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# Zeta-Function Regularization is Uniquely Defined and Well

Authors:
• Spanish National Research Council Universitat Autonoma de Barcelona

## Abstract

Hawking's zeta function regularization procedure is shown to be rigorously and uniquely defined, thus putting and end to the spreading lore about different difficulties associated with it. Basic misconceptions, misunderstandings and errors which keep appearing in important scientific journals when dealing with this beautiful regularization method ---and other analytical procedures--- are clarified and corrected. Comment: 7 pages, LaTeX file
arXiv:hep-th/9308028v1 6 Aug 1993
ZETA-FUNCTION REGULARIZATION IS UNIQUELY
DEFINED AND WELL
E. Elizalde1
Division of Applied Mechanics, Norwegian Institute of Technology,
University of Trondheim, N-7034 Trondheim, Norway
Abstract
Hawking’s zeta function regularization procedure is shown to be rigorously and uniquely
deﬁned, thus putting and end to the spreading lore about diﬀerent diﬃculties associated
with it. Basic misconceptions, misunderstandings and errors which keep appearing in im-
portant scientiﬁc journals when dealing with this beautiful regularization method —and
other analytical procedures— are clariﬁed and corrected.
PACS numbers: 03.70.+k, 04.20.Cv, 11.10.Gh
1On leave of absence from and permanent address: Department E.C.M., Faculty of Physics, Barcelona
University, Diagonal 647, 08028 Barcelona, Spain; e-mail: eli @ ebubecm1.bitnet
1
This letter is a defense of Hawking’s zeta function regularization method [1], against the
diﬀerent criticisms which have been published in important scientiﬁc journals and which
seem to conclude (sometimes with exagerated emphasis) that the procedure is ambiguous,
ill deﬁned, and that it possesses even more problems than the well known ones which aﬄict
e.g. dimensional regularization. Our main purpose is to clarify, once and for all, some basic
concepts, misunderstandings, and also errors which keep appearing in the physical litera-
procedures. The situation is such that, what is in fact a most elegant, well deﬁned, and
unique —in many respects— regularization method, may look now to the non-specialist as
just one more among many possible regularization procedures, plagued with diﬃculties and
illdeﬁniteness.
We shall not review here in detail the essentials of the method, nor give an exhaustive
list of the papers that have pointed out ‘diﬃculties with the zeta function procedure’. For
this purpose we refer the specialized reader to a book scheduled to appear towards the end
of 1993 [2]. Instead, we shall restrict ourselves to some speciﬁc points which are at the very
heart of the matter and which may be interesting to a much broader audience.
The method of zeta function regularization is uniquely deﬁned in the following way. Take
the Hamiltonian, H, corresponding to our quantum system, plus boundary conditions, plus
possible background ﬁeld and including a possibly non-trivial metric (because we may live
in a curved spacetime). In mathematical terms, all this boils down to a (second order,
elliptic) diﬀerential operator, A, plus corresponding boundary conditions. The spectrum of
this operator Amay or may not be calculable explicitly, and in the ﬁrst case may or may
not exhibit a beautiful regularity in terms of powers of natural numbers. Whatever be the
situation, it is a well stablished mathematical theorem that to any such operator a zeta
function,ζA, can be deﬁned in a rigorous way. The formal expression of this deﬁnition is:
ζA(s) = Tr eslnA.(1)
Let us stress that this is completely general. Moreover, the present is by no means the
only kind of zeta function known to the mathematicians, for whom this concept has a wider
character (see, for instance [3]). The zeta function ζA(s) is generally a meromorphic function
(develops only poles) in the complex plane, sC. Its calculation usually requires complex
integration around some circuit in the complex plane, the use of the Mellin transform, and
the calculation of invariants of the spacetime metric (the Hadamard-Minakshisundaram-
Pleyel-Seeley-DeWitt-Gilkey-... coeﬃcients) [4].
2
In the particular case when the eigenvalues of the diﬀerential operator A—what is
equivalent, the eigenvalues of the Hamiltonian (with the boundary conditions, etc. taken
into acount)— can be calculated explicitly (let us call them λnand assume they form a
discrete set), the expression of the zeta function is given by:
ζA(s) = X
n
λs
n.(2)
Now, as a particular case of this (already particular) case, when the eigenvalues are of one
of the forms: (i) an, (ii) a(n+b) or (iii) a(n2+b2), we obtain, respectively, the so-called (i)
(ordinary) Riemann zeta function ζR(or simply ζ), (ii) Hurwitz (or generalized Riemann)
zeta function ζH, and (iii) (a speciﬁc case of the) Epstein-Hurwitz zeta function ζE H [3].
