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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,

where tr stands for the functional trace: the determinant is then defined by

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:

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.

Contents

Defining formulaeEdit

Path integral versionEdit

For a positive selfadjoint 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 selfadjoint, 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 basisfunctions. 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 versionEdit

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 it just can 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.

Practical exampleEdit

 
The infinite potential well with A = 0.

The infinite potential wellEdit

We will compute the determinant of the following operator describing the motion of a quantum mechanical particle in an infinite potential well:

 

where A is the depth of the potential and L is the length of the well. We will compute this determinant by diagonalizing the operator and multiplying the eigenvalues. So as not to have to bother with the uninteresting divergent constant, we will compute the quotient between the determinants of the operator with depth A and the operator with depth A = 0. The eigenvalues of this potential are equal to

 

This means that

 

Now we can use Euler's infinite product representation for the sine function:

 

from which a similar formula for the hyperbolic sine function can be derived:

 

Applying this, we find that

 

Another way for computing the functional determinantEdit

For one-dimensional potentials, a short-cut yielding the functional determinant exists.[4] It is based on consideration of the following expression:

 

where m is a complex constant. This expression is a meromorphic function of m, having zeros when m equals an eigenvalue of the operator with potential V1(x) and a pole when m is an eigenvalue of the operator with potential V2(x). We now consider the functions ψm1 and ψm2 with

 

obeying the boundary conditions

 

If we construct the function

 

which is also a meromorphic function of m, we see that it has exactly the same poles and zeroes as the quotient of determinants we are trying to compute: if m is an eigenvalue of the operator number one, then ψm1(x) will be an eigenfunction thereof, meaning ψm1(L) = 0; and analogously for the denominator. By Liouville's theorem, two meromorphic functions with the same zeros and poles must be proportional to one another. In our case, the proportionality constant turns out to be one, and we get

 

for all values of m. For m = 0 we get

 

The infinite potential well revisitedEdit

The problem in the previous section can be solved more easily with this formalism. The functions ψ0i(x) obey

 

yielding the following solutions:

 

This gives the final expression

 

See alsoEdit

NotesEdit

  1. ^ (Branson 1993); (Osgood, Phillips & Sarnak 1988)
  2. ^ See Osgood, Phillips & Sarnak (1988). For a more general definition in terms of the spectral function, see Hörmander (1968) or Shubin (1987).
  3. ^ For the case of the generalized Laplacian, as well as regularity at zero, see Berline, Getzler & Vergne (2004, Proposition 9.35). For the general case of an elliptic pseudodifferential operator, see Seeley (1967).
  4. ^ S. Coleman, The uses of instantons, Int. School of Subnuclear Physics, (Erice, 1977)

ReferencesEdit

  • Berline, Nicole; Getzler, Ezra; Vergne, Michèle (2004), Heat Kernels and Dirac Operators, ISBN 978-3-540-20062-8
  • Branson, Thomas P. (2007), "Q-curvature, spectral invariants, and representation theory", SIGMA. Symmetry, Integrability and Geometry. Methods and Applications, 3: Paper 090, 31, arXiv:0709.2471, ISSN 1815-0659, MR 2366932
  • Branson, Thomas P. (1993), The functional determinant, Lecture Notes Series, 4, Seoul: Seoul National University Research Institute of Mathematics Global Analysis Research Center, MR 1325463
  • Hörmander, Lars (1968), "The spectral function of an elliptic operator", Acta Mathematica, 121: 193–218, doi:10.1007/BF02391913, ISSN 0001-5962, MR 0609014
  • Osgood, B.; Phillips, R.; Sarnak, Peter (1988), "Extremals of determinants of Laplacians", Journal of Functional Analysis, 80 (1): 148–211, doi:10.1016/0022-1236(88)90070-5, ISSN 0022-1236, MR 0960228
  • Ray, D. B.; Singer, I. M. (1971), "R-torsion and the Laplacian on Riemannian manifolds", Advances in Mathematics, 7 (2): 145–210, doi:10.1016/0001-8708(71)90045-4, MR 0295381
  • Seeley, R. T. (1967), "Complex powers of an elliptic operator", Singular Integrals (Proc. Sympos. Pure Math., Chicago, Ill., 1966), Providence, R.I.: American Mathematical Society, pp. 288–307, MR 0237943
  • Shubin, M. A. (1987), Pseudodifferential operators and spectral theory, Springer Series in Soviet Mathematics, Berlin, New York: Springer-Verlag, ISBN 978-3-540-13621-7, MR 0883081