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P, T, PT, and CPT invariance of Hermitian Hamiltonians

Zafar Ahmed

Published Jan 1, 2003• 19 citations
OpenAlex

Abstract

Currently, it has been claimed that certain Hermitian Hamiltonians have parity (P) and they are PT-invariant. We propose generalized definitions of time-reversal operator (T) and orthonormality such that all Hermitian Hamiltonians are P, T, PT, and CPT invariant. The PT-norm and CPT-norm are indefinite and definite respectively. The energy-eigenstates are either E-type (e.g., even) or O-type (e.g., odd). C mimics the charge-conjugation symmetry which is recently found to exist for a non-Hermitian Hamiltonian. For a Hermitian Hamiltonian it coincides with P. The Hamiltonians which are invariant under the joint transformation of Parity (x → −x) and Time-reversal (i → −i) are called PT-invariant. It has been conjectured [1] that such Hamiltonians possess real discrete energy-eigenvalues provided the PT symmetry is unbroken. PT-symmetry is called unbroken or exact if the energy-eigenstates are also simultaneous eigenstates of PT. On the other hand when PT-symmetry is spontaneously broken the energy-eigenvalues are complex conjugate pairs. Multipronged investigations supporting

Related papers

C-,PT- andCPT-invariance of pseudo-Hermitian Hamiltonians

Zafar Ahmed

We propose a construction of unique and definite metric, η + , time-reversal operator, T , and an inner-product of eigenstates such that the pseudo-Hermitian matrix Hamiltonians are C -, PT - and CPT -invariant and PT -norm ( CPT -norm) is indefinite (definite). Here, P and C denote the generalized parity and charge-conjugation symmetries, respectively. The limitations of the other current approaches are also indicated.

arXiv:quant-ph/0302141•Journal of Physics A Mathematical and General•Sep 2, 2003•27 citations

Pseudo-Hermiticity and generalized PT- and CPT-symmetries

Ali Mostafazadeh

We study certain linear and antilinear symmetry generators and involution operators associated with pseudo-Hermitian Hamiltonians and show that the theory of pseudo-Hermitian operators provides a simple explanation for the recent results of Bender, Brody and Jones (quant-ph/0208076) on the CPT-symmetry of a class of PT-symmetric non-Hermitian Hamiltonians. We present a natural extension of these results to the class of diagonalizable pseudo-Hermitian Hamiltonians H with a discrete spectrum. In particular, we introduce generalized parity (𝒫), time-reversal (𝒯), and charge-conjugation (𝒞) operators and establish the 𝒫𝒯- and 𝒞𝒫𝒯-invariance of H.

arXiv:math-ph/0209018•Journal of Mathematical Physics•Feb 28, 2003•230 citations

Geometry of PT-symmetric quantum mechanics

Carl M. Bender, Dorje C. Brody, Lane P. Hughston +1 more

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.

arXiv:0704.2959•Apr 23, 2007•2 citations

Antilinearity rather than Hermiticity as a guiding principle for quantum theory

Philip D. Mannheim

Abstract Currently there is much interest in Hamiltonians that are not Hermitian but instead possess an antilinear PT symmetry. Here we seek to put such PT symmetric theories into as general a context as possible. After providing a brief overview of the PT symmetry program, we show that having an antilinear symmetry that acts on a well-defined Hilbert space is the most general condition that one can impose on a quantum theory for which one can have a well-defined inner product that is time independent, have a Hamiltonian that is self-adjoint, and have energy eigenvalues that are all real. For each of these properties Hermiticity is only a sufficient condition but not a necessary one, with Hermiticity thus being the special case in which the Hamiltonian has both antilinearity and Hermiticity. As well as being the necessary condition for the reality of energy eigenvalues, antilinearity in addition allows for the physically interesting cases of manifestly non-Hermitian but nonetheless self-adjoint Hamiltonians that have energy eigenvalues that appear in complex conjugate pairs, or that are Jordan block and cannot be diagonalized at all. We show that one can extend these ideas to quantum field theory, with the dual requirements of the existence of time independent inner products and invariance under complex Lorentz transformations forcing the antilinear symmetry to uniquely be CPT . We thus extend the CPT theorem to non-Hermitian Hamiltonians. For theories that are separately charge conjugation invariant, PT symmetry then follows, with the case for the physical relevance of the PT -symmetry program thus being advanced. While CPT symmetry can be defined at the classical level for every classical path in a path integral quantization procedure, in contrast, in such a path integral there is no reference at all to the Hermiticity of the Hamiltonian or the quantum Hilbert space on which it acts, as they are strictly quantum-mechanical concepts that can only be defined after the path integral quantization has been performed and the quantum Hilbert space has been constructed. CPT symmetry thus goes beyond Hermiticity and has primacy over it, with our work raising the question of how Hermiticity ever comes into quantum theory at all. To this end we show that whether or not a CPT -invariant theory has a Hamiltonian that is Hermitian is a property of the solutions to the theory and not of the Hamiltonian itself. Hermiticity thus never needs to be postulated at all.

arXiv:1512.04915•Journal of Physics A Mathematical and Theoretical•Apr 26, 2018•43 citations

CPT-Frames for Non-Hermitian Hamiltonians

Cao Huai-xin, Zhihua Guo, Zhengli Chen

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.

arXiv:1212.3944•Communications in Theoretical Physics•Sep 1, 2013•11 citations

Generalized PT-, C-, and CPT-Symmetries, Position Operators, and Localized States of Klein-Gordon Fields

Mostafazadeh, A

We construct a concrete realization of the generalized parity (P), time-reversal (T), and charge-conjugation (C) operators, that were initially defined in the study of the PT-symmetric and pseudo-Hermitian Hamiltonians, for Klein-Gordon fields. We show that PT and C operators, that signify certain symmetries of the system, correspond to the ordinary time-reversal and charge-conjugation transformations, respectively. Furthermore, we construct a positive-definite and relativistically invariant inner product on the space of Klein-Gordon fields and show that the generator h of time-translations viewed as acting in the Hilbert space ${cal H}$ defined by this inner product is Hermitian. The quantum system having ${cal H}$ as its Hilbert space and h as its Hamiltonian is unitarily equivalent to the one defined by a Hermitian Hamiltonian acting in the Hilbert space $L^2(R^3)oplus L^2(R^3)$. We give the explicit form of the corresponding unitary transformation and use it to construct position operators, localized and coherent states, and a set of wave functions for the Klein-Gordon fields. Our approach provides a consistent quantum mechanical description of the Klein-Gordon fields that admits a genuine probabilistic interpretation.

