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

Progress of Theoretical and Experimental Physics

Anti-$\mathcal{PT}$ symmetry for a non-Hermitian Hamiltonian

Mustapha Maamache, Linda Kheniche

Published Sep 15, 2020• 9 citations
Publisher

Abstract

Abstract Anti-$\mathcal{PT}$ symmetry, $(\mathcal{PT})H=-H(\mathcal{PT})$, is a plausible variant of $\mathcal{PT}$ symmetry. Of particular interest is the situation when all the eigenstates of an anti-$\mathcal{PT}$-symmetric non-Hermitian Hamiltonian $H$ are also eigenstates of the $\mathcal{PT}$ operator; then, the quasi-energies are purely imaginary, which implies that the Hermitian conjugate $H^{+}=-H$, and thus they are connected via the relation $(\mathcal{PT})H=H^{+}\mathcal{PT}$, similar to the quasi-Hermiticity relation. Therefore, the eigenfunctions of the anti-$\mathcal{PT}$-symmetric $H$ form a complete orthonormal set with positive definite norms, and moreover the time evolution is unitary.

DOI
10.1093/ptep/ptaa143

Related papers

Pseudo-Hermiticity versus PT-symmetry III: Equivalence of pseudo-Hermiticity and the presence of antilinear symmetries

Ali Mostafazadeh

We show that a diagonalizable (non-Hermitian) Hamiltonian H is pseudo-Hermitian if and only if it has an antilinear symmetry, i.e., a symmetry generated by an invertible antilinear operator. This implies that the eigenvalues of H are real or come in complex conjugate pairs if and only if H possesses such a symmetry. In particular, the reality of the spectrum of H implies the presence of an antilinear symmetry. We further show that the spectrum of H is real if and only if there is a positive-definite inner-product on the Hilbert space with respect to which H is Hermitian or alternatively there is a pseudo-canonical transformation of the Hilbert space that maps H into a Hermitian operator.

arXiv:math-ph/0203005•Journal of Mathematical Physics•Aug 1, 2002•780 citations

A Possible Method for Non-Hermitian and Non-PT-Symmetric Hamiltonian Systems

Junqing Li, Yan-Gang Miao, Xue Zhu Zhao

A possible method to investigate non-Hermitian Hamiltonians is suggested through finding a Hermitian operator η+ and defining the annihilation and creation operators to be η+ -pseudo-Hermitian adjoint to each other. The operator η+ represents the η+ -pseudo-Hermiticity of Hamiltonians. As an example, a non-Hermitian and non-PT-symmetric Hamiltonian with imaginary linear coordinate and linear momentum terms is constructed and analyzed in detail. The operator η+ is found, based on which, a real spectrum and a positive-definite inner product, together with the probability explanation of wave functions, the orthogonality of eigenstates, and the unitarity of time evolution, are obtained for the non-Hermitian and non-PT-symmetric Hamiltonian. Moreover, this Hamiltonian turns out to be coupled when it is extended to the canonical noncommutative space with noncommutative spatial coordinate operators and noncommutative momentum operators as well. Our method is applicable to the coupled Hamiltonian. Then the first and second order noncommutative corrections of energy levels are calculated, and in particular the reality of energy spectra, the positive-definiteness of inner products, and the related properties (the probability explanation of wave functions, the orthogonality of eigenstates, and the unitarity of time evolution) are found not to be altered by the noncommutativity.

arXiv:1107.4972•PLoS ONE•Jun 4, 2014•10 citations

Symmetries and invariants for non-Hermitian Hamiltonians

Martínez, M. A. Simón, Álvaro Buendía, Juan Gonzalo Muga

We discuss Hamiltonian symmetries and invariants for quantum systems driven by non-Hermitian Hamiltonians. For time-independent Hermitian Hamiltonians, a unitary or antiunitary transformation $AHA^\dagger$ that leaves the Hamiltonian $H$ unchanged represents a symmetry of the Hamiltonian, which implies the commutativity $[H,A]=0$, and a conservation law, namely the invariance of expectation values of $A$. For non-Hermitian Hamiltonians, $H^\dagger$ comes into play as a distinct operator that complements $H$ in generalized unitarity relations. The above description of symmetries has to be extended to include also $A$-pseudohermiticity relations of the form $AH=H^\dagger A$. A superoperator formulation of Hamiltonian symmetries is provided and exemplified for Hamiltonians of a particle moving in one-dimension considering the set of $A$ operators forming Klein's 4-group: parity, time-reversal, parity\&time-reversal, and unity. The link between symmetry and conservation laws is discussed and shown to be more subtle for non-Hermitian than for Hermitian Hamiltonians.

