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 $H=p2+x2(ix)ε$ 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 $−1<ε<0,$ 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 $ε>−N;$ 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.