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].