APPLIED MATH SEMINAR

Title: Extending Quantum Mechanics into the Comnplex Domain


Speaker: Qinghai Want, Washington University



When/where: Tuesday, March 8th, 4:15PM, AKW 200


Abstract:
In this talk I will show that one can replace the axiom in quantum
mechanics that the Hamiltonian must be Hermitian by the alternative
requirement that the Hamiltonian be PT-symmetric (that is, invariant
under combined space and time reflection). I will begin by reviewing
the background and motivation and I will illustrate by using an
explicit 2x2 matric Hamiltonian. I will show how to calculate the
spectral zeta function exactly for a class of PT-symmetric
Hamiltonians, and I will verify the result numerically by using a 10th-
order WKB calculation. Using matched asyumptotic expansions, I will
show how to calculate the 1-point correlation function for a -x4
theory. I will then demonstrate a fourth-order perturbative 
calculation of the C operator that is needed to define the inner
product in PT-symmetric quantum mechanics. I will show that this C
operator is a Lorentz scalar. (By contrast, the parity operator is not
a Lorentz scalar, and instead it transforms as a direct product of 
tensorial representations. This direct sum is deeply related to the
wilson polynomials.) Finally, I will apply the notions of PT-symmetric
quantum mechanics to a well known quantum-field-theoretical model called
the Lee model. I will give a simple physical interpretation for the
ghost states, which have been known and misinterpreted for the last 50
years.