Summary: Physics 633 Final Exam
Please show all significant steps clearly in all problems.
1. In class we considered two applications of Bogoliubov transformations: in the BCS
theory of superconductivity and in treating weakly-interacting excitations of a superfluid. A
third application is particle creation in a rapidly changing gravitational field, either in the
expansion of the early universe (considered by Stephen Fulling of our own Texas A&M Math
Department) or near a black hole (considered by Stephen Hawking of Cambridge University).
References are Aspects of Quantum Field Theory in Curved Space-Time, by S. A. Fulling,
and Quantum Fields in Curved Space, by N. D. Birrell and P. C. W. Davies. The treatment
of these problems involves nontrivial applications of Green's functions etc., but the simple
ideas below are relevant.
(a) (6) Suppose that we have a simple Bogoliubov transformation for spin 1/2 fermions,
of the form
ak = ukak + vka
k = u
k + v