•Jul 8, 2003•2 citations
Currently, it has been claimed that certain Hermitian Hamiltonians have parity (P) and they are PT-invariant. We propose generalized definitions of time-reversal operator (T) and orthonormality such that all Hermitian Hamiltonians are P, T, PT, and CPT invariant. The PT-norm and CPT-norm are indefinite and definite respectively. The energy-eigenstates are either E-type (e.g., even) or O-type (e.g., odd). C mimics the charge-conjugation symmetry which is recently found to exist for a non-Hermitian Hamiltonian. For a Hermitian Hamiltonian it coincides with P. The Hamiltonians which are invariant under the joint transformation of Parity (x → −x) and Time-reversal (i → −i) are called PT-invariant. It has been conjectured [1] that such Hamiltonians possess real discrete energy-eigenvalues provided the PT symmetry is unbroken. PT-symmetry is called unbroken or exact if the energy-eigenstates are also simultaneous eigenstates of PT. On the other hand when PT-symmetry is spontaneously broken the energy-eigenvalues are complex conjugate pairs. Multipronged investigations supporting
We study certain linear and antilinear symmetry generators and involution operators associated with pseudo-Hermitian Hamiltonians and show that the theory of pseudo-Hermitian operators provides a simple explanation for the recent results of Bender, Brody and Jones (quant-ph/0208076) on the CPT-symmetry of a class of PT-symmetric non-Hermitian Hamiltonians. We present a natural extension of these results to the class of diagonalizable pseudo-Hermitian Hamiltonians H with a discrete spectrum. In particular, we introduce generalized parity (𝒫), time-reversal (𝒯), and charge-conjugation (𝒞) operators and establish the 𝒫𝒯- and 𝒞𝒫𝒯-invariance of H.
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.
Abstract Currently there is much interest in Hamiltonians that are not Hermitian but instead possess an antilinear PT symmetry. Here we seek to put such PT symmetric theories into as general a context as possible. After providing a brief overview of the PT symmetry program, we show that having an antilinear symmetry that acts on a well-defined Hilbert space is the most general condition that one can impose on a quantum theory for which one can have a well-defined inner product that is time independent, have a Hamiltonian that is self-adjoint, and have energy eigenvalues that are all real. For each of these properties Hermiticity is only a sufficient condition but not a necessary one, with Hermiticity thus being the special case in which the Hamiltonian has both antilinearity and Hermiticity. As well as being the necessary condition for the reality of energy eigenvalues, antilinearity in addition allows for the physically interesting cases of manifestly non-Hermitian but nonetheless self-adjoint Hamiltonians that have energy eigenvalues that appear in complex conjugate pairs, or that are Jordan block and cannot be diagonalized at all. We show that one can extend these ideas to quantum field theory, with the dual requirements of the existence of time independent inner products and invariance under complex Lorentz transformations forcing the antilinear symmetry to uniquely be CPT . We thus extend the CPT theorem to non-Hermitian Hamiltonians. For theories that are separately charge conjugation invariant, PT symmetry then follows, with the case for the physical relevance of the PT -symmetry program thus being advanced. While CPT symmetry can be defined at the classical level for every classical path in a path integral quantization procedure, in contrast, in such a path integral there is no reference at all to the Hermiticity of the Hamiltonian or the quantum Hilbert space on which it acts, as they are strictly quantum-mechanical concepts that can only be defined after the path integral quantization has been performed and the quantum Hilbert space has been constructed. CPT symmetry thus goes beyond Hermiticity and has primacy over it, with our work raising the question of how Hermiticity ever comes into quantum theory at all. To this end we show that whether or not a CPT -invariant theory has a Hamiltonian that is Hermitian is a property of the solutions to the theory and not of the Hamiltonian itself. Hermiticity thus never needs to be postulated at all.
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.
We construct a concrete realization of the generalized parity (P), time-reversal (T), and charge-conjugation (C) operators, that were initially defined in the study of the PT-symmetric and pseudo-Hermitian Hamiltonians, for Klein-Gordon fields. We show that PT and C operators, that signify certain symmetries of the system, correspond to the ordinary time-reversal and charge-conjugation transformations, respectively. Furthermore, we construct a positive-definite and relativistically invariant inner product on the space of Klein-Gordon fields and show that the generator h of time-translations viewed as acting in the Hilbert space calH{cal H}calH defined by this inner product is Hermitian. The quantum system having calH{cal H}calH as its Hilbert space and h as its Hamiltonian is unitarily equivalent to the one defined by a Hermitian Hamiltonian acting in the Hilbert space L2(R3)oplusL2(R3)L^2(R^3)oplus L^2(R^3)L2(R3)oplusL2(R3). We give the explicit form of the corresponding unitary transformation and use it to construct position operators, localized and coherent states, and a set of wave functions for the Klein-Gordon fields. Our approach provides a consistent quantum mechanical description of the Klein-Gordon fields that admits a genuine probabilistic interpretation.