Physics Department,

Campus Box 1105,

Washington University,

St. Louis, MO 63130 USA

E-mail: cmb@howdy.wustl.edu

**Complex extension of quantum mechanics**

**Abstract:**

This talk examines Hamiltonians *H* that are not Hermitian but
do exhibit space-time reflection (*PT*) symmetry. If the (*PT*)
symmetry of *H* is not spontaneously broken, then the spectrum of
*H* is entirely real and positive. Examples of *PT*-symmetric
non-Hermitian Hamiltonians are *H *= *p*^{2} + *ix*^{3}
and *H* = *p*^{2} - *x*^{4}. The apparent
shortcoming of quantum theories arising from *PT*-symmetric Hamiltonians
is that the *PT* norm is not positive definite. This suggests that
it may be difficult to develop a quantum theory based on such Hamiltonians.
In this talk it is shown that these difficulties can be overcome by introducing
a previously unnoticed underlying physical symmetry *C* of Hamiltonians
having an unbroken *PT* symmetry. Using *C*, it is shown how
to construct an inner product whose associated norm *is* positive
definite. The result is a new class of fully consistent complex quantum
theories. Observables in these theories exhibit *CPT* symmetry, probabilities
are positive, and the dynamics is governed by unitary time evolution.