Zeta-function method for regularization

2020 Mathematics Subject Classification: Primary: 11M Secondary: 58G [MSN][ZBL]

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 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, $H$, has a spectral decomposition of the form (think, as simplest case, of a quantum harmonic oscillator): $\{\def\l{\lambda}\l_i,\phi_i\}_{i\in I}$, with $I$ some set of indices (which can be discrete, continuous, mixed, multiple, etc.), then the quantum vacuum energy is obtained as follows [ElOdRoByZe], [El2]:

$\def\phi{\varphi}$ $$\sum_{i\in I}(\phi_i,H\phi_i) = {\rm tr}\; H = \sum_{i\in I}\l_i = \sum_{i\in I}\l_i^{-s}\Big|_{s=-1} = \zeta_H(-1), $$

where $\zeta_H$ is the zeta-function corresponding to the operator $H$. 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, 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 ($\Psi$DOs, cf. also Pseudo-differential operator) [El]. It is thus no surprise that the preferred definition of the determinant for such operators is obtained through the corresponding zeta-function.

The zeta function regularization may be extended in order to include divergent integrals \begin{equation} \int_{a}^{\infty}x^{m}dx \qquad m >0 \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 \qquad \int_{a}^{\infty}dxx^{m-2r}= -\frac{a^{m-2r+1}}{m-2r+1} \end{equation} the integrals inside the recurrence equation are convergent, this zeta regularization approach is used in physics (see renormalization) in order to get finite amplitudes to the divergent integral.

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 [El]. 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 [ElOdRoByZe], [El2]. [ByCoVaZe].

References

[ByCoVaZe]	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
[El]	E. Elizalde, "Multidimensional extension of the generalized Chowla–Selberg formula" Commun. Math. Phys., 198 (1998) pp. 83–95 MR1657369
[El2]	E. Elizalde, "Ten physical applications of spectral zeta functions", Springer (1995) MR1448403
[ElOdRoByZe]	E. Elizalde, S.D. Odintsov, A. Romeo, A.A. Bytsenko, S. Zerbini, "Zeta regularization techniques with applications", World Sci. (1994) MR1346490
[Ha]	S.W. Hawking, "Zeta function regularization of path integrals in curved space time" Commun. Math. Phys., 55 (1977) pp. 133–148 MR0524257
[Na]	M. Nakahara, "Geometry, topology, and physics", Inst. Phys. (1995) pp. 7–8