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arXiv:0908.2645

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

Abouzeid M. Shalaby

Published Aug 18, 2009• 0 citations
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Abstract

This paper has been withdrawn by the author(s), due a mistake of factor 1/2.
DOI
10.48550/arxiv.0908.2645

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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

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

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.
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.