PT-symmetric quantum mechanics

This paper proposes to broaden the canonical formulation of quantum mechani cs. Ordinarily, one imposes the condition H dagger = H on the Hamiltonian, where dagger represents the mathematical operation of complex conjugation a nd matrix transposition. This conventional Hermiticity condition is suffici ent to ensure that the Hamiltonian H has a real spectrum. However, replacin g this mathematical condition by the weaker and more physical requirement H double dagger = H, where double dagger represents combined parity reflecti on and time reversal PT, one obtains new classes of complex Hamiltonians wh ose spectra are still real and positive. This generalization of Hermiticity is investigated using a complex deformation H = p(2) + x(2)(ix)(epsilon) o f the harmonic oscillator Hamiltonian, where e is a real parameter. The sys tem exhibits two phases: When epsilon greater than or equal to 0, the energ y spectrum of H is real and positive as a consequence of PT symmetry. Howev er, when -1 < epsilon < 0, the spectrum contains an infinite number of comp lex eigenvalues and a finite number of real, positive eigenvalues because P T symmetry is spontaneously broken. The phase transition that occurs at eps ilon = 0 manifests itself in both the quantum-mechanical system and the und erlying classical system. Similar qualitative features are exhibited by com plex deformations of other standard real Hamiltonians H = p(2) + x(2)(ix)(e psilon) with N integer and epsilon > -N; each of these complex Hamiltonians exhibits a phase transition at epsilon = 0. These PT symmetric theories ma y be viewed as analytic continuations of conventional theories from real to complex phase space. (C) 1999 American Institute of Physics. [S0022-2488(9 9)00105-X].