Functional Integrals

E. S. Kolpakov

Abstract The paper is focused on the study of functional integrals defined on discontinuous trajectories. A special type of transformations was proposed for a class of such functional integrals in papers on quantum physics, which makes it possible to reduce integrals of this kind to integrals over continuous trajectories.

Zbigniew Haba

Dalcio Kisling Dacol

A functional integral algorithm for the partition function and for the temperature Green’s function generating functional of spin 1/2 systems is developed. Starting from the representation of the spin operators in terms of fermion creation and destruction operators we write the above mentioned objects as functional integrals over an infinite dimensional Grassmann algebra. We show how to eliminate the integration over the Grassmann algebra in favor of an integration over function space. The use of the formalism is illustrated with two simple examples.

Cécile DeWitt-Morette

Joseph I. Kapusta, Charles Gale

Functional integral representation of the partition functionThe customary approach to nonrelativistic many-body theory is to proceed with the method of second quantization begun in the first chapter.There is another approach, the method of functional integrals, which we shall follow here.Of course, what can be done in one formalism can always be done in another.Nevertheless, functional integrals seem to be the method of choice for most elementary particle theorists these days, and they seem to lend themselves more readily to nonperturbative phenomena such as tunneling, instantons, lattice gauge theory, etc.For gauge theories they are practically indispensable.However, there is a certain amount of formalism that must be developed before we can start to discuss physical applications.In this chapter, we shall derive the functional integral representation of the partition function for interacting relativistic non-gauge field theories.As a check on the formalism, as well as to obtain some feeling for how functional integrals work, we shall then rederive some well-known results on relativistic ideal gases for bosons and fermions.

Sergio Alheverio, Philippe Blanchard, Raphael J. Høegh-Krohn