PT-Symmetric Quantum Mechanics

Carl Bender; Stefan Boettcher; Peter Meisinger

doi:10.1063/1.532860

paper

PT-Symmetric Quantum Mechanics

arXiv:quant-ph/9809072 · doi:10.1063/1.532860

Abstract

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 $P T$ , one obtains new classes of complex Hamiltonians whose spectra are still real and positive. This generalization of Hermiticity is investigated using a complex deformation $H = p^{2} + x^{2} (i x)^{ε}$ of the harmonic oscillator Hamiltonian, where $ε$ is a real parameter. The system exhibits two phases: When $ε \geq 0$ , the energy spectrum of $H$ is real and positive as a consequence of $P T$ symmetry. However, when $- 1 < ε < 0$ , the spectrum contains an infinite number of complex eigenvalues and a finite number of real, positive eigenvalues because $P T$ symmetry is spontaneously broken. The phase transition that occurs at $ε = 0$ 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 $H = p^{2} + x^{2 N} (i x)^{ε}$ with $N$ integer and $ε > - N$ ; each of these complex Hamiltonians exhibits a phase transition at $ε = 0$ . These $P T$ -symmetric theories may be viewed as analytic continuations of conventional theories from real to complex phase space.

20 pages RevTex, 23 ps-figures

PT-Symmetric Quantum Mechanics

Abstract

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