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\begin{document}

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\section*{zeta function regularization}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{physics}{}\paragraph*{{Physics}}\label{physics}

[[!include physicscontents]]

\hypertarget{theta_functions}{}\paragraph*{{Theta functions}}\label{theta_functions}

[[!include theta functions - contents]]

\hypertarget{complex_geometry}{}\paragraph*{{Complex geometry}}\label{complex_geometry}

[[!include complex geometry - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{analytic_regularization_of_propagators}{Analytic regularization of propagators}\dotfill \pageref*{analytic_regularization_of_propagators} \linebreak
\noindent\hyperlink{functional_determinants}{Functional determinants}\dotfill \pageref*{functional_determinants} \linebreak
\noindent\hyperlink{higher_amplitudes}{Higher amplitudes}\dotfill \pageref*{higher_amplitudes} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{of_laplace_operator_on_complex_torus_and_dedekind_eta_function}{Of Laplace operator on complex torus and Dedekind eta function}\dotfill \pageref*{of_laplace_operator_on_complex_torus_and_dedekind_eta_function} \linebreak
\noindent\hyperlink{zeta_regularization_for_divergent_integrals}{Zeta regularization for divergent integrals}\dotfill \pageref*{zeta_regularization_for_divergent_integrals} \linebreak
\noindent\hyperlink{analytic_torsion}{Analytic torsion}\dotfill \pageref*{analytic_torsion} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

In the context of [[regularization (physics)|regularization in physics]], \emph{zeta function regularization} is a method/prescription for extracing finite values for [[traces]] of powers of [[Laplace operators]]/[[Dirac operators]] by

\begin{enumerate}%
\item considering $s$-powers for all values of $s$ in the [[complex plane]] where the naive trace does make sense and then


\item using [[analytic continuation]] to obtain the desired [[special values of L-functions|special value]] at $s = 1$ -- as for [[zeta functions]].



\end{enumerate}
\hypertarget{analytic_regularization_of_propagators}{}\subsubsection*{{Analytic regularization of propagators}}\label{analytic_regularization_of_propagators}

One speaks of \emph{analytic regularization} (\hyperlink{Speer71}{Speer 71}) or \emph{zeta function regularization} (e.g. \hyperlink{M99}{M 99}, \hyperlink{BCEMZ03}{BCEMZ 03, section 2}) if a [[Feynman propagator]]/[[Green's function]] for a [[boson|bosonic]] [[field (physics)|field]], which is naively given by the expression ``$Tr\left(\frac{1}{H}\right)$'' (for $H$ the given [[wave operator]]/[[Laplace operator]]) is made well defined by interpreting it as the [[principal value]] of the [[special values of L-functions|special value]] at $s= 1$

\begin{displaymath}
Tr_{reg}
  \left(\frac{1}{H}\right)
  \coloneqq
  pv\, \zeta_H(1)
\end{displaymath}
of the [[zeta function of an elliptic differential operator|zeta function]] which is given by the expression

\begin{displaymath}
\zeta_H(s) 
    \coloneqq 
  Tr\left(
    \frac{1}{H}
  \right)^s
\end{displaymath}
for all values of $s \in \mathbb{C}$ for which the right hand side exists, and is defined by [[analytic continuation]] elsewhere.

Analogously the zeta function regularization of the [[Dirac propagator]] for a [[fermion]] [[field (physics)|field]] with [[Dirac operator]] $D$ is defined by

\begin{displaymath}
Tr_{reg}
  \left(\frac{D}{D^2} \right)
  \coloneqq
  pv\, \eta_D(1)
\end{displaymath}
where $\eta$ is the [[eta function]] of $D$.

\hypertarget{functional_determinants}{}\subsubsection*{{Functional determinants}}\label{functional_determinants}

Notice that the first [[derivative]] $\zeta^\prime_H$ of this [[zeta function of an elliptic differential operator|zeta function]] is, where the original series converges, given by

\begin{displaymath}
\zeta_H^\prime(s)
  =
  \sum_{n = 1}^\infty \frac{- \ln \lambda_n}{ (\lambda_n)^s}
  \,.
\end{displaymath}
Therefore the \emph{[[functional determinant]]} of $H$ (\hyperlink{RaySinger71}{Ray-Singer 71}) is the exponential of the zeta function of $H$ at 0:

\begin{displaymath}
Det_{reg} H
  \coloneqq
  \exp(- \zeta_H^\prime(0))
  \,.
\end{displaymath}
(see also \hyperlink{BCEMZ03}{BCEMZ 03, section 2.3})

Via the [[analytic continuation]] involved in defining $\zeta_H(0)$ in the first place, this may be thought of as a \emph{[[regularization (physics)|regularization]]} of the ill-defined naive definition ``$\prod_n \lambda_n$'' of the [[determinant]] of $H$. As such functional determinants often appear in [[quantum field theory]] as what is called \emph{[[zeta function regularization]]}.

\hypertarget{higher_amplitudes}{}\subsubsection*{{Higher amplitudes}}\label{higher_amplitudes}

Accordingly, more general [[scattering amplitudes]] are controled by [[multiple zeta functions]] (\ldots{}).

