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arXiv:2305.05814

Isospectral Local Hamiltonians for Perturbative PT-symmetric Hamiltonians

Yi-Da Li, Qing Wang

Published May 9, 2023• 0 citations
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Abstract

A new method to work out the Hermitian correspondence of a PT-symmetric quantum mechanical Hamiltonian is proposed. In contrast to the conventional method, the new method ends with a local Hamiltonian of the form p^2/2+m^2x^2/2+v(x) without any higher-derivative terms. This method is demonstrated in the perturbative regime. Possible extensions to multi-variable quantum mechanics and quantum field theories are discussed.

DOI
10.48550/arxiv.2305.05814

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A new method to work out the Hermitian correspondence of a PT-symmetric quantum mechanical Hamiltonian is proposed. In contrast to the conventional method, the new method ends with a local Hamiltonian of the form p^2/2+m^2x^2/2+v(x) without any higher-derivative terms. This method is demonstrated in the perturbative regime. Possible extensions to multi-variable quantum mechanics and quantum field theories are discussed.
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.
If a HamiltonianH has an unbroken PT symmetry, then it also possesses a hidden symmetry represented by the linear operator C. The operator C commutes with both H and PT . The inner product with respect to CPT is associated with a positive norm and the quantum theory built on the associated Hilbert space is unitary. In this paper it is shown how to construct the operator C for the non-Hermitian PT -symmetric Hamiltonian H = 12p + 1 2x 2 + i x using perturbative techniques. It is also shown how to construct the operator C for H = 12p 2 + 12x 2 − x using nonperturbative methods [1].
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.
Two non-Hermitian PT-symmetric Hamiltonian systems are reconsidered by means of the algebraic method which was originally proposed for the pseudo-Hermitian Hamiltonian systems rather than for the PT-symmetric ones. Compared with the way converting a non-Hermitian Hamiltonian to its Hermitian counterpart, this method has the merit that keeps the Hilbert space of the non-Hermitian PT-symmetric Hamiltonian unchanged. In order to give the positive definite inner product for the PT-symmetric systems, a new operator V, instead of C, can be introduced. The operator V has the similar function to the operator C adopted normally in the PT-symmetric quantum mechanics, however, it can be constructed, as an advantage, directly in terms of Hamiltonians. The spectra of the two non-Hermitian PT-symmetric systems are obtained, which coincide with that given in literature, and in particular, the Hilbert spaces associated with positive definite inner products are worked out.