PT-symmetric quantum mechanics: Physics off the real axis
The average physicist on the street would say that to have a real
energy spectrum and unitary time evolution a quantum Hamiltonian must be
Dirac Hermitian; that is, invariant under complex conjugation + matrix
transposition. However, the non-Dirac-Hermitian Hamiltonian $H=p^2+ix^3$
has a positive discrete spectrum and generates unitary time evolution, so
$H$ defines a consistent physical quantum theory. Thus, Hermiticity
symmetry is too restrictive. While $H$ is not Dirac Hermitian, it is PT
symmetric
(spacetime-reflection symmetric); that is, invariant under parity P + time
reversal T. The quantum mechanics defined by a PT-symmetric Hamiltonian is a
complex generalization of ordinary quantum mechanics. If quantum mechanics
is extended to the complex domain, new theories having remarkable properties
emerge. For example, the Hamiltonian $H=p^2-x^4$, which has an upside-down
potential, defines two distinct phases, an unstable P-symmetric phase having
complex eigenvalues and a stable PT-symmetric phase whose energy levels are
positive and discrete. The properties of PT-symmetric classical and quantum
systems are under intense study by theorists and experimentalists; many
theoretical predictions have been verified in laboratory experiments.