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Journal of Xi'an University of Arts and Science

PT-symmetry and PT-frame

Zhengli Chen

Published Jan 1, 2013• 0 citations
OpenAlex

Abstract

In recent years,some elementary points of the PT-symmetry and its mathematical or physical descriptions have been studied.This paper gives a discussion of PT-symmetric quantum theory by applying the methods of operator algebra and matrix.Some important results and properties are provided.

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PT-symmetric quantum mechanics

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It is generally assumed that a Hamiltonian for a physically acceptable quantum system (one that has a positive-definite spectrum and obeys the requirement of unitarity) must be Hermitian. However, a PT-symmetric Hamiltonian can also define a physically acceptable quantum-mechanical system even if the Hamiltonian is not Hermitian. The study of PT-symmetric quantum systems is a young and extremely active research area in both theoretical and experimental physics. The purpose of this Review is to provide established scientists as well as graduate students with a compact, easy-to-read introduction to this field that will enable them to understand more advanced publications and to begin their own theoretical or experimental research activity. The ideas and techniques of PT symmetry have been applied in the context of many different branches of physics. This Review introduces the concepts of PT symmetry by focusing on elementary one-dimensional PT-symmetric quantum and classical mechanics and relies in particular on oscillator models to illustrate and explain the basic properties of PT-symmetric quantum theory.

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CPT-Frames for Non-Hermitian Hamiltonians

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Based on the PT-symmetric quantum theory, the concepts of PT-frame, PT-symmetric operator and CPT-frame on a Hilbert space K and for an operator on K are proposed. It is proved that the spectrum and point spectrum of a PT-symmetric linear operator are both symmetric with respect to the real axis and the eigenvalues of an unbroken PT-symmetric operator are real. For a linear operator H on C d , it is proved that H has unbroken PT-symmetry if and only if it has d different eigenvalues and the corresponding eigenstates are eigenstates of PT. Given a CPT-frame on K, a new positive inner product on K is induced and called CPT-inner product. Te relationship between the CPT-adjoint and the Dirac adjoint of a densely defined linear operator is derived, and it is proved that an operator which has a bounded CPT-frame is CPT-Hermitian if and only if it is T-symmetric, in that case, it is similar to a Hermitian operator. The existence of an operator C consisting of a CPT-frame is discussed. These concepts and results will serve a mathematical discussion about PT-symmetric quantum mechanics.

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Quantum Measurement of PT-Symmetric Quantum Theory

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量子计算机是一类遵循量子力学规律进行高速数学和逻辑运算、存储及处理量子信息的物理装置，量子测量是量子信息和量子计算中的一类重要研究课题，但是在PT-对称量子系统中还没有相关的研究。本文给出PT-对称量子系统中一般量子测量的概念，并运用矩阵和算子论的方法，对这个问题进行了详细的讨论，得到了PT-对称量子系统中关于量子测量的两个结果。 Quantum computer is a kind of physical device which carries out mathematical and logical calculations, and stores and manages the quantum information efficiently under the rule of the quantum mechanics. Quantum measurement is an important research topic in quantum information and quantum computing realm, however, the relative research has not appeared in the PT-symme- trical quantum system. In this paper, we give the general concept of quantum measurement in the PT-symmetrical quantum system. Moreover, the detailed discussion is described by using the methods of matrix and operation theory, and we obtain two results about quantum measurement in the PT-symmetrical quantum system.

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In this report we examine the concept of PT -symmetric quantum mechanics and analyze two such systems. PT -symmetric quantum mechanics is a new branch of quantum mechanics, first proposed by Bender and Boettcher in 1998 and has been the subject of much research since then. Various important aspects of PT -symmetric quantum mechanics is explained, such as asymptotic behaviour of solutions of the Schrödinger equation, as well as the concept of the C operator that is unique to PT -symmetric quantum mechanics. We briefly discuss both the concept of broken and unbroken PT -symmetry, and the relation between PT -symmetric and Hermitian quantum mechanics through similarity transforms. We specifically direct our attention to two theoretical quantum mechanical systems described by Hamiltonians with the property of being PT -symmetric. For these systems we determine the energy levels, one by analytical means and the other by use of numerical methods. To determine the energy levels numerically we have used the builtin ODE-solver ode45 in MATLAB in conjunction with the equation solver fzero. Our results are compared to those of previous studies and our methods and results are discussed.