arXiv:1805.04968•May 13, 2018•2 citations

Quantum Mechanics Based on Non-Hermitian Hamiltonians (Recent Trends in Exponential Asymptotics)

Carl M. Bender

A physical theory of quantum mechanics can be based on a complex Hamiltonian that is not Hermitian but instead satisfies the physical condition of space-time reflection symmetry ( $P\mathcal{T}$ symmetry).Thus, there are infinitely many new Hamiltonians that one can construct that might explain experimental data.One would think that a non-Hermitian Hamiltonian would give a quantum theory that violates unitarity.However, if $P\mathcal{T}$ symmetry is not broken, it is possible to use a physical symmetry of the Hamiltonian to construct an inner product whose associated norm is positive definite.This construction is general and works for any $P\mathcal{T}$ -symmetric Hamiltonian.The dynamics is governed by unitary time evolution.This formulation does not conflict with the requirements of conventional quantum mechanics.There are many possible observable and experimental consequences of extending quantum mechanics into the complex domain, both in particle physics and in solid state physics.

•Apr 1, 2005•0 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

On the η pseudo P T symmetry theory for non-Hermitian Hamiltonians: time-dependent systems

Mustapha Maamache

Abstract In the context of non-Hermitian quantum mechanics, many systems are known to possess a pseudo P T symmetry, i.e. the non-Hermitian Hamiltonian H is related to its adjoint H † via the relation, H † = P T H P T . We propose a derivation of pseudo P T symmetry and η -pseudo-Hermiticity simultaneously for the time dependent non-Hermitian Hamiltonians by intoducing a new symmetry operator η ˜ ( t ) = P T η ( t ) that not satisfy the time-dependent quasi-Hermiticity relation but obeys the Heisenberg evolution equation. Here, we solve the SU(1, 1) time-dependent non-Hermitian Hamiltonian and we construct a time-dependent solutions by employing this new symmetry operator and discuss a concrete physical applications of our results. Dedicated to the memory of my mother Djabou Zoulikha and to Djabi Smail

