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Zbl 1078.58019
Asada, Akira
Regularized calculus: an application of zeta regularization to infinite dimensional geometry and analysis. (English)
[J] Int. J. Geom. Methods Mod. Phys. 1, No. 1-2, 107-157 (2004). ISSN 0219-8878

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 $L^2(M)$ where $M$ is a compact manifold. Then observe that for the radial coordinate variable $r = \Vert x\Vert = (x_1^2 + x_2^2 + \cdots )^{1/2}$, $$ \Delta\, r^2 = \sum_{n = 1}^\infty \frac{\partial^2}{\partial x_n^2} r^2 = 2 \sum_{n = 1}^\infty 1 $$ 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 $\{\mu_n\}$ denote the eigenvalues of $P$, and consider $$ \Delta(s) = \sum_{n = 1}^\infty \mu_n^{- 2s} \frac{\partial^2}{\partial x_n^2}. $$ For a function $f$ we define the regularized Laplacian of $f$ by the formula $$ : \Delta : f = \Delta(s) f\vert _{s = 0} $$ provided, of course, that the right-hand side exists. Now consider our function $r^2$ again. In this case, $$ \Delta(s) r^2 = \sum_{n = 1}^\infty \mu_n^{- 2s} \frac{\partial^2}{\partial x_n^2} r^2 = 2 \sum_{n = 1}^\infty \mu_n^{2s} = 2 \zeta(P,s), $$ where $\zeta(P,s)$ is the zeta function of $P$ defined by $\zeta(P,s) = \sum_{n = 1}^\infty \mu_n^{-2s}$. Therefore, $$ : \Delta : r^2 = 2 \nu, $$ where $\nu = \zeta(P,0)$ represents a generalized dimension of the Hilbert space.\par 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.
[Paul Loya (Binghamton)]

MSC 2000:: *58J52 Determinants and determinant bundles, analytic torsion
81T16 Nonperturbative methods of renormalization
47G10 Integral operators
47G30 Pseudodifferential operators

Keywords: $\zeta$-regularization

Cited in: Zbl 1187.58031

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