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arXiv:quant-ph/0302141 · Journal of Physics A Mathematical and General

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

Zafar Ahmed

Published Sep 2, 2003• 27 citations
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Abstract

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.

DOI
10.1088/0305-4470/36/37/309

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arXiv:quant-ph/0307059•International Journal of Modern Physics A•May 10, 2006•39 citations
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.
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.
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.
We extend the definition of generalized parity P , charge-conjugation C and time-reversal T operators to nondiagonalizable pseudo-Hermitian Hamiltonians, and use these generalized operators to describe the full set of symmetries of a pseudo-Hermitian Hamiltonian according to a fourfold classification. In particular, we show that T P and CT P are the generators of the P -antiunitary symmetries; moreover, a necessary and sufficient condition is provided for a pseudo-Hermitian Hamiltonian H to admit a P -reflecting symmetry which generates the P -pseudounitary and the P -pseudoantiunitary symmetries. Finally, a physical example is considered and some hints on the P -unitary evolution of a physical system are also given.
We extend the definition of generalized parity PPP, charge-conjugation CCC and time-reversal TTT operators to nondiagonalizable pseudo-Hermitian Hamiltonians, and we use these generalized operators to describe the full set of symmetries of a pseudo-Hermitian Hamiltonian according to a fourfold classification. In particular we show that TPTPTP and CTPCTPCTP are the generators of the antiunitary symmetries; moreover, a necessary and sufficient condition is provided for a pseudo-Hermitian Hamiltonian HHH to admit a PPP-reflecting symmetry which generates the PPP-pseudounitary and the PPP-pseudoantiunitary symmetries. Finally, a physical example is considered and some hints on the PPP-unitary evolution of a physical system are also given.
Generalized parity [Formula: see text], time-reversal [Formula: see text], and charge-conjugation [Formula: see text] operators were initially defined in the study of the pseudo-Hermitian Hamiltonians. We construct a concrete realization of these operators for Klein–Gordon fields and show that in this realization [Formula: see text] and [Formula: see text] operators respectively correspond to the ordinary time-reversal and charge-grading operations. Furthermore, we present a complete description of the quantum mechanics of Klein–Gordon fields that is based on the construction of a Hilbert space with a relativistically invariant, positive-definite, and conserved inner product. In particular we offer a natural construction of a position operator and the corresponding localized and coherent states. The restriction of this position operator to the positive-frequency fields coincides with the Newton–Wigner operator. Our approach does not rely on the conventional restriction to positive-frequency fields. Yet it provides a consistent quantum mechanical description of Klein–Gordon fields with a genuine probabilistic interpretation.