Finally, depending on the physical magnitude to be calculated, the zeta function must be
evaluated at a certain particular value of s. For instance, if we are interested in the vacuum
or Casimir energy, which is simply given as the sum over the spectrum:
EC=¯h
2X
n
λn,(3)
this will be given by the corresponding zeta function evaluated at s=1:
EC=¯h
2ζA(1).(4)
Normally, the series (3) will be divergent, and this will involve an analytic continuation
through the zeta function. That is why such regularization can be termed as a particular
case of analytic continuation procedure. In summary, the zeta function of the quantum
system is a very general, uniquely deﬁned, rigorous mathematical concept, which does not
Let us now come down to the concrete situations which have motivated this article. In
Ref. [5], when calculating the Casimir energy of a piecewise uniform closed string, Brevik
and Nielsen where confronted with the following expression ([5], Eq. (52))
X
n=0
(n+β),(5)
which is clearly inﬁnite. Here, the zeta-function regularization procedure consists in the
following. This expressions comes about as the sum over the eigenvalues n+βof the
Hamiltonian of a certain quantum system (here the transverse oscillations of the mentioned
string), i.e. λn=n+β. There is little doubt about what to do: as clearly stated above, the
corresponding zeta function is
ζA(s) =
X
n=0
(n+β)s.(6)
3
Now, for Re s > 1 this is the expression of the Hurwitz zeta function ζH(s;β), which can be
analytically continued as a meromorphic function to the whole complex plane. Thus, the zeta
function regularization method unambiguously prescribes that the sum under consideration
should be assigned the following value
X
n=0
(n+β) = ζH(1; β).(7)
The wrong alternative (for obvious reasons, after all what has been said before), would be
to argue that ‘we might as well have written
X
n=0
(n+β) = ζR(1) + βζR(0),(8)
what gives a diﬀerent result’. In fact, that ‘the Hurwitz zeta function (and not the ordinary
Riemann)’ was the one ‘to be used’ was recognized by Li et al. [6], who reproduced in this
way the correct result obtained by Brevik and Nielsen by means of a (more conventional)
exponential cutoﬀ regularization. However, the authors of [6] were again misunderstanding
the main issue when they considered their method as being a generalization of the zeta reg-
ularization procedure (maybe just because the generalized Riemann zeta function appears!).
Quite on the contrary, this is just a speciﬁc and particularly simple application of the zeta
function regularization procedure .
Of course, the method can be viewed as just one of the many possibilities of analytic
continuation in order to give sense to (i.e., to regularize) inﬁnite expressions. From this
point of view, it is very much related with the standard dimensional regularization method.
In a very recent paper, [7], Svaiter and Svaiter have argued that, being so close relatives,
these two procedures even share the same type of diseases. But precisely to cure the problem
of the dependence of the regularized result on the kind of the extra dimensions (artiﬁcially
introduced in dimensional regularization) was —let us recall— one of the main motivations
of Hawking for the introduction of a new procedure, i.e. zeta function regularization, in
physics [1]. So we seem to have been caught in a devil’s staircase.
The solution to this paradox is the following. Actually, there is no error in the examples
of Ref. [7] and the authors know perfectly what they are doing, but their interpretation
of the results may originate a big deal of confusion among non-specialists. To begin with,
it might look at ﬁrst sight as if the concept itself of analytical continuation would not be
uniquely deﬁned. Given a function in some domain of the complex plane (here, normally, a
part of the real line or the half plane Re s > a, being asome abscisa of convergence), its
4
analytic continuation to the rest of the complex plane (in our case, usually as a meromorphic
function, but this need not in general be so) is uniquely deﬁned. Put it plain, a function
cannot have two diﬀerent analytic continuations. What Svaiter and Svaiter do in their
examples is simply to start in each case from two diﬀerent functions of sand then continue
each of them analytically. Of course, the result is diﬀerent. In particular, these functions
are
f1(s) =
X
n=0
ns(9)
vs.
f2(s) =
X
n=0
nn
a+ 1(s+1)
(10)
continued to s=1, in the ﬁrst example, which corresponds to a Hermitian massless
conformal scalar ﬁeld in 2d Minkowski spacetime with a compactiﬁed dimension, and
g1(s) =
X
n=0
n3s(11)
continued to s=1, vs.
g2(s) =
X
n=0
n3n
a+ 1s
,(12)
continued to s= 0, in the second example, in which the vacuum energy corresponding
to a conformally coupled scalar ﬁeld in an Einstein universe is studied. Needless to say,
the number of posibilites to deﬁne ‘diﬀerent analytic continuations’ in this way is literally
inﬁnite. What use can one make of them remains to be seen.
