Path integral version
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 L2 inner product, and 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:
Hence the inner product in the exponential can be written as
In the basis of the functions fi, the functional integration reduces to an integration over all basis functions. Formally, assuming our intuition from the finite dimensional case carries over into the infinite dimensional setting, the measure should then be equal to
This makes the functional integral a product of Gaussian integrals:
The integrals can then be evaluated, giving
where N is an infinite constant that needs to be dealt with by some regularization procedure. The product of all eigenvalues is equal to the determinant for finite-dimensional spaces, and we formally define this to be the case in our infinite-dimensional case also. 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). Otherwise, it is necessary to perform some kind of regularization. The most popular of which for computing functional determinants is the zeta function regularization.[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. Otherwise, it is also possible to consider the quotient of two determinants, making the divergent constants cancel.
Zeta function version
Let S be an elliptic differential operator with smooth coefficients which is positive on functions of compact support. That is, there exists a constant c > 0 such that
for all compactly supported smooth functions φ. Then S has a self-adjoint extension to an operator on L2 with lower bound c. The eigenvalues of S can be arranged in a sequence
Then the zeta function of S is defined by the series:[2]
It is known that ζS has a meromorphic extension to the entire plane.[3] Moreover, although one can define the zeta function in more general situations, the zeta function of an elliptic differential operator (or pseudodifferential operator) is regular at .
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 . Integration over a gives which can just be considered as the logarithm of the determinant for a Harmonic oscillator. This last value is just equal to , where is the Hurwitz zeta function.