𝓟𝓣-symmetric quantum mechanics

TLDR

Conventional quantum mechanics requires Hermitian Hamiltonians (H†=H) to guarantee real spectra, while PT‑symmetric theories extend this by analytically continuing to complex phase space. The paper aims to broaden the canonical formulation of quantum mechanics. The authors replace Hermiticity with PT symmetry (H‡=H) and study complex deformations of the harmonic oscillator, H=p²+x²(i x)^ε, to explore the resulting spectra. For ε≥0 the PT‑symmetric Hamiltonian has a real, positive spectrum, whereas for −1<ε<0 it develops infinitely many complex eigenvalues and a finite set of real ones, with a phase transition at ε=0 that also appears in the corresponding classical system and in analogous deformations of other real Hamiltonians.

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 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=p2+x2(ix)ε of the harmonic oscillator Hamiltonian, where ε is a real parameter. The system exhibits two phases: When ε⩾0, the energy spectrum of H is real and positive as a consequence of 𝒫𝒯 symmetry. However, when −1&amp;lt;ε&amp;lt;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 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=p2+x2N(ix)ε with N integer and ε&amp;gt;−N; each of these complex Hamiltonians exhibits a phase transition at ε=0. These 𝒫𝒯-symmetric theories may be viewed as analytic continuations of conventional theories from real to complex phase space.