However, what is absolutely misleading is to conclude from those examples that analytical
regularizations ‘suﬀer from the same problem as dimensional regularization’, precisely the
one that Hawking wanted to cure!. This has no meaning. In the end, also dimensional
regularization is an analytical procedure! One must realize that zeta function regularization
is perfectly well deﬁned, and has little to do with these arbitrary analytic continuations ‘`a
la ζ’ in which one changes at will any exponent at any place with the only restriction that
one recovers the starting expression for a particular value of the exponent s.
The facts are as follows. (i) There exist inﬁnitely many diﬀerent analytic regularization
procedures, being dimensional regularization and zeta function regularization just two of
them. (ii) Zeta function regularization is, as we have seen, a speciﬁcally deﬁned procedure,
provides a unique analytical continuation and (sometimes) a ﬁnite result. (iii) Therefore,
zeta function regularization does not suﬀer, in any way, from the same kind of problem (or
a related one) as dimensional regularization. (iv) This does not mean, however, that zeta
5
function regularization has no problems, but they are of a diﬀerent kind; the ﬁrst appears
already when it turns out that the point (let say s=1 or s= 0) at which the zeta function
must be evaluated turns out to be precisely a pole of the analytic continuation. This and
similar diﬃculties can be solved, as discussed in detail in Ref. [8]. Eventually, as a ﬁnal step
one has to resort to renormalization group techniques [9]. (v) Zeta function regularization
has been extended to higher loop order by McKeon and Sherry under the name of operator
regularization and there also some diﬃculties (concerning the breaking of gauge invariance)
appear [10]. (vi) But, in the end, the fundamental question is: which of the regularizations
that are being used is the one choosen by nature? In practice, one always tries to avoid
answering this question, by cheking the ﬁnite results obtained with diﬀerent regularizations
and by comparing them with classical limits which provide well-known, physically meaningful
values. However, one would be led to believe that in view of its uniqueness, naturalness and
mathematical elegance, zeta function regularization could well be the one. Those properties
are certainly to be counted among their main virtues, but (oddly enough) in some sense
also as its drawbacks: we do not manage to see clearly how and what inﬁnites are thrown
away, something that is evident in other more pedestrial regularizations (which are actually
equivalent in some cases to the zeta one, as pointed out, e.g., in [7]).
The ﬁnal issue of this paper will concern the practical application of the procedure.
Actually, aside from some very simple cases (among those, the ones reviewed here), the use
of the procedure of analytic continuation through the zeta function requires a good deal of
mathematical work [2]. It is no surprise that has been so often associated with mistakes and
errors [11]. One which often repeates itself can be traced back to Eq. (1.70) of the celebrated
book by Mostepanenko and Trunov [12] on the Casimir eﬀect:
a2
π2
X
n=1 n2+a2m2
π2!1
=1
2m21 + am
πcoth am
π,(13)
in other words (for a=πand m=c),
X
n=1 n2+c21=1
2c2(1 + ccoth c).(14)
That Eq. (13) is not right can be observed by simple inspection. The corrected formula
X
n=1 π2n2+c21=1
2c2(1 + ccoth c).(15)
The integrated version of this equality, namely,
X
n=−∞
ln n2+c22= 2c+ 2 ln 1e2c,(16)
6
under the speciﬁc form
T
X
n=−∞
ln h(ωn)2+ (ql)2i=ql+ 2Tln 1eql/T ,(17)
with ωn= 2πnT and ql=πl/R, has been used by Antill´on and Germ´an in a very recent paper
([13], Eq. (2.20)), when studying the Nambu-Goto string model at ﬁnite length and non-zero
temperature. Now this equality is again formal. It involves an analytic continuation, since
it has no sense to integrate the lhs term by term: we get a divergent series.
A rigorous way to proceed is as follows. The expression on the left hand side happens
to be the most simple form of the inhomogeneous Epstein zeta function (called usually
Epstein-Hurwitz zeta function [4]). This function is quite involved and diﬀerent expresions
for it (including asymptotical expansions very useful for accurate numerical calculations)
ζEH (s;c2) =
X
n=1 n2+c2s
=c2s
2+πΓ(s1/2)
2Γ(s)c2s+1 +2πscs+1/2
Γ(s)
X
n=1
ns1/2Ks1/2(2πnc),(18)
which is reminiscent of the famous Chowla-Selberg formula (see [3], p. 1379). Derivatives
can be taken here and the analytical continuation in spresents again no problem.
