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What is PT symmetry

Miloslav Znojil

Published Mar 12, 2001• 7 citations
OpenAlex

Abstract

The recently proposed complexification of Hamiltonians which keeps the spectra real (and is usually called PT symmetry) is re-interpreted here as a certain natural linear-algebraic alternative to Hermiticity. The juxtaposition is mediated by the Feshbachian projection on a model space which reduces the difference just to a sign of a propagator in the effective Hamiltonian. The problem of norms and orthogonality is clarified and a few perturbation aspects are mentioned. A remarkable simplicity of the secular polynomial is conjectured. The mechanism of the PT symmetry breaking is clarified via a separable approximation scheme.

Related papers

A generalization of the concept of PT symmetry

Miloslav Znojil

We show that the PT symmetric Hamiltonians (and their generalizations defined in the text) may be all assigned the projected (so called Feshbach or effective) nonlinear Hamiltonians which are "locally" Hermitian. This implies that many (if not all) of the bound-state energies may be real in a broad domain of Hermiticity-violating interactions. A complexification of a Winternitz's super-integrable example in two dimensions is conjectured as an illustration.

arXiv:math-ph/0106021•Jun 21, 2001•1 citation

PT-Symmetric Quantum Mechanics

This paper proposes to broaden the canonical formulation of quantum mechanics.Ordinarily, one imposes the condition H † = H on the Hamiltonian, where † represents the mathematical operation of complex conjugation and matrix transposition.This conventional Hermiticity condition is sufficient to ensure that the Hamiltonian H has a real spectrum.However, replacing this mathematical condition by the weaker and more physical requirement H ‡ = H, where ‡ represents combined parity reflection and time reversal PT , one obtains new classes of complex Hamiltonians whose spectra are still real and positive.This generalization of Hermiticity is investigated using a complex deformation H = p 2 + x 2 (ix) ǫ of the harmonic oscillator Hamiltonian, where ǫ is a real parameter.The system exhibits two phases: When ǫ ≥ 0, the energy spectrum of H is real and positive as a consequence of PT symmetry.However, when -1 < ǫ < 0, the spectrum contains an infinite number of complex eigenvalues and a finite number of real, positive eigenvalues because PT symmetry is spontaneously broken.The phase transition that occurs at ǫ = 0 manifests itself in both the quantum-mechanical system and the underlying classical system.Similar qualitative features are exhibited by complex deformations of other standard real Hamiltonians H = p 2 + x 2N (ix) ǫ with N integer and ǫ > -N ; each of these complex Hamiltonians exhibits a phase transition at ǫ = 0.These PT -symmetric theories may be viewed as analytic continuations of conventional theories from real to complex phase space.

WORLD SCIENTIFIC (EUROPE) eBooks•Nov 21, 2018•1 citation

PT-symmetry in conventional quantum physics

Zafar Ahmed

Investigations during the last few years show that complex PT-symmetric or pseudo-Hermitian Hamiltonians possess a real discrete spectrum and several other features akin to a Hermitian Hamiltonian. These developments not only show us new directions but also demand a revisit to the conventional physics wherein PT-symmetry could have been invoked. After a review of some very interesting results, we present the super-barrier reflectionlessness and the cranking model of the nucleus. These two instances are being viewed as the meeting ground of the two: PT-symmetry and Hermiticity. The former presents an agreement and the latter is presently a dichotomy.

Journal of Physics A Mathematical and General•Jul 26, 2006•17 citations

Should PT symmetric quantum mechanics be interpreted as nonlinear?

Miloslav Znojil

PT symmetric quantum mechanics admits non-Hermitian Hamiltonians with real spectra but, as long as the evolution is pseudo-unitary, the probabilistic interpretation of wave functions is lost. We offer an effective Hermitization of the theory using the Feshbach-type elimination of ghosts. The energy-dependence of the resulting Hermitian effective Hamiltonians Heff is a nonlinearity which, in a strong-coupling regime, explains the spontaneous PT symmetry breaking and the simultaneous loss of the Hermiticity of Heff while the pseudo-unitary evolution law persists. PACS 03.65.Bz, 03.65.Ca 03.65.Fd The concept of extended, non-Hermitian quantum mechanics [1] with real spectra and PT symmetric Hamiltonians (P means parity and T denotes time reversal) grew from several sources. The oldest root of its appeal is the Rayleigh-Schrödinger perturbation theory. Within its framework, Caliceti et al [2] have discovered that a

•Jan 1, 2001•20 citations

Generalized PT symmetry and real spectra

Carl M. Bender, Michael Berry, Aikaterini Mandilara

The fact that eigenvalues of PT -symmetric Hamiltonians H can be real for some values of a parameter and complex for others is explained by showing that the matrix elements of H , and hence the secular equation, are real, not only for PT but also for any antiunitary operator A satisfying A 2 k = 1 with k odd. The argument is illustrated by a 2 × 2 matrix Hamiltonian, and two examples of the generalization are given.

