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''zeta-function regularization''

Regularization and renormalization procedures are essential issues in contemporary physics — without which it would simply not exist, at least in the form known today (2000). They are also essential in supersymmetry calculations. Among the different methods, zeta-function regularization — which is obtained by [[Analytic continuation|analytic continuation]] in the complex plane of the zeta-function of the relevant physical operator in each case — might well be the most beautiful of all. Use of this method yields, for instance, the vacuum energy corresponding to a quantum physical system (with constraints of any kind, in principle). Assuming the corresponding Hamiltonian operator, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130090/z1300901.png" />, has a spectral decomposition of the form (think, as simplest case, of a quantum harmonic oscillator): <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130090/z1300902.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130090/z1300903.png" /> some set of indices (which can be discrete, continuous, mixed, multiple, etc.), then the quantum vacuum energy is obtained as follows [[#References|[a5]]], [[#References|[a6]]]:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130090/z1300904.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130090/z1300905.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130090/z1300906.png" /> is the zeta-function corresponding to the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130090/z1300907.png" />. The formal sum over the eigenvalues is usually ill-defined, and the last step involves analytic continuation, inherent to the definition of the zeta-function itself. These mathematically simple-looking relations involve very deep physical concepts (no wonder that understanding them took several decades in the recent history of [[Quantum field theory|quantum field theory]], QFT). The zeta-function method is unchallenged at the one-loop level, where it is rigorously defined and where many calculations of QFT reduce basically (from a mathematical point of view) to the computation of determinants of elliptic pseudo-differential operators (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130090/z1300908.png" />DOs, cf. also [[Pseudo-differential operator|Pseudo-differential operator]]) [[#References|[a2]]]. It is thus no surprise that the preferred definition of determinant for such operators is obtained through the corresponding zeta-function.

When one comes to specific calculations, the zeta-function regularization method relies on the existence of simple formulas for obtaining the analytic continuation above. These consist of the reflection formula of the corresponding zeta-function in each case, together with some other fundamental expressions, as the Jacobi theta-function identity, Poisson's resummation formula and the famous Chowla–Selberg formula [[#References|[a2]]]. However, some of these formulas are restricted to very specific zeta-functions, and it often turned out that for some physically important cases the corresponding formulas did not exist in the literature. This has required a painful process (it has taken over a decade already) of generalization of previous results and derivation of new expressions of this kind [[#References|[a5]]], [[#References|[a6]]]. [[#References|[a1]]].

'' zeta regularization for integrals ''

Elizalde and others have extended the Zeta regularization algorithm to include divergent integrals \begin{equation} \int_{a}^{\infty}x^{m}dx \end{equation} by using the recurrence equation

\begin{equation} \begin{array}{l}
\int\nolimits_{a}^{\infty }x^{m-s} dx =\frac{m-s}{2} \int\nolimits_{a}^{\infty }x^{m-1-s} dx +\zeta (s-m)-\sum\limits_{i=1}^{a}i^{m-s}  +a^{m-s}  \\
-\sum\limits_{r=1}^{\infty }\frac{B_{2r} \Gamma (m-s+1)}{(2r)!\Gamma (m-2r+2-s)}  (m-2r+1-s)\int\nolimits_{a}^{\infty }x^{m-2r-s} dx \end{array} \end{equation}

this is the natural extension to integrals of the Zeta regularization algorithm , this recurrence equation is finite since for \begin{equation} m-2r < -1 \end{equation} the integrals inside the recurrence equation are convergents and can be inmediatly evaluated

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.A. Bytsenko,   G. Cognola,   L. Vanzo,   S. Zerbini,   "Quantum fields and extended objects in space-times with constant curvature spatial section"  ''Phys. Rept.'' , '''266'''  (1996)  pp. 1–126</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  E. Elizalde,   "Multidimensional extension of the generalized Chowla–Selberg formula"  ''Commun. Math. Phys.'' , '''198'''  (1998)  pp. 83–95</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  S.W. Hawking,   "Zeta function regularization of path integrals in curved space time"  ''Commun. Math. Phys.'' , '''55'''  (1977)  pp. 133–148</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  M. Nakahara,   "Geometry, topology, and physics" , Inst. Phys.  (1995)  pp. 7–8</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  E. Elizalde,   S.D. Odintsov,   A. Romeo,   A.A. Bytsenko,   S. Zerbini,   "Zeta regularization techniques with applications" , World Sci.  (1994)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  E. Elizalde,   "Ten physical applications of spectral zeta functions" , Springer  (1995)</TD></TR></table>