Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:quant-ph/0310106 · Journal of Physics A Mathematical and General

Pseudo-Hermitian Hamiltonians, indefinite inner product spaces and their symmetries

A Blasi, G Scolarici, L Solombrino

Published Mar 29, 2004• 18 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/0305-4470/37/15/003

Related papers

Pseudo-Hermitian Hamiltonians, indefinite inner product spaces and their symmetries

Alberto Blasi, G. Scolarici, Luigi Solombrino

We extend the definition of generalized parity $P$, charge-conjugation $C$ and time-reversal $T$ 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 $TP$ and $CTP$ are the generators of the 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.

•Oct 16, 2003•15 citations

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

On the pseudo-Hermitian nondiagonalizable Hamiltonians

G. Scolarici, Luigi Solombrino

We consider a class of (possibly nondiagonalizable) pseudo-Hermitian operators with discrete spectrum, and we establish for such a class the equivalence between the pseudo-Hermiticity property and the existence of an antilinear involutory symmetry. Moreover, we prove that this class actually coincides with the one of (possibly nondiagonalizable) weak pseudo-Hermitian operators, and that in no case (unless they are diagonalizable and have a real spectrum) they are Hermitian with respect to a definite inner product. Finally, we show that a typical degeneracy of the real eigenvalues (which reduces to the well-known Kramers degeneracy in the Hermitian case) occurs whenever a fermionic (possibly nondiagonalizable) pseudo-Hermitian Hamiltonian admits an antilinear symmetry like the time-reversal operator T. Some consequences and applications are briefly discussed.

arXiv:quant-ph/0211161•Journal of Mathematical Physics•Sep 25, 2003•44 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

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

Pseudo-Hermiticity for a class of nondiagonalizable Hamiltonians

Ali Mostafazadeh

We give two characterization theorems for pseudo-Hermitian (possibly nondiagonalizable) Hamiltonians with a discrete spectrum that admit a block-diagonalization with finite-dimensional diagonal blocks. In particular, we prove that for such an operator H the following statements are equivalent: (1) H is pseudo-Hermitian; (2) the spectrum of H consists of real and/or complex-conjugate pairs of eigenvalues and the geometric multiplicity and the dimension of the diagonal blocks for the complex-conjugate eigenvalues are identical; (3) H is Hermitian with respect to a positive-semidefinite inner product. We further discuss the relevance of our findings for the merging of a complex-conjugate pair of eigenvalues of diagonalizable pseudo-Hermitian Hamiltonians in general, and the PT-symmetric Hamiltonians and the effective Hamiltonian for a certain closed FRW minisuperspace quantum cosmological model in particular.

arXiv:math-ph/0207009•Journal of Mathematical Physics•Nov 25, 2002•102 citations
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.
We consider a class of (possibly nondiagonalizable) pseudo-Hermitian operators with discrete spectrum, and we establish for such a class the equivalence between the pseudo-Hermiticity property and the existence of an antilinear involutory symmetry. Moreover, we prove that this class actually coincides with the one of (possibly nondiagonalizable) weak pseudo-Hermitian operators, and that in no case (unless they are diagonalizable and have a real spectrum) they are Hermitian with respect to a definite inner product. Finally, we show that a typical degeneracy of the real eigenvalues (which reduces to the well-known Kramers degeneracy in the Hermitian case) occurs whenever a fermionic (possibly nondiagonalizable) pseudo-Hermitian Hamiltonian admits an antilinear symmetry like the time-reversal operator T. Some consequences and applications are briefly discussed.
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 give two characterization theorems for pseudo-Hermitian (possibly nondiagonalizable) Hamiltonians with a discrete spectrum that admit a block-diagonalization with finite-dimensional diagonal blocks. In particular, we prove that for such an operator H the following statements are equivalent: (1) H is pseudo-Hermitian; (2) the spectrum of H consists of real and/or complex-conjugate pairs of eigenvalues and the geometric multiplicity and the dimension of the diagonal blocks for the complex-conjugate eigenvalues are identical; (3) H is Hermitian with respect to a positive-semidefinite inner product. We further discuss the relevance of our findings for the merging of a complex-conjugate pair of eigenvalues of diagonalizable pseudo-Hermitian Hamiltonians in general, and the PT-symmetric Hamiltonians and the effective Hamiltonian for a certain closed FRW minisuperspace quantum cosmological model in particular.