Journal of Physics A Mathematical and General•Jul 25, 2002•255 citations

𝒫𝒯-symmetrically regularized Eckart, Pöschl-Teller and Hulthén potentials

Miloslav Znojil

𝒫𝒯 ( = parity times time-reversal) symmetry of complex Hamiltonians with real spectra is usually interpreted as a weaker mathematical substitute for Hermiticity. Perhaps an equally important role is played by the related strengthened analyticity assumptions. In a constructive illustration we complexify a few potentials solvable only in s -wave. Then we continue their domain from the semi-axis to the whole axis and obtain new exactly solvable models. Their energies turn out to be real as expected. The new one-dimensional spectra themselves differ quite significantly from their s -wave predecessors.

arXiv:quant-ph/9912027•Journal of Physics A Mathematical and General•Jun 9, 2000•58 citations
The recently proposed complexification of Hamiltonians which keeps the spectra real (and is usually called PT symmetry) is re-interpreted here as a certain natural linear-algebraic alternative to Hermiticity. The juxtaposition is mediated by the Feshbachian projection on a model space which reduces the difference just to a sign of a propagator in the effective Hamiltonian. The problem of norms and orthogonality is clarified and a few perturbation aspects are mentioned. A remarkable simplicity of the secular polynomial is conjectured. The mechanism of the PT symmetry breaking is clarified via a separable approximation scheme.
This paper proposes to broaden the canonical formulation of quantum mechanics.Ordinarily, one imposes the condition H † = H on the Hamiltonian, where † represents the mathematical operation of complex conjugation and matrix transposition.This conventional Hermiticity condition is sufficient to ensure that the Hamiltonian H has a real spectrum.However, replacing this mathematical condition by the weaker and more physical requirement H ‡ = H, where ‡ represents combined parity reflection and time reversal PT , one obtains new classes of complex Hamiltonians whose spectra are still real and positive.This generalization of Hermiticity is investigated using a complex deformation H = p 2 + x 2 (ix) ǫ of the harmonic oscillator Hamiltonian, where ǫ is a real parameter.The system exhibits two phases: When ǫ ≥ 0, the energy spectrum of H is real and positive as a consequence of PT symmetry.However, when -1 < ǫ < 0, the spectrum contains an infinite number of complex eigenvalues and a finite number of real, positive eigenvalues because PT symmetry is spontaneously broken.The phase transition that occurs at ǫ = 0 manifests itself in both the quantum-mechanical system and the underlying classical system.Similar qualitative features are exhibited by complex deformations of other standard real Hamiltonians H = p 2 + x 2N (ix) ǫ with N integer and ǫ > -N ; each of these complex Hamiltonians exhibits a phase transition at ǫ = 0.These PT -symmetric theories may be viewed as analytic continuations of conventional theories from real to complex phase space.
Investigations during the last few years show that complex PT-symmetric or pseudo-Hermitian Hamiltonians possess a real discrete spectrum and several other features akin to a Hermitian Hamiltonian. These developments not only show us new directions but also demand a revisit to the conventional physics wherein PT-symmetry could have been invoked. After a review of some very interesting results, we present the super-barrier reflectionlessness and the cranking model of the nucleus. These two instances are being viewed as the meeting ground of the two: PT-symmetry and Hermiticity. The former presents an agreement and the latter is presently a dichotomy.
PT symmetric quantum mechanics admits non-Hermitian Hamiltonians with real spectra but, as long as the evolution is pseudo-unitary, the probabilistic interpretation of wave functions is lost. We offer an effective Hermitization of the theory using the Feshbach-type elimination of ghosts. The energy-dependence of the resulting Hermitian effective Hamiltonians Heff is a nonlinearity which, in a strong-coupling regime, explains the spontaneous PT symmetry breaking and the simultaneous loss of the Hermiticity of Heff while the pseudo-unitary evolution law persists. PACS 03.65.Bz, 03.65.Ca 03.65.Fd The concept of extended, non-Hermitian quantum mechanics [1] with real spectra and PT symmetric Hamiltonians (P means parity and T denotes time reversal) grew from several sources. The oldest root of its appeal is the Rayleigh-Schrödinger perturbation theory. Within its framework, Caliceti et al [2] have discovered that a
𝒫𝒯 ( = parity times time-reversal) symmetry of complex Hamiltonians with real spectra is usually interpreted as a weaker mathematical substitute for Hermiticity. Perhaps an equally important role is played by the related strengthened analyticity assumptions. In a constructive illustration we complexify a few potentials solvable only in s -wave. Then we continue their domain from the semi-axis to the whole axis and obtain new exactly solvable models. Their energies turn out to be real as expected. The new one-dimensional spectra themselves differ quite significantly from their s -wave predecessors.