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{of_laplace_operator_on_complex_torus_and_dedekind_eta_function}{}\subsubsection*{{Of Laplace operator on complex torus and Dedekind eta function}}\label{of_laplace_operator_on_complex_torus_and_dedekind_eta_function}

For $\mathbb{C}/(\mathbb{Z}\oplus \tau \mathbb{Z})$ a [[complex torus]] (complex [[elliptic curve]]) equipped with its standard flat [[Riemannian metric]], then the [[zeta function of an elliptic differential operator|zeta function]] of the corresponding [[Laplace operator]] $\Delta$ is

\begin{displaymath}
\zeta_{\Delta} = (2\pi)^{-2 s} E(s)
   \coloneqq
   (2\pi)^{-2 s} \underset{(k,l)\in \mathbb{Z}\times\mathbb{Z}-(0,0)}{\sum}
  \frac{1}{{\vert k +\tau l\vert}^{2s}}
  \,.
\end{displaymath}
The corresponding [[functional determinant]] is

\begin{displaymath}
\exp(
    E^\prime_{\Delta}(0)
  )
  = (Im \tau)^2 {\vert \eta(\tau)\vert}^4
  \,,
\end{displaymath}
where $\eta$ is the [[Dedekind eta function]].

(recalled e.g. in \hyperlink{Todorov03}{Todorov 03, page 3})

\hypertarget{zeta_regularization_for_divergent_integrals}{}\subsubsection*{{Zeta regularization for divergent integrals}}\label{zeta_regularization_for_divergent_integrals}

the zeta regularizatio method can be extended to include also a regularization for the divergent integrals $\int_{a}^{\infty}x^{m}dx$ which appears in QFT, this is made by means of the identity

\begin{displaymath}
\begin{array}{l}
\int_{a}^{\infty }x^{m-s} dx =\frac{m-s}{2} \int_{a}^{\infty }x^{m-1-s} dx +\zeta (s-m)-\sum_{i=1}^{a}i^{m-s}  +a^{m-s}  \\
-\sum_{r=1}^{\infty }\frac{B_{2r} \Gamma (m-s+1)}{(2r)!\Gamma (m-2r+2-s)}  (m-2r+1-s)\int_{a}^{\infty }x^{m-2r-s} dx \end{array}
\end{displaymath}
for the case of $m=-1$ although the harmonic series has a pole we can regularize by the 2 possibilities

$\sum_{n=0}^{\infty} \frac{1}{n+a} = -\Psi (a)$ or $\sum_{n=0}^{\infty} \frac{1}{n+a} = -\Psi (a)+log(a)$ in particular \newline $\sum_{n=1}^{\infty} \frac{1}{n} = \gamma$ Euler-Mascheroni constant, and $\Psi(a)= -\frac{\Gamma '(a)}{\Gamma (a)}$

So within this reuglarization there wouldn't be any UV ultraviolet divergence

\hypertarget{analytic_torsion}{}\subsubsection*{{Analytic torsion}}\label{analytic_torsion}

The functional determinant of a [[Laplace operator]] of a [[Riemannian manifold]] acting on [[differential n-forms]] is up to a sign in the exponent a factor in what is called the \emph{[[analytic torsion]]} of the manifold.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

[[!include zeta-functions and eta-functions and theta-functions and L-functions -- table]]

\hypertarget{references}{}\subsection*{{References}}\label{references}

Original articles include

\begin{itemize}%
\item [[Eugene Speer]], \emph{On the structure of Analytic Renormalization}, Comm. Math. Phys. 23, 23-36 (1971) (\href{http://projecteuclid.org/euclid.cmp/1103857549}{Euclid})


\item D. Ray, [[Isadore Singer]], \emph{R-torsion and the Laplacian on Riemannian manifolds}, Advances in Math. 7: 145--210, (1971) doi:10.1016/0001-8708(71)90045-4, MR 0295381



\end{itemize}
Modern accounts and reviews include

\begin{itemize}%
\item [[Daniel Freed]], page 8 of \emph{On determinant line bundles}, Math. aspects of [[string theory]], ed. S. T. Yau, World Sci. Publ. 1987, (revised \href{http://www.math.utexas.edu/~dafr/Index/determinants.pdf}{pdf}, \href{http://arxiv.org/abs/dg-ga/9505002}{dg-ga/9505002})


\item [[Emilio Elizalde]], \emph{Ten Physical Applications of Spectral Zeta Functions} (1995)


\item [[Valter Moretti]], \emph{Local z-function techniques vs point-splitting procedures: a few rigorous results} Commun. Math. Phys. 201, 327 (1999).


\item A. Bytsenko, G. Cognola, [[Emilio Elizalde]], [[Valter Moretti]], S. Zerbini, section 2 of \emph{Analytic Aspects of Quantum Fields}, World Scientific Publishing, 2003, ISBN 981-238-364-6


\item [[Nicolas Robles]], \emph{Zeta function regularization}, 2009 ([[RoblesZetaRegularization.pdf:file]])



\end{itemize}
See also

\begin{itemize}%
\item [[Andrey Todorov]], \emph{The analogue of the Dedekind eta function for CY threefolds}, 2003 \href{http://www.ma.huji.ac.il/conf/crelle.pdf}{pdf}

\end{itemize}
[[!redirects zeta function regularizations]]

[[!redirects functional determinant]] [[!redirects functional determinants]]

[[!redirects zeta-function regularization]] [[!redirects zeta-function regularizations]]



\end{document}