Simple Applications of Generalized Functions in Theoretical Physics: The Case of Many–Body Perturbation Expansions

Let Ĥ = Ĥ₀ + $${\rm{\hat V}}$$ be a self-adjoint operator, bounded from below and defined on a Hilbert space, representing the Himiltonian of an interacting physical system, and Ĥ₀ the one for a simpler system with known spectrum and eigenstates. Typically, physicists want to evaluate the (grand–) canonical partition function Tr exp(–βĤ), where β-1 > 0 is Boltzmann’s constant times temperature, and Tr stands for the trace, in powers of $${\rm{\hat V}}$$. For a fixed power of $${\rm{\hat V}}$$, the expansion is unique and consists of a sum of terms, interpreted as physical processes. An individual term can be calculated only if generalized functions are introduced. This is a somewhat arbitrary procedure, however. Different schemes are presented an partial summations of individual terms through all the orders of the expansion in $${\rm{\hat V}}$$ are discussed.