nLab zeta function regularization

Idea

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

1. considering $s$-powers for all values of $s$ in the complex plane where the naive trace does make sense and then

2. using analytic continuation to obtain the desired special value at $s = 1$ – as for zeta functions.

Analytic regularization of propagators

One speaks of analytic regularization (Speer 71) or zeta function regularization (e.g. M 99, 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 $s= 1$

$$Tr_{reg} \left(\frac{1}{H}\right) \coloneqq pv\, \zeta_H(1)$$

of the zeta function which is given by the expression

$$\zeta_H(s) \coloneqq Tr\left( \frac{1}{H} \right)^s$$

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

$$Tr_{reg} \left(\frac{D}{D^2} \right) \coloneqq pv\, \eta_D(1)$$

where $\eta$ is the eta function of $D$.

Functional determinants

Notice that the first derivative $\zeta^\prime_H$ of this zeta function is, where the original series converges, given by

$$\zeta_H^\prime(s) = \sum_{n = 1}^\infty \frac{- \ln \lambda_n}{ (\lambda_n)^s} \,.$$

Therefore the functional determinant of $H$ (Ray-Singer 71) is the exponential of the zeta function of $H$ at 0:

$$Det_{reg} H \coloneqq \exp(- \zeta_H^\prime(0)) \,.$$

(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.

Higher amplitudes

Accordingly, more general scattering amplitudes are controled by multiple zeta functions (…).

Examples

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 the corresponding Laplace operator $\Delta$ is

$$\zeta_{\Delta} = (2\pi)^{-2 s} E(s) \coloneqq (2\pi)^{-2 s} \underset{(k,l)\in \mathbb{Z}\times\mathbb{Z}-(0,0)}{\sum} \frac{1}{{\vert k +\tau l\vert}^{2s}} \,.$$

The corresponding functional determinant is

$$\exp( E^\prime_{\Delta}(0) ) = (Im \tau)^2 {\vert \eta(\tau)\vert}^4 \,,$$

where $\eta$ is the Dedekind eta function.

(recalled e.g. in Todorov 03, page 3)

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

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

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

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 analytic torsion of the manifold.

Related concepts

context/function field analogy	theta function $\theta$	zeta function $\zeta$ (= Mellin transform of $\theta(0,-)$)	L-function $L_{\mathbf{z}}$ (= Mellin transform of $\theta(\mathbf{z},-)$)	eta function $\eta$	special values of L-functions
physics/2d CFT	partition function $\theta(\mathbf{z},\mathbf{\tau}) = Tr(\exp(-\mathbf{\tau} \cdot (D_\mathbf{z})^2))$ as function of complex structure $\mathbf{\tau}$ of worldsheet $\Sigma$ (hence polarization of phase space) and background gauge field/source $\mathbf{z}$	analytically continued trace of Feynman propagator $\zeta(s) = Tr_{reg}\left(\frac{1}{(D_{0})^2}\right)^s = \int_{0}^\infty \tau^{s-1} \,\theta(0,\tau)\, d\tau$	analytically continued trace of Feynman propagator in background gauge field $\mathbf{z}$: $L_{\mathbf{z}}(s) \coloneqq Tr_{reg}\left(\frac{1}{(D_{\mathbf{z}})^2}\right)^s = \int_{0}^\infty \tau^{s-1} \,\theta(\mathbf{z},\tau)\, d\tau$	analytically continued trace of Dirac propagator in background gauge field $\mathbf{z}$ $\eta_{\mathbf{z}}(s) = Tr_{reg} \left(\frac{sgn(D_{\mathbf{z}})}{ { \vert D_{\mathbf{z}} } \vert }\right)^s$	regularized 1-loop vacuum amplitude $pv\, L_{\mathbf{z}}(1) = Tr_{reg}\left(\frac{1}{(D_{\mathbf{z}})^2}\right)$ / regularized fermionic 1-loop vacuum amplitude $pv\, \eta_{\mathbf{z}}(1)= Tr_{reg} \left( \frac{D_{\mathbf{z}}}{(D_{\mathbf{z}})^2} \right)$ / vacuum energy $-\frac{1}{2}L_{\mathbf{z}}^\prime(0) = Z_H = \frac{1}{2}\ln\;det_{reg}(D_{\mathbf{z}}^2)$
Riemannian geometry (analysis)		zeta function of an elliptic differential operator	zeta function of an elliptic differential operator	eta function of a self-adjoint operator	functional determinant, analytic torsion
complex analytic geometry	section $\theta(\mathbf{z},\mathbf{\tau})$ of line bundle over Jacobian variety $J(\Sigma_{\mathbf{\tau}})$ in terms of covering coordinates $\mathbf{z}$ on $\mathbb{C}^g \to J(\Sigma_{\mathbf{\tau}})$	zeta function of a Riemann surface	Selberg zeta function		Dedekind eta function
arithmetic geometry for a function field		Goss zeta function (for arithmetic curves) and Weil zeta function (in higher dimensional arithmetic geometry)
arithmetic geometry for a number field	Hecke theta function, automorphic form	Dedekind zeta function (being the Artin L-function $L_{\mathbf{z}}$ for $\mathbf{z} = 0$ the trivial Galois representation)	Artin L-function $L_{\mathbf{z}}$ of a Galois representation $\mathbf{z}$, expressible “in coordinates” (by Artin reciprocity) as a finite-order Hecke L-function (for 1-dimensional representations) and generally (via Langlands correspondence) by an automorphic L-function (for higher dimensional reps)		class number $\cdot$ regulator
arithmetic geometry for $\mathbb{Q}$	Jacobi theta function ($\mathbf{z} = 0$)/ Dirichlet theta function ($\mathbf{z} = \chi$ a Dirichlet character)	Riemann zeta function (being the Dirichlet L-function $L_{\mathbf{z}}$ for Dirichlet character $\mathbf{z} = 0$)	Artin L-function of a Galois representation $\mathbf{z}$ , expressible “in coordinates” (via Artin reciprocity) as a Dirichlet L-function (for 1-dimensional Galois representations) and generally (via Langlands correspondence) as an automorphic L-function

References

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)

• Valter Moretti, Local z-function techniques vs point-splitting procedures: a few rigorous results

  Commun. Math. Phys. 201, 327 (1999).

• 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

• Andrey Todorov, The analogue of the Dedekind eta function for CY threefolds, 2003 pdf