The usefulness of zeta function regularization is without question [16,2,4]. It can give
immediate sense to expressions such as 1+1 + 1+···=1/2, which turn out to be invaluable
for the construction of new physical theories, as diﬀerent as Pauli-Villars regularization with
inﬁnite constants (advocated by Slavnov [17]) and mass generation in cosmology through
Landau poles (used by Yndurain [18]). The Riemann zeta function was termed by Hilbert in
his famous 1900 lecture as the most important function of whole mathematics [19]. Probably
it will remain so in the Paris Congress of AD 2000, but now maybe with quantum ﬁeld physics
Acknowledgments
I am very grateful to Prof. I. Brevik and to Prof. K. Olaussen for many illuminating
discussions and also to them and to Prof. L. Brink for the hospitality extended to me at
the Universities of Trondheim and G¨oteborg, respectively. This work has been supported by
DGICYT (Spain) and by CIRIT (Generalitat de Catalunya).
7
References
[1] S.W. Hawking, Commun. Math Phys. 55, 133 (1977); J.S. Dowker and R. Critchley,
Phys. Rev. D13, 3224 (1976); L.S. Brown and G.J. MacLay, Phys. Rev. 184, 1272
(1969).
[2] E. Elizalde, S.D. Odintsov, A. Romeo, A.A. Bytsenko, and S. Zerbini, Zeta regularization
techniques with applications, World Sci., to be published.
[3] S. Iyanaga and Y. Kawada, Eds., Encyclopedic dictionary of mathematics, Vol. II (The
MIT press, Cambridge, 1977), p. 1372 ﬀ.
[4] E. Elizalde, S. Leseduarte, and S. Zerbini, Mellin transform techniques for zeta-function
resummations, Univ. of Barcelona preprint, UB-ECM-PF 93/7 (1993).
[5] I. Brevik and H.B. Nielsen, Phys. Rev. D41, 1185 (1990).
[6] X. Li, X, Shi, and J. Zhang, Phys. Rev. D44, 560 (1991).
[7] B.F. Svaiter and N.F. Svaiter, Phys. Rev. D47, 4581 (1993).
[8] S.K. Blau, M. Visser, and A. Wipf, Nucl. Phys. B310, 163 (1988).
[9] E. Elizalde and K. Kirsten, to be published.
[10] D.G.C. McKeon and T.N. Sherry, Phys. Rev. Lett. 59, 532 (1987); Phys. Rev. D35,
3854 (1987); A. Rheban, Phys. Rev. D39, 3101 (1989).
[11] E. Elizalde and A. Romeo, Phys. Rev. D40, 436 (1989).
[12] Mostepanenko and Trunov, The casimir eﬀect (in russian), Atomiadz, Moscow, 1991.
[13] A. Antill´on and G. Germ´an, Phys. Rev. D47, 4567 (1993).
[14] E. Elizalde, J. Math. Phys. 31, 170 (1990).
[15] I. Brevik and I. Clausen, PÆS3hys. Rev. D39, 603 (1989).
[16] A. Actor, J. Phys. A24, 3741 (1991); D.M. McAvity and H. Osborn, Cambridge preprint
DAMTP/92-31 (1992), to be published in Nucl. Phys. B.
[17] A. Slavnov, to be published.
8
[18] F. Yndurain, to be published.
[19] C. Reid, Hilbert (Springer, Berlin, 1970), p. 82.
9
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The Casimir surface force on a solid ball is calculated, assuming the material to be dispersive and to be satisfying the condition ε(ω)μ(ω)=1, ε(ω) being the spectral permittivity and μ(ω) the spectral permeability. A recent paper by Brevik and Einevoll [Phys. Rev. D 37, 2977 (1988)] discussed this problem at zero temperature. In this paper, the case of finite temperatures is covered. The analysis is based upon use of the Debye expansion of the modified Bessel functions. This expansion, being an asymptotic one, is limited in accuracy. At general temperatures, the expansion shows an oscillatory variation on the second-order level. Such a variation is unphysical, so that the physical information that one can extract from second-order theory is limited. On the zeroth-order level, the formalism is, however, found to work well. We discuss the zeroth-order approximation in complete form and comment upon the second-order correction. Although numerical methods are in general necessary, useful analytic approximations can be obtained in the limiting cases of very low, or very high, nondimensional temperatures. At low temperatures, the dispersive effect induces a strong, attractive contribution to the force. At high temperatures, the force becomes small.
