PCT, a fundamental symmetry of particle physics, is derived assuming Lorentz invariance and positivity of the spectrum of the Hamiltonian. What happens if we assume only Lorentz invariance and PCT symmetry? Hamiltonians having this property need not be Hermitian but, except when PCT is spontaneously broken, the energy levels of such Hamiltonians are all real and positive!

In this talk I examine some elementary quantum systems whose Hamiltonians
are non-Hermitian but PCT-symmetric. These systems have weird and
remarkable properties, and the classical theories underlying these
quantum systems also have strange and interesting behaviors. Examples
of such Hamiltonians are
*H*=*p*^{2}+*ix*^{3} and
*H*=*p*^{2}-*x*^{4}. Hamiltonians
such as these may be regarded as *complex deformations*
of conventional Hermitian Hamiltonians. Thus, in this talk I will be
studying the analytic continuation of conventional classical mechanics
and quantum mechanics into the complex plane.

I will also examine the corresponding non-Hermitian quantum field
theories. Field theories whose self-interaction terms are
*ig*φ^{3} and -*g*φ^{4} have strange
and interesting properties and may have possible experimental
applications in such areas as band theory and in Higgs particle physics.

George T. Fleming ( gfleming@mps.ohio-state.edu ), last updated 23 October 2000.