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. 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 the determinant of a finite matrix is equal to the product of the eigenvalues of the matrix. A mathematically rigorous definition is via the zeta function of the operator,
ζ S ( a ) = tr S − a , {\displaystyle \zeta _{S}(a)=\operatorname {tr} \,S^{-a}\,,}
where tr stands for the functional trace: the determinant is then defined by
det S = e − ζ S ′ ( 0 ) , {\displaystyle \det S=e^{-\zeta _{S}'(0)}\,,}
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 (QFT), uses a functional integration:
det S ∝ ( ∫ V D ϕ e − ⟨ ϕ , S ϕ ⟩ ) − 2 . {\displaystyle \det S\propto \left(\int _{V}{\mathcal {D}}\phi \;e^{-\langle \phi ,S\phi \rangle }\right)^{-2}\,.}
This path integral is only well defined up to some divergent multiplicative constant. To give it a rigorous meaning it must be divided by another functional determinant, thus effectively cancelling the problematic 'constants'. 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.
Defining formulae
Path integral version For a positive self-adjoint operator S on a finite-dimensional Euclidean space V, the formula
1 det S = ∫ V e − π ⟨ x , S x ⟩ d x {\displaystyle {\frac {1}{\sqrt {\det S}}}=\int _{V}e^{-\pi \langle x,Sx\rangle }\,dx}
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
∫ V e − π ⟨ ϕ , S ϕ ⟩ D ϕ {\displaystyle \int _{V}e^{-\pi \langle \phi ,S\phi \rangle }\,{\mathcal {D}}\phi }
where V is the function space and ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot \rangle } the L2 inner product, and D ϕ {\displaystyle {\mathcal {D}}\phi } the Wiener measure. 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 roughly means all functions φ can be written as linear combinations of the functions fi:
| ϕ ⟩ = ∑ i c i | f i ⟩ with c i = ⟨ f i | ϕ ⟩ . {\displaystyle |\phi \rangle =\sum _{i}c_{i}|f_{i}\rangle \quad {\text{with }}c_{i}=\langle f_{i}|\phi \rangle .}
Hence the inner product in the exponential can be written as
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