•Jan 1, 2016•0 citations
It is generally assumed that a Hamiltonian for a physically acceptable quantum system (one that has a positive-definite spectrum and obeys the requirement of unitarity) must be Hermitian. However, a PT-symmetric Hamiltonian can also define a physically acceptable quantum-mechanical system even if the Hamiltonian is not Hermitian. The study of PT-symmetric quantum systems is a young and extremely active research area in both theoretical and experimental physics. The purpose of this Review is to provide established scientists as well as graduate students with a compact, easy-to-read introduction to this field that will enable them to understand more advanced publications and to begin their own theoretical or experimental research activity. The ideas and techniques of PT symmetry have been applied in the context of many different branches of physics. This Review introduces the concepts of PT symmetry by focusing on elementary one-dimensional PT-symmetric quantum and classical mechanics and relies in particular on oscillator models to illustrate and explain the basic properties of PT-symmetric quantum theory.
Based on the PT-symmetric quantum theory, the concepts of PT-frame, PT-symmetric operator and CPT-frame on a Hilbert space K and for an operator on K are proposed. It is proved that the spectrum and point spectrum of a PT-symmetric linear operator are both symmetric with respect to the real axis and the eigenvalues of an unbroken PT-symmetric operator are real. For a linear operator H on C d , it is proved that H has unbroken PT-symmetry if and only if it has d different eigenvalues and the corresponding eigenstates are eigenstates of PT. Given a CPT-frame on K, a new positive inner product on K is induced and called CPT-inner product. Te relationship between the CPT-adjoint and the Dirac adjoint of a densely defined linear operator is derived, and it is proved that an operator which has a bounded CPT-frame is CPT-Hermitian if and only if it is T-symmetric, in that case, it is similar to a Hermitian operator. The existence of an operator C consisting of a CPT-frame is discussed. These concepts and results will serve a mathematical discussion about PT-symmetric quantum mechanics.
A major mathematical problem in PT-symmetric quantum mechanics is to determine whether or not the spectrum of any given non self-adjoint but PT-symmetric Schroedinger operator is real. In this case the PT-symmetry is called "proper". Clearly in this connection an equally important issue is the spontaneous violation of the PT-symmetry which might occur in a PT-symmetric operator family. The spontaneous violation of the PT-symmetry is defined as the transition from real values of the spectrum to complex ones at the variation of the parameter labeling the family. Its occurrence is referred to also as the PTPTPT-symmetric phase transition. \nThis chapter intends to present a review of recent results concerning these two mathematical points, within the standard notions of spectral theory for Hilbert space operators. The main technical instrument is represented by perturbation theory.
量子计算机是一类遵循量子力学规律进行高速数学和逻辑运算、存储及处理量子信息的物理装置，量子测量是量子信息和量子计算中的一类重要研究课题，但是在PT-对称量子系统中还没有相关的研究。本文给出PT-对称量子系统中一般量子测量的概念，并运用矩阵和算子论的方法，对这个问题进行了详细的讨论，得到了PT-对称量子系统中关于量子测量的两个结果。 Quantum computer is a kind of physical device which carries out mathematical and logical calculations, and stores and manages the quantum information efficiently under the rule of the quantum mechanics. Quantum measurement is an important research topic in quantum information and quantum computing realm, however, the relative research has not appeared in the PT-symme- trical quantum system. In this paper, we give the general concept of quantum measurement in the PT-symmetrical quantum system. Moreover, the detailed discussion is described by using the methods of matrix and operation theory, and we obtain two results about quantum measurement in the PT-symmetrical quantum system.
This dissertation concerns the concept of PT symmetry as a replacement of the conventional symmetry of the Dirac Hermiticity of the operators, in particular, for the Hamiltonian. After solving a controversial issue surrounding the claimed violation of the special relativity by PT-symmetric quantum mechanics, we move further to discuss PT symmetry for the systems whose associated time reversal symmetry is odd, i.e., T^2 = −1. Our results, as published papers, are incorporated in the subsequent chapters. Then we conclude by suggestions for future work.
In this report we examine the concept of PT -symmetric quantum mechanics and analyze two such systems. PT -symmetric quantum mechanics is a new branch of quantum mechanics, first proposed by Bender and Boettcher in 1998 and has been the subject of much research since then. Various important aspects of PT -symmetric quantum mechanics is explained, such as asymptotic behaviour of solutions of the Schrödinger equation, as well as the concept of the C operator that is unique to PT -symmetric quantum mechanics. We briefly discuss both the concept of broken and unbroken PT -symmetry, and the relation between PT -symmetric and Hermitian quantum mechanics through similarity transforms. We specifically direct our attention to two theoretical quantum mechanical systems described by Hamiltonians with the property of being PT -symmetric. For these systems we determine the energy levels, one by analytical means and the other by use of numerical methods. To determine the energy levels numerically we have used the builtin ODE-solver ode45 in MATLAB in conjunction with the equation solver fzero. Our results are compared to those of previous studies and our methods and results are discussed.