1998/09/24 by Carl Bender, Stefan Boettcher, Peter Meisinger · 4 citations
Physics and Astronomy · #quant-ph #cond-mat #hep-th
paper · pdf · doi:10.1063/1.532860
published as J.Math.Phys. 40 (1999) 2201-2229 · 20 pages RevTex, 23 ps-figures
arxiv created 1998/09/24 · arxiv updated 2009/11/30
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 \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=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 \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=p2+x2N(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.