PT-Symmetric Quantum Mechanics

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 $H^‡=H$, where $‡$ 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 $H=p^2+x^2(ix)^ε$ of the harmonic oscillator Hamiltonian, where $ε$ is a real parameter. The system exhibits two phases: When $ε\geq0$, the energy spectrum of $H$ is real and positive as a consequence of ${\cal PT}$ symmetry. However, when $-1<ε<0$, 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 $ε=0$ 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 $H=p^2+x^{2N}(ix)^ε$ with $N$ integer and $ε>-N$; each of these complex Hamiltonians exhibits a phase transition at $ε=0$. These ${\cal PT}$-symmetric theories may be viewed as analytic continuations of conventional theories from real to complex phase space.