dorsal/arxiv
View SchemaPT-Symmetric Quantum Mechanics
| Authors | Carl Bender, Stefan Boettcher, Peter Meisinger |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/9809072 |
| URL | https://arxiv.org/abs/quant-ph/9809072 |
| DOI | 10.1063/1.532860 |
| Journal | J.Math.Phys. 40 (1999) 2201-2229 |
Abstract
This paper proposes to broaden the canonical formulation of quantum mechanics. Ordinarily, one imposes the condition $H^\dagger=H$ on the Hamiltonian, where $\dagger$ 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 $H^\ddag=H$, where $\ddag$ represents combined parity reflection and time reversal ${\cal PT}$, one obtains new classes of complex Hamiltonians whose spectra are still real and positive. This generalization of Hermiticity is investigated using a complex deformation $H=p^2+x^2(ix)^\epsilon$ of the harmonic oscillator Hamiltonian, where $\epsilon$ is a real parameter. The system exhibits two phases: When $\epsilon\geq0$, the energy spectrum of $H$ is real and positive as a consequence of ${\cal PT}$ symmetry. However, when $-1<\epsilon<0$, the spectrum contains an infinite number of complex eigenvalues and a finite number of real, positive eigenvalues because ${\cal PT}$ symmetry is spontaneously broken. The phase transition that occurs at $\epsilon=0$ 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 $H=p^2+x^{2N}(ix)^\epsilon$ with $N$ integer and $\epsilon>-N$; each of these complex Hamiltonians exhibits a phase transition at $\epsilon=0$. These ${\cal PT}$-symmetric theories may be viewed as analytic continuations of conventional theories from real to complex phase space.
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"abstract": "This paper proposes to broaden the canonical formulation of quantum\nmechanics. Ordinarily, one imposes the condition $H^\\dagger=H$ on the\nHamiltonian, where $\\dagger$ represents the mathematical operation of complex\nconjugation and matrix transposition. This conventional Hermiticity condition\nis sufficient to ensure that the Hamiltonian $H$ has a real spectrum. However,\nreplacing this mathematical condition by the weaker and more physical\nrequirement $H^\\ddag=H$, where $\\ddag$ represents combined parity reflection\nand time reversal ${\\cal PT}$, one obtains new classes of complex Hamiltonians\nwhose spectra are still real and positive. This generalization of Hermiticity\nis investigated using a complex deformation $H=p^2+x^2(ix)^\\epsilon$ of the\nharmonic oscillator Hamiltonian, where $\\epsilon$ is a real parameter. The\nsystem exhibits two phases: When $\\epsilon\\geq0$, the energy spectrum of $H$ is\nreal and positive as a consequence of ${\\cal PT}$ symmetry. However, when\n$-1\u003c\\epsilon\u003c0$, the spectrum contains an infinite number of complex\neigenvalues and a finite number of real, positive eigenvalues because ${\\cal\nPT}$ symmetry is spontaneously broken. The phase transition that occurs at\n$\\epsilon=0$ manifests itself in both the quantum-mechanical system and the\nunderlying classical system. Similar qualitative features are exhibited by\ncomplex deformations of other standard real Hamiltonians\n$H=p^2+x^{2N}(ix)^\\epsilon$ with $N$ integer and $\\epsilon\u003e-N$; each of these\ncomplex Hamiltonians exhibits a phase transition at $\\epsilon=0$. These ${\\cal\nPT}$-symmetric theories may be viewed as analytic continuations of conventional\ntheories from real to complex phase space.",
"arxiv_id": "quant-ph/9809072",
"authors": [
"Carl Bender",
"Stefan Boettcher",
"Peter Meisinger"
],
"categories": [
"quant-ph",
"cond-mat",
"hep-th"
],
"doi": "10.1063/1.532860",
"journal_ref": "J.Math.Phys. 40 (1999) 2201-2229",
"title": "PT-Symmetric Quantum Mechanics",
"url": "https://arxiv.org/abs/quant-ph/9809072"
},
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