Article
The effective Lagrangian and vacuum energy-momentum tensor 〈Tμν〉 due to a scalar field in a de Sitter-space background are calculated using the dimensional-regularization method. For generality the scalar field equation is chosen in the form (□2+ξR+m2)ϕ=0. If ξ=1/6 and m=0, the renormalized 〈Tμν〉 equals gμν(960π2a4)-1, where a is the radius of de Sitter space. More formally, a general zeta-function method is developed. It yields the renormalized effective Lagrangian as the derivative of the zeta function on the curved space. This method is shown to be virtually identical to a method of dimensional regularization applicable to any Riemann space.
Article
The energy due to zero point fluctuations of the dual string is calculated and shown to be divergent. We make it finite by introducing a cut-off. The result is non-covariant but invariant under reparametrization, so one has to modify the classical Lagrangian so as to get a covariant result. It is then shown that this leads to the well-known relation relating the mass squared of the lowest state to the number of dimensions space-time. This result is independent of the cut-off and shows clearly the physical significance of the critical dimension of space-time. Similar considerations for the Neveu-Schwarz model are also discussed.
Article
A framework allowing for perturbative calculations to be carried out for quantum field theories with arbitrary smoothly curved boundaries is described. It is based on an expansion of the heat kernel derived earlier for arbitrary mixed Dirichlet and Neumann boundary conditions. The method is applied to a general renormalisable scalar field theory in four dimensions using dimensional regularisation to two loops and expanding about arbitrary background fields. Detailed results are also specialised to an $O(n)$ symmetric model with a single coupling constant. Extra boundary terms are introduced into the action which give rise to either Dirichlet or generalised Neumann boundary conditions for the quantum fields. For plane boundaries the resulting renormalisation group functions are in accord with earlier results but here the additional terms depending on the extrinsic curvature of the boundary are found. Various consistency relations are also checked and the implications of conformal invariance at the critical point where the $\beta$ function vanishes are also derived. The local Scr\"odinger equation for the wave functional defined by the functional integral under deformations of the boundary is also verified to two loops. Its consistency with the renormalisation group to all orders in perturbation theory is discussed.
Article
We use zeta function techniques to give a finite definition for the Casimir energy of an arbitrary ultrastatic spacetime with or without boundaries. We find that the Casimir energy is intimately related to, but not identical to, the one-loop effective energy. We show that in general the Casimir energy depends on a normalization scale. This phenomenon has relevance to applications of the Casimir energy in bag models of QCD.Within the framework of Kaluza-Klein theories we discuss the one-loop corrections to the induced cosmological and Newton constants in terms of a Casimir like effect. We can calculate the dependence of these constants on the radius of the compact dimensions, without having to resort to detailed calculations.
Article
This paper describes a technique for regularizing quadratic path integrals on a curved background spacetime. One forms a generalized zeta function from the eigenvalues of the differential operator that appears in the action integral. The zeta function is a meromorphic function and its gradient at the origin is defined to be the determinant of the operator. This technique agrees with dimensional regularization where one generalises ton dimensions by adding extra flat dimensions. The generalized zeta function can be expressed as a Mellin transform of the kernel of the heat equation which describes diffusion over the four dimensional spacetime manifold in a fith dimension of parameter time. Using the asymptotic expansion for the heat kernel, one can deduce the behaviour of the path integral under scale transformations of the background metric. This suggests that there may be a natural cut off in the integral over all black hole background metrics. By functionally differentiating the path integral one obtains an energy momentum tensor which is finite even on the horizon of a black hole. This energy momentum tensor has an anomalous trace.
Article
In this paper we present an alternate way of computing amplitudes in quantum field theory in the context of background-field quantization. We concentrate mainly on one-loop effects. The Feynman diagrams of the usual perturbation series are avoided by first performing the functional integration and then using a perturbative expansion due to Schwinger. In this approach we regulate operators rather than the initial Lagrangian. To one-loop order our scheme reduces to a perturbative expansion of the well-known ζ function associated with the superdeterminant of an operator. This technique preserves all symmetries present in the initial theory and does not lead to any explicit divergences as the regulating parameter approaches its limiting value. For illustration, we apply our approach to a toy (φ3)6 scalar theory, to Yang-Mills theory in the covariant gauge, and to quantum electrodynamics. This method reproduces the usual axial anomaly in the three-point functions VVA and AAA. Operator regularization is used in a dimensionally regulated theory reproducing the usual results obtained in the dimensionally regulated Feynman-diagram approach. An outline of how operator regularization is applied beyond one-loop order is provided. Other possible applications of operator regularization are discussed.