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''zeta-function regularization''
 
''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]]]:
+
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, $H$, 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
 +
{{Cite|ElOdRoByZe}},
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{{Cite|El2}}:
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 +
$\def\phi{\varphi}$
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$$\sum_{i\in I}(\phi_i,H\phi_i) = {\rm tr}\; H = \sum_{i\in I}\l_i = \sum_{i\in I}\l_i^{-s}\Big|_{s=-1} = \zeta_H(-1), $$
 +
 
  
<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>
+
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|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|Pseudo-differential operator]])
 +
{{Cite|El}}. It is thus no surprise that the preferred definition of the determinant for such operators is obtained through the corresponding zeta-function.
  
<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>
+
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
 +
{{Cite|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
 +
{{Cite|ElOdRoByZe}},
 +
{{Cite|El2}}.
 +
{{Cite|ByCoVaZe}}.
  
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.
+
Based on methods of Elizalde {{Cite|El2}},zeta regularization can  also be generalized to regularize divergent integrals , so we can regularize the UV divergences in QFT theories
  
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]]].
+
$$\begin{array}{l}
 +
\int\nolimits_{a}^{\infty }x^{m-s} dx =\frac{m-s}{2} \int\nolimits_{a}^{\infty }x^{m-1-s} dx +\zeta (s-m)-\sum\limits_{i=1}^{a}i^{m-s}  +a^{m-s}  \\
 +
-\sum\limits_{r=1}^{\infty }\frac{B_{2r} \Gamma (m-s+1)}{(2r)!\Gamma (m-2r+2-s)}  (m-2r+1-s)\int\nolimits_{a}^{\infty }x^{m-2r-s} dx \end{array}  $$
  
 
====References====
 
====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>
+
{|
 +
|-
 +
|valign="top"|{{Ref|ByCoVaZe}}||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
 +
|-
 +
|valign="top"|{{Ref|El}}||valign="top"| E. Elizalde,  "Multidimensional extension of the generalized Chowla–Selberg formula"  ''Commun. Math. Phys.'', '''198'''  (1998)  pp. 83–95 {{MR|1657369}}
 +
|-
 +
|valign="top"|{{Ref|El2}}||valign="top"| E. Elizalde,  "Ten physical applications of spectral zeta functions", Springer (1995) {{MR|1448403}}
 +
|-
 +
|valign="top"|{{Ref|ElOdRoByZe}}||valign="top"| E. Elizalde,  S.D. Odintsov,  A. Romeo,  A.A. Bytsenko,  S. Zerbini,  "Zeta regularization techniques with applications", World Sci.  (1994) {{MR|1346490}}
 +
|-
 +
|valign="top"|{{Ref|Ha}}||valign="top"| S.W. Hawking,  "Zeta function regularization of path integrals in curved space time"  ''Commun. Math. Phys.'', '''55'''  (1977)  pp. 133–148 {{MR|0524257}}
 +
|-
 +
|valign="top"|{{Ref|Na}}||valign="top"|  M. Nakahara,  "Geometry, topology, and physics", Inst. Phys. (1995) pp. 7–8
 +
|-
 +
|}

Latest revision as of 00:03, 30 December 2015

2010 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.

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].

Based on methods of Elizalde [El2],zeta regularization can also be generalized to regularize divergent integrals , so we can regularize the UV divergences in QFT theories

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

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
How to Cite This Entry:
Zeta-function method for regularization. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Zeta-function_method_for_regularization&oldid=13101
This article was adapted from an original article by E. Elizalde (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article