# High-Energy Physics & RIKEN Theory Seminar

## "Making sense of non-Hermitian Hamiltonians"

#### Presented by Carl Bender, Washington University

Wednesday, December 16, 2009, 2:00 pm — Small Seminar Room, Bldg. 510

ABSTRACT:

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.

Hosted by: Sally Dawson

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