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Regularized calculus: an application of zeta regularization to infinite dimensional geometry and analysis. (English) Zbl 1078.58019

The article under review is a survey article and is a rich source of information about zeta function regularization and its applications to infinite dimensional geometry. To give an example about the topics discussed in the paper under review, consider an infinite-dimensional Laplacian

${\Delta }=\sum _{n=1}^{\infty }\frac{{\partial }^{2}}{\partial {x}_{n}^{2}}$

on some Hilbert space, say ${L}^{2}\left(M\right)$ where $M$ is a compact manifold. Then observe that for the radial coordinate variable $r=\parallel x\parallel ={\left({x}_{1}^{2}+{x}_{2}^{2}+\cdots \right)}^{1/2}$,

${\Delta }\phantom{\rule{0.166667em}{0ex}}{r}^{2}=\sum _{n=1}^{\infty }\frac{{\partial }^{2}}{\partial {x}_{n}^{2}}{r}^{2}=2\sum _{n=1}^{\infty }1$

diverges. The function ${r}^{2}$ is such a useful function in geometry (for example, in polar coordinates) so we would like this sum to converge. To overcome this divergence as well as divergences of other infinite-dimensional geometric quantities, the author uses a [systematic] zeta-regularized calculus. The neat idea is as follows. Take an elliptic (pseudo) differential operator $P$ on $M$ and let $\left\{{\mu }_{n}\right\}$ denote the eigenvalues of $P$, and consider

${\Delta }\left(s\right)=\sum _{n=1}^{\infty }{\mu }_{n}^{-2s}\frac{{\partial }^{2}}{\partial {x}_{n}^{2}}·$

For a function $f$ we define the regularized Laplacian of $f$ by the formula

${:{\Delta }:f={\Delta }\left(s\right)f|}_{s=0}$

provided, of course, that the right-hand side exists. Now consider our function ${r}^{2}$ again. In this case,

${\Delta }\left(s\right){r}^{2}=\sum _{n=1}^{\infty }{\mu }_{n}^{-2s}\frac{{\partial }^{2}}{\partial {x}_{n}^{2}}{r}^{2}=2\sum _{n=1}^{\infty }{\mu }_{n}^{2s}=2\zeta \left(P,s\right),$

where $\zeta \left(P,s\right)$ is the zeta function of $P$ defined by $\zeta \left(P,s\right)={\sum }_{n=1}^{\infty }{\mu }_{n}^{-2s}$. Therefore,

$:{\Delta }:{r}^{2}=2\nu ,$

where $\nu =\zeta \left(P,0\right)$ represents a generalized dimension of the Hilbert space.

By a systematic use of zeta function regularization, the paper defines infinite-dimensional regularized volume forms, integrals, fractional derivatives, logarithmic derivatives, exterior derivatives, and many other quantities. An important application is that the author gives a mathematical justification of the appearance of the Ray-Singer determinant in the evaluation of a Gaussian path integral.

MSC:
 58J52 Determinants and determinant bundles, analytic torsion 81T16 Nonperturbative methods of renormalization (quantum theory) 47G10 Integral operators 47G30 Pseudodifferential operators