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arXiv:2412.10732 · Physical review. D/Physical review. D.

Isospectral local Hermitian theory for the P T -symmetric i ϕ 3 quantum field theory

Yi-Da Li, Qing Wang

Published Jan 21, 2025• 2 citations
arXivHTMLPDF

Abstract

We propose a new method to calculate perturbatively the isospectral Hermitian theory for the P T -symmetric i ϕ 3 quantum field theory in d dimensions, whose result is local. The result of the new method in 1 dimension reproduces our previous result in the i x 3 quantum mechanics, and the new method can be seen as a generalization of our previous method to quantum field theory. We also find the isospectral local Hermitian theory has the same form in all dimensions and differs in coefficients only, and our previous results in quantum mechanics can be used directly to determine the form of the isospectral local Hermitian quantum field theory.

DOI
10.1103/physrevd.111.025016

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We propose a new method to calculate perturbatively the isospectral Hermitian theory for the P T -symmetric i ϕ 3 quantum field theory in d dimensions, whose result is local. The result of the new method in 1 dimension reproduces our previous result in the i x 3 quantum mechanics, and the new method can be seen as a generalization of our previous method to quantum field theory. We also find the isospectral local Hermitian theory has the same form in all dimensions and differs in coefficients only, and our previous results in quantum mechanics can be used directly to determine the form of the isospectral local Hermitian quantum field theory.
A new method to work out the Hermitian correspondence of a PT\mathcal{PT}PT-symmetric quantum mechanical Hamiltonian is proposed. In contrast to the conventional method, the new method ends with a local Hamiltonian of the form 12p2+12m2x2+v(x)\frac{1}{2}{p}^{2}+\frac{1}{2}{m}^{2}{x}^{2}+v(x)21​p2+21​m2x2+v(x) without any higher-derivative terms. This method is available for Hamiltonians with general antilinear symmetries in the perturbative regime. Possible extensions to multivariable quantum mechanics and quantum field theories are discussed.
It has recently been shown that a non-Hermitian Hamiltonian H possessing an unbroken PT\mathcal{PT}PT symmetry (i) has a real spectrum that is bounded below, and (ii) defines a unitary theory of quantum mechanics with positive norm. The proof of unitarity requires a linear operator C,\mathcal{C},C, which was originally defined as a sum over the eigenfunctions of H. However, using this definition to calculate C\mathcal{C}C is cumbersome in quantum mechanics and impossible in quantum field theory. An alternative method is devised here for calculating C\mathcal{C}C directly in terms of the operator dynamical variables of the quantum theory. This method is general and applies to a variety of quantum mechanical systems having several degrees of freedom. More importantly, this method is used to calculate the C\mathcal{C}C operator in quantum field theory. The C\mathcal{C}C operator is a time-independent observable in PT\mathcal{PT}PT-symmetric quantum field theory.
In my talk at the Seventh QCD Workshop held in Villefranche in January 2003 I showed that a non-Hermitian Hamiltonian H possessing an unbroken [Formula: see text] symmetry (i) has a real spectrum that is bounded below, and (ii) defines a unitary theory of quantum mechanics with positive norm. The proof of unitarity requires a linear operator [Formula: see text], which was originally defined as a sum over the eigenfunctions of H. However, using this definition to calculate [Formula: see text] is cumbersome in quantum mechanics and impossible in quantum field theory. I describe here an alternative method for calculating [Formula: see text] directly in terms of the operator dynamical variables of the quantum theory. This new method is general and applies to a variety of quantum mechanical systems having several degrees of freedom. More importantly, this method gives the [Formula: see text] operator in quantum field theory. The [Formula: see text] operator is a new time-independent observable in [Formula: see text]-symmetric quantum field theory.
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.
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