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Physical review. D/Physical review. D.

Isospectral local Hamiltonians for perturbative PT -symmetric Hamiltonians

Yi-Da Li, Qing Wang

Published Oct 11, 2023• 1 citation
Publisher

Abstract

A new method to work out the Hermitian correspondence of a $\mathcal{PT}$-symmetric quantum mechanical Hamiltonian is proposed. In contrast to the conventional method, the new method ends with a local Hamiltonian of the form $\frac{1}{2}{p}^{2}+\frac{1}{2}{m}^{2}{x}^{2}+v(x)$ 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.

DOI
10.1103/physrevd.108.085008

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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.
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 transformation of the form x to iy; x,y in R, or an equivalent similarity transformation with a metric operator ηηη are shown to transform non-Hermitian PT-symmetric Hamiltonians into Hermitian partner Hamiltonians in Hilbert space. Isospectrality and mass signature are discussed.
In 1998 it was discovered that the requirement that a Hamiltonian be Dirac Hermitian (H = H†) can be weakened and generalized to the requirement that a Hamiltonian be PT symmetric ([H,PT] = 0); that is, invariant under combined space reflection and time reversal. Weakening the constraint of Hermiticity allows one to consider new kinds of physically acceptable Hamiltonians and, in effect, it amounts to extending quantum mechanics from the real (Hermitian) domain into the complex domain. Much work has been done on the analysis of various PT‐symmetric quantum‐mechanical models. However, only very little analysis has been done on PT‐symmetric quantum‐field‐theoretic models. Here, we describe some of what has been done in the context of PT‐symmetric quantum field theory and describe some possible fundamental applications.
Recently, much research has been carried out on Hamiltonians that are not Hermitian but are symmetric under space-time reflection, that is, Hamiltonians that exhibit PT symmetry. Investigations of the Sturm-Liouville eigenvalue problem associated with such Hamiltonians have shown that in many cases the entire energy spectrum is real and positive and that the eigenfunctions form an orthogonal and complete basis. Furthermore, the quantum theories determined by such Hamiltonians have been shown to be consistent in the sense that the probabilities are positive and the dynamical trajectories are unitary. However, the geometrical structures that underlie quantum theories formulated in terms of such Hamiltonians have hitherto not been fully understood. This paper studies in detail the geometric properties of a Hilbert space endowed with a parity structure and analyses the characteristics of a PT-symmetric Hamiltonian and its eigenstates. A canonical relationship between a PT-symmetric operator and a Hermitian operator is established. It is shown that the quadratic form corresponding to the parity operator, in particular, gives rise to a natural partition of the Hilbert space into two halves corresponding to states having positive and negative PT norm. The indefiniteness of the norm can be circumvented by introducing a symmetry operator C that defines a positive definite inner product by means of a CPT conjugation operation.
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.