Functional determinant

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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 \rangle}

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 \rangle and the vector S|\phi \rangle, which is the result of applying the operator S to the vector |\phi \rangle.

Motivation for the terminology comes from the following. We shall consider the case when S possesses only discrete spectrum {λi} with a corresponding complete set of eigenvectors {fi} and 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 Ω). 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/(2\pi) (which would be an infinite product in infinite dimensional spaces such as L2[Ω]). Hence the inner product in the exponential is written as

\langle\phi|S|\phi\rangle=\sum_{i,j}\langle \phi|f_i\rangle\langle f_i|S|f_j\rangle\langle f_j|\phi\rangle= \sum_i\phi_i^2 \lambda_i

where φi are the components of the vector φ in the spectral basis and the integral becomes Gaussian

\int \mathcal{D} \phi e^{-\langle \phi | S|\phi \rangle}=\int \prod_i\frac{ df_i}{2\pi} e^{-\sum_i \phi_i^2\lambda_i}

which evaluates to

\int \mathcal{D}\phi e^{-\langle\phi|S|\phi\rangle}\propto\left( \prod_i \lambda_i\right)^{-1/2}.

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




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