Functional Determinant

In mathematics, if S is a linear operator mapping a function space V to itself, it is sometimes possible to define an infinite-dimensional generalization of the determinant. The corresponding quantity det(S) is called the functional determinant of S.

There are several formulas for the functional determinant. They are all based on the fact that, for diagonalizable finite-dimensional matrices, the determinant is equal to the product of the eigenvalues. A mathematically rigorous definition is via the zeta function of the operator,

where tr stands for the functional trace: the determinant is then defined by

where the zeta function in the point s = 0 is defined by analytic continuation. Another possible generalization, often used by physicists when using the Feynman path integral formalism in quantum field theory, uses a functional integration:

This path integral is only well defined up to some divergent multiplicative constant. In order to give it a rigorous meaning, it must be divided by another functional determinant, making the spurious constants cancel.

These are now, ostensibly, two different definitions for the functional determinant, one coming from quantum field theory and one coming from spectral theory. Each involves some kind of regularization: in the definition popular in physics, two determinants can only be compared with one another; in mathematics, the zeta function was used. Osgood, Phillips & Sarnak (1988) have shown that the results obtained by comparing two functional determinants in the QFT formalism agree with the results obtained by the zeta functional determinant.