Some Remarks on Hamiltonian Systems and Quantum Mechanics

Carl M. Bender

PCT , a fundamental symmetry of quantum field theory, is derived from the assumption of Lorentz invariance and positivity of the spectrum.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 quantum mechanical and quantum field theoretic systems whose Hamiltonians are non-Hermitian but obey PCT symmetry.These systems have weird and remarkable properties.Examples of such Hamiltonians are H = p 2 + ix 3 and H = p 2x 4 .Hamiltonians such as these are complex deformations of conventional Hermitian Hamiltonians.Thus, in this talk I study the analytic continuation of conventional quantum mechanics into the complex plane.

P A M Dirac

Dariusz Chruściński, Giuseppe Marmo

It is shown how to introduce a geometric formulation of the algebraic approach to the standard non-relativistic quantum mechanics. It turns out that the GNS representation provides not only symplectic but even Hermitian realization of a 'quantum Poisson algebra'. We discuss alternative Hamiltonian structures emerging out of different GNS representations which provide a natural setting for quantum bi-Hamiltonian systems.

D. E. Bilhorn, Leslie L. Foldy, Roy M. Thaler +2 more

A new derivation of the reciprocity theorem is given. The general invariance property of the Hamiltonian which leads to symmetry of the Green's function for a quantum mechanical system is exhibited. It is found that reciprocity does not necessarily imply Hermiticity of the Hamiltonian, so that the ``complex optical model potential,'' for example, satisfies the reciprocity relations. The concept of reciprocity is then generalized to include a somewhat wider class of symmetry properties. Some properties of antiunitary transformations are discussed.

Nicholas Chako