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$dim = 1$: ,
In the context of regularization in physics, zeta function regularization is a method/prescription for extracing finite values for traces of powers of Laplace operators/Dirac operators by
considering $s$-powers for all values of $s$ in the complex plane where the naive trace does make sense and then
using analytic continuation to obtain the desired special value at $s = 1$ – as for zeta functions.
One speaks of analytic regularization (Speer 71) or zeta function regularization (e.g. BCEMZ 03, section 2) if a Feynman propagator/Green's function for a bosonic 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 value at $s= 1$
of the zeta function which is given by the expression
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 with Dirac operator $D$ is defined by
where $\eta$ is the eta function of $D$.
Notice that the first derivative $\zeta^\prime_H$ of this zeta function is, where the original series converges, given by
Therefore the functional determinant of $H$ (Ray-Singer 71) is the exponential of the zeta function of $H$ at 0:
(see also 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 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 zeta function regularization.
Accordingly, more general scattering amplitudes are controled by multiple zeta functions (…).
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 the corresponding Laplace operator $\Delta$ is
The corresponding functional determinant is
where $\eta$ is the Dedekind eta function.
(recalled e.g. in Todorov 03, page 3)
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
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
$\sum_{n=1}^{\infty} \frac{1}{n} = \gamma$ Euler-Mascheroni constant, and $\Psi(a)= -\frac{\Gamma '(a)}{\Gamma (a)}$
So within this reuglarization there wouldn’t be any UV ultraviolet divergence
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 analytic torsion of the manifold.
context/ | $\theta$ | $\zeta$ (= of $\theta(0,-)$) | $L_{\mathbf{z}}$ (= of $\theta(\mathbf{z},-)$) | $\eta$ | |
---|---|---|---|---|---|
/ | $\theta(\mathbf{z},\mathbf{\tau}) = Tr(\exp(-\mathbf{\tau} \cdot (D_\mathbf{z})^2))$ as function of $\mathbf{\tau}$ of $\Sigma$ (hence of ) and / $\mathbf{z}$ | analytically continued of $\zeta(s) = Tr_{reg}\left(\frac{1}{(D_{0})^2}\right)^s = \int_{0}^\infty \tau^{s-1} \,\theta(0,\tau)\, d\tau$ | analytically continued of in $\mathbf{z}$: $L_{\mathbf{z}}(s) \coloneqq Tr_{reg}\left(\frac{1}{(D_{\mathbf{z}})^2}\right)^s = \int_{0}^\infty \tau^{s-1} \,\theta(\mathbf{z},\tau)\, d\tau$ | analytically continued of in $\mathbf{z}$ $\eta_{\mathbf{z}}(s) = Tr_{reg} \left(\frac{sgn(D_{\mathbf{z}})}{ { \vert D_{\mathbf{z}} } \vert }\right)^s$ | $pv\, L_{\mathbf{z}}(1) = Tr_{reg}\left(\frac{1}{(D_{\mathbf{z}})^2}\right)$ / fermionic $pv\, \eta_{\mathbf{z}}(1)= Tr_{reg} \left( \frac{D_{\mathbf{z}}}{(D_{\mathbf{z}})^2} \right)$ / $-\frac{1}{2}L_{\mathbf{z}}^\prime(0) = Z_H = \frac{1}{2}\ln\;det_{reg}(D_{\mathbf{z}}^2)$ |
() | , | ||||
$\theta(\mathbf{z},\mathbf{\tau})$ of over $J(\Sigma_{\mathbf{\tau}})$ in terms of covering coordinates $\mathbf{z}$ on $\mathbb{C}^g \to J(\Sigma_{\mathbf{\tau}})$ | |||||
for a | (for ) and (in ) | ||||
for a | , | (being the $L_{\mathbf{z}}$ for $\mathbf{z} = 0$ the ) | $L_{\mathbf{z}}$ of a $\mathbf{z}$, expressible “in coordinates” (by ) as a finite-order (for 1-dimensional representations) and generally (via ) by an (for higher dimensional reps) | $\cdot$ | |
for $\mathbb{Q}$ | ($\mathbf{z} = 0$)/ ($\mathbf{z} = \chi$ a ) | (being the $L_{\mathbf{z}}$ for $\mathbf{z} = 0$) | of a $\mathbf{z}$ , expressible “in coordinates” (via ) as a (for 1-dimensional Galois representations) and generally (via ) as an |
Original articles include
Eugene Speer, On the structure of Analytic Renormalization, Comm. Math. Phys. 23, 23-36 (1971) (Euclid)
D. Ray, Isadore Singer, 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
Modern accounts and reviews include
Daniel Freed, page 8 of On determinant line bundles, Math. aspects of string theory, ed. S. T. Yau, World Sci. Publ. 1987, (revised pdf, dg-ga/9505002)
Emilio Elizalde, Ten Physical Applications of Spectral Zeta Functions (1995)
A. Bytsenko, G. Cognola, Emilio Elizalde, Valter Moretti, S. Zerbini, section 2 of Analytic Aspects of Quantum Fields, World Scientific Publishing, 2003, ISBN 981-238-364-6
Nicolas Robles, Zeta function regularization, 2009 (pdf)
See also
Last revised on January 15, 2016 at 11:13:38. See the history of this page for a list of all contributions to it.