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理论物理通讯：英文版

Comparison of Different Approaches of Finding the Positive Definite Metric in Pseudo-Hermitian Theories

Ananya Ananya, Ghatak, Bhabani, Prasad, Mandal

Published Jan 1, 2013• 0 citations
OpenAlex

Abstract

为了为 pseudo-Hermitian 系统与概率的描述开发单一的量子论，一个人需要在赋予以一个积极明确的公制的操作符的一个不同 Hilbert 空格考虑理论。有不同途径发现如此的公制的操作员。我们比较在几个明确的例子的帮助下在 pseudo-Hermitian 理论计算积极明确的公制的操作员在的不同途径象处于相对论的状况一样非相对论。非凡的点并且自发的对称碎也在这些模型被讨论。

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为了为 pseudo-Hermitian 系统与概率的描述开发单一的量子论，一个人需要在赋予以一个积极明确的公制的操作符的一个不同 Hilbert 空格考虑理论。有不同途径发现如此的公制的操作员。我们比较在几个明确的例子的帮助下在 pseudo-Hermitian 理论计算积极明确的公制的操作员在的不同途径象处于相对论的状况一样非相对论。非凡的点并且自发的对称碎也在这些模型被讨论。
To develop a unitary quantum theory with probabilistic description for pseudo-Hermitian systems one needs to consider the theories in a different Hilbert space endowed with a positive definite metric operator. There are different approaches to find such metric operators. We compare the different approaches of calculating positive definite metric operators in pseudo-Hermitian theories with the help of several explicit examples in non-relativistic as well as in relativistic situations. Exceptional points and spontaneous symmetry breaking are also discussed in these models.
Pseudo-Hermitian quantum theories are those in which the Hamiltonian H satisfies H† = ηHη-1, where η = e-Q is a positive-definite Hermitian operator, rather than the usual H† = H. In the operator formulation of such theories the standard Hilbert-space metric must be modified by the inclusion of η in order to ensure their probabilistic interpretation. With possible generalizations to quantum field theory in mind, it is important to ask how the functional integral formalism for pseudo-Hermitian theories differs from that of standard theories. It turns out that here Q plays quite a different role, serving primarily to implement a canonical transformation of the variables. It does not appear explicitly in the expression for the vacuum generating functional. Instead, the relation to the Hermitian theory is encoded via the dependence of Z on the external source j(t). These points are illustrated and amplified in various versions of the Swanson model, a non-Hermitian transform of the simple harmonic oscillator.
A class of pseudo-Hermitian quantum system with an explicit form of the positive-definite metric in the Hilbert space is presented. The general method involves a realization of the basic canonical commutation relations defining the quantum system in terms of operators that are Hermitian with respect to a pre-determined positive-definite metric in the Hilbert space. Appropriate combinations of these operators result in a large number of pseudo-Hermitian quantum systems admitting entirely real spectra and unitary time evolution. The examples considered include simple harmonic oscillators with complex angular frequencies, Stark (Zeeman) effect with non-Hermitian interaction, non-Hermitian general quadratic form of N boson (fermion) operators, symmetric and asymmetric XXZ spin chain in the complex magnetic field, non-Hermitian Haldane–Shastry spin chain and Lipkin–Meshkov–Glick model.