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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-
ture, about this method of zeta function regularization and other analytical regularization
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 e−slnA.(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, s∈C. 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].
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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
λ−s
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
admit either interpretations nor ambiguities.
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)
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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
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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
n−s(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
n−3s(11)
continued to s=−1, vs.
g2(s) =
∞
X
n=0
n3n
a+ 1−s
,(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
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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
2m2−1 + am
πcoth am
π,(13)
in other words (for a=πand m=c),
∞
X
n=1 n2+c2−1=1
2c2(−1 + ccoth c).(14)
That Eq. (13) is not right can be observed by simple inspection. The corrected formula
reads
∞
X
n=1 π2n2+c2−1=1
2c2(−1 + ccoth c).(15)
The integrated version of this equality, namely,
∞
X
n=−∞
ln n2+c2/π2= 2c+ 2 ln 1−e−2c,(16)
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under the specific form
T
∞
X
n=−∞
ln h(ωn)2+ (ql)2i=ql+ 2Tln 1−e−ql/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)
have been given in [14] (see also [15]). In particular
ζEH (s;c2) =
∞
X
n=1 n2+c2−s
=−c−2s
2+√πΓ(s−1/2)
2Γ(s)c−2s+1 +2πsc−s+1/2
Γ(s)
∞
X
n=1
ns−1/2Ks−1/2(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
adhered to.
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).
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References
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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).
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3854 (1987); A. Rheban, Phys. Rev. D39, 3101 (1989).
[11] E. Elizalde and A. Romeo, Phys. Rev. D40, 436 (1989).
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[13] A. Antill´on and G. Germ´an, Phys. Rev. D47, 4567 (1993).
[14] E. Elizalde, J. Math. Phys. 31, 170 (1990).
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DAMTP/92-31 (1992), to be published in Nucl. Phys. B.
[17] A. Slavnov, to be published.
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[18] F. Yndurain, to be published.
[19] C. Reid, Hilbert (Springer, Berlin, 1970), p. 82.
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