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 Quantum Mechanics 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 $|\phi \rangle$ .