Dr. Carl Bender

Physics Department

Washington University in St. Louis

**Monday,
February 1, 2010**

**4:30 p.m.**

**Cardwell 102**

Making Sense of non-Hermitian Hamiltonians

The average quantum physicist on the street believes that a
quantum-mechanical Hamiltonian must be Dirac Hermitian (invariant under combined
matrix transposition and complex conjugation) in order to guarantee that the
energy eigenvalues are real and that time evolution is unitary. However, the
Hamiltonian H = p^{2} + ix^{3}, which is obviously not Dirac
Hermitian, has a real positive discrete spectrum and generates unitary time
evolution, and thus it defines a fully consistent and physical quantum theory.

Evidently, the axiom of Dirac Hermiticity is too restrictive.
While H = p^{2} + ix^{3} is not Dirac Hermitian, it is PT
symmetric; that is, invariant under combined space reflection P and time
reversal T. The quantum mechanics defined by a PT-symmetric Hamiltonian is a
complex generalization of ordinary quantum mechanics. When quantum mechanics is
extended into the complex domain, new kinds of theories having strange and
remarkable properties emerge. Some of these properties have recently been
verified in laboratory experiments. If one generalizes classical mechanics into
the complex domain, the resulting theories have equally remarkable properties.

This talk will be presented at an elementary colloquium-style level and will be easy to understand and broadly accessible.