Mathematics Colloquium
Carl M. Bender
Washington University in Saint Louis
Title: PT-symmetric quantum mechanics: Physics off the real axis
Date: Friday, March 21, 2025
Place and Time: Love 101, 3:05-3:55 pm
Abstract. The average physicist on the street believes 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 (space-time-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.