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arXiv:0908.1802

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

Abouzeid M. Shalaby

Published Aug 12, 2009• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.0908.1802

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 ``Time-dependent quasi-Hermitian Hamiltonians and the unitary quantum evolution''

Miloslav Znojil

In his fresh "Comment" (arXiv:0711.0137v1), A. Mostafazadeh reacts on my very recent letter (arXiv:0710.5653v1) where I tried to clarify certain misunderstandings which occurred in A. M., Phys. Lett. B \textbf{650}, 208 (2007) [arXiv:0706.1872v2, "Paper"]. As long as the "Comment" offers a new support of the original assertions made in the "Paper", I feel obliged to re-clarify the matter by extending my argumentation. I insist that it is possible to escape the main conclusion of the "Paper", indeed. In particular, I point out a gap in the new calculations in "Comment", add a few remarks on the notation and reconfirm that the unitarity of the time-evolution DOES NOT require the time-independence of the metric operator.

arXiv:0711.0514•Nov 5, 2007•6 citations

On the validity of the 'New ansatz for metric operator calculation in pseudo-Hermitian field theory'"

Abouzeid M. Shalaby

This paper has been withdrawn by the author(s), due a mistake of factor 1/2.

arXiv:0908.2645•Aug 18, 2009•0 citations

Comment on "Reply to Comment on Time-dependent Quasi-Hermitian Hamiltonians and the Unitary Quantum Evolution"

Mostafazadeh, Ali

I point out that if one defines the operator $U_R(t)$ as done by M. Znojil in his reply [arXiv:0711.0514v1] to my comment [arXiv:0711.0137v1] and also accepts the validity of the defining relation of $U_R(t)$ as given in his paper [arXiv:0710.5653v1], one finds that the time-evolution of the associated quantum system is not governed by the Schrödinger equation for the Hamiltonian operator $H$ but an operator $H'$ which differs from $H$ if the metric operator is time-dependent. In the latter case this effective Hamiltonian $H'$ is not observable. This is consistent with the conclusions of my paper [Phys. Lett. B 650, 208 (2007), arXiv:0706.1872v2] which allow for unitary time-evolution generated by unobservable Hamiltonians.

arXiv:0711.1078•Nov 7, 2007•4 citations

Comment on ``Time-dependent quasi-Hermitian Hamiltonians and the unitary quantum evolution''

Mostafazadeh, Ali

In arXiv:0710.5653v1 M. Znojil claims that he has found and corrected an error in my paper: [Phys. Lett. B \textbf{650}, 208 (2007), arXiv:0706.1872v2] and that it is possible to escape its main conclusion, namely that the unitarity of the time-evolution and observability of the Hamiltonian imply time-independence of the metric operator. In this note I give a very short calculation showing that the analysis given by M. Znojil also leads to the same conclusion as of my above-mentioned paper. This is actually a reconfirmation of the validity of the results of the latter paper.

arXiv:0711.0137•Nov 1, 2007•7 citations

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

Abouzeid M. Shalaby

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.

arXiv:0901.1889•Jan 13, 2009•0 citations
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.
In his fresh "Comment" (arXiv:0711.0137v1), A. Mostafazadeh reacts on my very recent letter (arXiv:0710.5653v1) where I tried to clarify certain misunderstandings which occurred in A. M., Phys. Lett. B \textbf{650}, 208 (2007) [arXiv:0706.1872v2, "Paper"]. As long as the "Comment" offers a new support of the original assertions made in the "Paper", I feel obliged to re-clarify the matter by extending my argumentation. I insist that it is possible to escape the main conclusion of the "Paper", indeed. In particular, I point out a gap in the new calculations in "Comment", add a few remarks on the notation and reconfirm that the unitarity of the time-evolution DOES NOT require the time-independence of the metric operator.
I point out that if one defines the operator UR(t)U_R(t)UR​(t) as done by M. Znojil in his reply [arXiv:0711.0514v1] to my comment [arXiv:0711.0137v1] and also accepts the validity of the defining relation of UR(t)U_R(t)UR​(t) as given in his paper [arXiv:0710.5653v1], one finds that the time-evolution of the associated quantum system is not governed by the Schrödinger equation for the Hamiltonian operator HHH but an operator H′H'H′ which differs from HHH if the metric operator is time-dependent. In the latter case this effective Hamiltonian H′H'H′ is not observable. This is consistent with the conclusions of my paper [Phys. Lett. B 650, 208 (2007), arXiv:0706.1872v2] which allow for unitary time-evolution generated by unobservable Hamiltonians.
In arXiv:0710.5653v1 M. Znojil claims that he has found and corrected an error in my paper: [Phys. Lett. B \textbf{650}, 208 (2007), arXiv:0706.1872v2] and that it is possible to escape its main conclusion, namely that the unitarity of the time-evolution and observability of the Hamiltonian imply time-independence of the metric operator. In this note I give a very short calculation showing that the analysis given by M. Znojil also leads to the same conclusion as of my above-mentioned paper. This is actually a reconfirmation of the validity of the results of the latter paper.
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.