Physica Scripta•Dec 19, 2023•0 citations
Anti-PT\mathcal{PT}PT symmetry for a non-Hermitian Hamiltonian
Abstract Anti-PT\mathcal{PT}PT symmetry, (PT)H=−H(PT)(\mathcal{PT})H=-H(\mathcal{PT})(PT)H=−H(PT), is a plausible variant of PT\mathcal{PT}PT symmetry. Of particular interest is the situation when all the eigenstates of an anti-PT\mathcal{PT}PT-symmetric non-Hermitian Hamiltonian HHH are also eigenstates of the PT\mathcal{PT}PT operator; then, the quasi-energies are purely imaginary, which implies that the Hermitian conjugate H+=−HH^{+}=-HH+=−H, and thus they are connected via the relation (PT)H=H+PT(\mathcal{PT})H=H^{+}\mathcal{PT}(PT)H=H+PT, similar to the quasi-Hermiticity relation. Therefore, the eigenfunctions of the anti-PT\mathcal{PT}PT-symmetric HHH form a complete orthonormal set with positive definite norms, and moreover the time evolution is unitary.
We show that a diagonalizable (non-Hermitian) Hamiltonian H is pseudo-Hermitian if and only if it has an antilinear symmetry, i.e., a symmetry generated by an invertible antilinear operator. This implies that the eigenvalues of H are real or come in complex conjugate pairs if and only if H possesses such a symmetry. In particular, the reality of the spectrum of H implies the presence of an antilinear symmetry. We further show that the spectrum of H is real if and only if there is a positive-definite inner-product on the Hilbert space with respect to which H is Hermitian or alternatively there is a pseudo-canonical transformation of the Hilbert space that maps H into a Hermitian operator.
A possible method to investigate non-Hermitian Hamiltonians is suggested through finding a Hermitian operator η+ and defining the annihilation and creation operators to be η+ -pseudo-Hermitian adjoint to each other. The operator η+ represents the η+ -pseudo-Hermiticity of Hamiltonians. As an example, a non-Hermitian and non-PT-symmetric Hamiltonian with imaginary linear coordinate and linear momentum terms is constructed and analyzed in detail. The operator η+ is found, based on which, a real spectrum and a positive-definite inner product, together with the probability explanation of wave functions, the orthogonality of eigenstates, and the unitarity of time evolution, are obtained for the non-Hermitian and non-PT-symmetric Hamiltonian. Moreover, this Hamiltonian turns out to be coupled when it is extended to the canonical noncommutative space with noncommutative spatial coordinate operators and noncommutative momentum operators as well. Our method is applicable to the coupled Hamiltonian. Then the first and second order noncommutative corrections of energy levels are calculated, and in particular the reality of energy spectra, the positive-definiteness of inner products, and the related properties (the probability explanation of wave functions, the orthogonality of eigenstates, and the unitarity of time evolution) are found not to be altered by the noncommutativity.
We discuss Hamiltonian symmetries and invariants for quantum systems driven by non-Hermitian Hamiltonians. For time-independent Hermitian Hamiltonians, a unitary or antiunitary transformation AHA†AHA^\daggerAHA† that leaves the Hamiltonian HHH unchanged represents a symmetry of the Hamiltonian, which implies the commutativity [H,A]=0[H,A]=0[H,A]=0, and a conservation law, namely the invariance of expectation values of AAA. For non-Hermitian Hamiltonians, H†H^\daggerH† comes into play as a distinct operator that complements HHH in generalized unitarity relations. The above description of symmetries has to be extended to include also AAA-pseudohermiticity relations of the form AH=H†AAH=H^\dagger AAH=H†A. A superoperator formulation of Hamiltonian symmetries is provided and exemplified for Hamiltonians of a particle moving in one-dimension considering the set of AAA operators forming Klein's 4-group: parity, time-reversal, parity\&time-reversal, and unity. The link between symmetry and conservation laws is discussed and shown to be more subtle for non-Hermitian than for Hermitian Hamiltonians.
A physical theory of quantum mechanics can be based on a complex Hamiltonian that is not Hermitian but instead satisfies the physical condition of space-time reflection symmetry ( PTP\mathcal{T}PT symmetry).Thus, there are infinitely many new Hamiltonians that one can construct that might explain experimental data.One would think that a non-Hermitian Hamiltonian would give a quantum theory that violates unitarity.However, if PTP\mathcal{T}PT symmetry is not broken, it is possible to use a physical symmetry of the Hamiltonian to construct an inner product whose associated norm is positive definite.This construction is general and works for any PTP\mathcal{T}PT -symmetric Hamiltonian.The dynamics is governed by unitary time evolution.This formulation does not conflict with the requirements of conventional quantum mechanics.There are many possible observable and experimental consequences of extending quantum mechanics into the complex domain, both in particle physics and in solid state physics.
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 In the context of non-Hermitian quantum mechanics, many systems are known to possess a pseudo P T symmetry, i.e. the non-Hermitian Hamiltonian H is related to its adjoint H † via the relation, H † = P T H P T . We propose a derivation of pseudo P T symmetry and η -pseudo-Hermiticity simultaneously for the time dependent non-Hermitian Hamiltonians by intoducing a new symmetry operator η ˜ ( t ) = P T η ( t ) that not satisfy the time-dependent quasi-Hermiticity relation but obeys the Heisenberg evolution equation. Here, we solve the SU(1, 1) time-dependent non-Hermitian Hamiltonian and we construct a time-dependent solutions by employing this new symmetry operator and discuss a concrete physical applications of our results. Dedicated to the memory of my mother Djabou Zoulikha and to Djabi Smail