4−ϕ4 Scalar Field Theory
In this work, a new ansatz is introduced to make the calculations of the metric operator in Pseudo-Hermitian field theory simpler. The idea is to assume that the metric operator is not only a functional of the field operators ϕ\phiϕ and its conjugate field π\piπ but also on the field gradient ∇ϕ\nabla\phi∇ϕ. Rather than the locality of the metric operator obtained, the ansatz enables one to calculate the metric operator just once for all dimensions of the space-time. We calculated the metric operator of the iϕ3i\phi^{3}iϕ3 scalar field theory up to first order in the coupling. The higher orders can be conjectured from their corresponding operators in the quantum mechanical case available in the literature. We assert that, the calculations existing in literature for the metric operator in field theory are cumbersome and are done case by case concerning the dimension of space-time in which the theory is investigated. Moreover, while the resulted metric operator in this work is local, the existing calculations for the metric operator leads to a non-local one. Indeed, we expect that the new results introduced in this work will greatly lead to the progress of the studies in Pseudo-Hermitian field theories where there exist a lack of such kind of studies in the literature. In fact, with the aid of this work a rigorous study of a PT\mathcal{PT}PT-symmetric Higgs mechanism can be reached.
In this report, we reply to a recent comment by Carl M. Bender, Gregorio Benincasa and Hugh F. Jones on our work 'New ansatz for metric operator calculation in pseudo-Hermitian field theory (Phys. Rev. D. 79, 107702 (2009)). In fact, they figured out that there exist sign errors in our work which leaded to the conclusion of the invalidity of the ansatz introduced in our work. Here, we show that, the ansatz is valid in d+1 space-time dimensions, which by itself is a new and very important result. The importance of the work comes from the fact that it is the first time to have a metric operator for a quantum field theory which is local in the fields as well as valid in 3+1 space-time dimensions. Moreover, it is composed of the operators in the Hamiltonian itself which makes the Feynmann diagram calculations for the physical amplitudes go the same way as in conventional theories.
For a given pseudo-Hermitian Hamiltonian of the standard form: H=p2∕2m+v(x), we reduce the problem of finding the most general (pseudo-)metric operator η satisfying H†=ηHη−1 to the solution of a differential equation. If the configuration space is R, this is a Klein-Gordon equation with a nonconstant mass term. We obtain a general series solution of this equation that involves a pair of arbitrary functions. These characterize the arbitrariness in the choice of η. We apply our general results to calculate η for the PT-symmetric square well, an imaginary scattering potential, and a class of imaginary delta-function potentials. For the first two systems, our method reproduces the known results in a straightforward and extremely efficient manner. For all these systems we obtain the most general η up to second-order terms in the coupling constants.
A new chapter in quantum mechanics has opened over the past 20 years with the fact that time-independent (TI) non-Hermitian Hamiltonians have a real spectrum and unitary time evolution when they exhibit an unbroken PT -symmetry and satisfy the pseudo-Hermiticity relation.In this Master's thesis, we first propose a method for the derivation of a general continuous antilinear time-dependent (TD) symmetry operator I(t) for non-Hermitian Hamiltonian H(t) and metric operator ρ(t) in a TD scenario.Assuming H(t) to be simultaneously ρ(t)-pseudo-Hermitian and Ξ(t)-anti-pseudo-Hermitian, we also derive the antilinear symmetry I(t) = Ξ -1 (t)ρ(t), which retrieves an important result obtained by Mostafazadeh [J.Math, Phys.43, 3944 (2002)] for the time-independent (TI) scenario.We apply our method for the derivation of the symmetry associated with TD non-Hermitian linear and quadratic Hamiltonians.In the TI scenario, we retrieve the well-known Bender-Berry-Mandilara result for the symmetry operator:].The results here derived allow us to propose a useful symmetry-metric relation for TD non-Hermitian Hamiltonians.From this relation, the TD metric is automatically derived from the TD symmetry of the problem.Our results reinforce the prospects of going beyond PT -symmetric quantum mechanics making the field of pseudo-Hermiticity even more comprehensive and promising.
We generalized a class of non-Hermitian Hamiltonians which introduced previously by us in such a way in which every member in the class is non-\textit{PT}-symmetric. For every member of the class, the ground state is a constant with zero energy eigen value. Instead of using an infinite set of coupled operator equations to calculate the metric operator we used a simple realization to obtain the class of closed form metric operators corresponding to the class of non-Hermitian and non-\textit{PT}-symmetric Hamiltonians introduced. The trick is that, if ψψψ is an eigen function of HHH, then ϕ=ηψϕ=ηψϕ=ηψ is an eigen function of H†H^{\dagger}H† with the same eigen value. Thus, knowing any pair (ψ,ϕ)(ψ,ϕ)(ψ,ϕ) one can deduce the form of the exact metric operator. We note that, the class of Hamiltonians generalized in this work has the form of that of imaginary magnetic field which can be absorbed by the quasi-gauge transformations represented by metric operators. Accordingly, it is expected that the QQQ operators will disappear for the whole members in the class in the path integral formulation. However, the detailed analysis of this issue will appear in another work.
Unitarity is a cornerstone of quantum theory, ensuring the conservation of probability and information. Although non-Hermitian Hamiltonians are typically associated with open or dissipative systems, pseudo-Hermitian quantum mechanics shows that real spectra and unitary evolution can still emerge through a suitably defined inner product. Motivated by this insight, we extend the pseudo-Hermitian framework to relativistic quantum field theory and construct a consistent formulation of scattering processes. A novel structural feature of this theory is the presence of distinct metric operators for the in and out sectors, connected through a nontrivial metric projector that guarantees global probability conservation under pseudo-unitary time evolution. We further develop a general symmetry formalism, showing that each symmetry generally corresponds to two pseudo-unitary operators associated with the in and out metrics, respectively. Within this framework, the scattering matrix admits a perturbative expansion through the Dyson series and remains Lorentz invariant and unitary, remarkably in complete agreement with the conventional Hermitian case. The fundamental CPT theorem is also shown to hold. Our results provide a rigorous foundation for interacting pseudo-Hermitian quantum field theories and open new directions for exploring their possible physical implications beyond the standard Hermitian paradigm.