Functional determinant

In functional analysis, a branch of mathematics, it is sometimes possible to generalize the notion of the determinant of a square matrix of finite order (representing a linear transformation from a finite-dimensional vector space to itself) to the infinite-dimensional case of a linear operator S mapping a function space V to itself.

To give it a rigorous meaning it must be divided by another functional determinant, thus effectively cancelling the problematic 'constants'.

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.

For a positive self-adjoint operator S on a finite-dimensional Euclidean space V, the formula holds.

The problem is to find a way to make sense of the determinant of an operator S on an infinite dimensional function space.

One approach, favored in quantum field theory, in which the function space consists of continuous paths on a closed interval, is to formally attempt to calculate the integral where V is the function space and

The basic assumption on S is that it should be self-adjoint, and have discrete spectrum λ1, λ2, λ3, ... with a corresponding set of eigenfunctions f1, f2, f3, ... which are complete in L2 (as would, for example, be the case for the second derivative operator on a compact interval Ω).

This results in the formula If all quantities converge in an appropriate sense, then the functional determinant can be described as a classical limit (Watson and Whittaker).

[1] For instance, this allows for the computation of the determinant of the Laplace and Dirac operators on a Riemannian manifold, using the Minakshisundaram–Pleijel zeta function.

Let S be an elliptic differential operator with smooth coefficients which is positive on functions of compact support.

Formally, differentiating this series term-by-term gives and so if the functional determinant is well-defined, then it should be given by Since the analytic continuation of the zeta function is regular at zero, this can be rigorously adopted as a definition of the determinant.

This kind of Zeta-regularized functional determinant also appears when evaluating sums of the form

In our case, the proportionality constant turns out to be one, and we get for all values of m. For m = 0 we get The problem in the previous section can be solved more easily with this formalism.