Functional determinant

In mathematics, if "S" is a linear operator mapping a function space V to itself, it is possible to define an infinite-dimensional generalization of the determinant in some cases.

The corresponding quantity det("S") is called the functional determinant of "S". There is a "functional integral definition" of it and the functional Pfaffian Pf:

:$frac\{1\}\{sqrt\{mbox\{det\}(S)=frac\{1\}\{mbox\{Pf\}(S)\}propto\; int\_\{V\}\; !\; mathcal\{D\}\; phi\; ;\; e^\{-langle\; phi\; |\; S|phi\; angle\}$

in analogy with the finite dimensional case.

The argument of the exponential inside the integral is written in Dirac notation and its meaning is the scalar product between the (ket) vector $|phi\; angle$ and the vector $S|phi\; angle$, which is the result of applying the operator $S$ to the vector $|phi\; angle$.

Motivation for the terminology comes from the following. We shall consider the case when "S" possesses only discrete spectrum $\{lambda\_i\}$ with a corresponding complete set of eigenvectors $\{f\_i\}$ and let further the spectrum consist of a discrete set of points (as would be the case for the second derivavtive operator on a compact interval $Omega$). The functional measure $mathcal\{D\}phi$ is equivalent to the spectral measure associated with "S" which in this case is simply the scaled product Lebesgue measure $mathcal\{D\}phi=prod\_i\; df\_i/(2pi)$ (which would be an infinite product in infinite dimensional spaces such as $L^\{2\}\; [Omega]$). Hence the inner product in the exponential is written as

:$langlephi|S|phi\; angle=sum\_\{i,j\}langle\; phi|f\_i\; anglelangle\; f\_i|S|f\_j\; anglelangle\; f\_j|phi\; angle=\; sum\_iphi\_i^2lambda\_i$

where $phi\_i$ are the components of the vector $phi$ in the spectral basis and the integral becomes Gaussian

:$int\; mathcal\{D\}\; phi\; e^\{-langle\; phi\; |\; S|phi\; angle\}=int\; prod\_ifrac\{\; df\_i\}\{2pi\}\; e^\{-sum\_i\; phi\_i^2lambda\_i\}$

which evaluates to

:$int\; mathcal\{D\}phi\; e^\{-langlephi|S|phi\; angle\}proptoleft(\; prod\_i\; lambda\_i\; ight)^\{-1/2\}.$

Hence as in finite dimensions, the generalized determinant may be interpreted as the product of the eigenvalues.