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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

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

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

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

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).

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.

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