Complete determination of the singularity structure of zeta functions

arXiv:hep-th/9608056v2 12 Aug 1996

DESY 96-155 hep-th/9608056 August 1996

Complete determination of the singularity structure of zeta functions E. Elizalde 1 II. Institut f¨ ur Theoretische Physik der Universit¨at Hamburg Luruper Chaussee 149, D-22761 Hamburg, Germany

Abstract Series of extended Epstein type provide examples of non-trivial zeta functions with important physical applications. The regular part of their analytic continuation is seen to be a convergent or an asymptotic series. Their singularity structure is completely determined in terms of the Wodzicki residue in its generalized form, which is proven to yield the residua of all the poles of the zeta function, and not just that of the rightmost pole (obtainable from the Dixmier trace). The calculation is a very down-to-earth application of these powerful functional analytical methods in physics.

PACS: 11.10.Gh, 02.30.Tb, 02.30.Dk, 02.30.Mv 1

Permanent address: Center for Advanced Studies CEAB, CSIC, Cam´ı de Santa B`arbara, E-17300 Blanes, and Dept. ECM, Facultat de F´ısica, Universitat de Barcelona, Diagonal 647, E-08028 Barcelona, Spain. Email: [email protected]

1

A most important issue in the application of the zeta-function regularization method [1] in physics is the precise determination of the pole structre of the analytical continuation of the corresponding zeta function. In the recent mathematical literature, there are precise results which partially characterize the meromorphic structure of the analytical continuation of the zeta function of any elliptic pseudodifferential operator (ΨDO) [2], even of complex order [3]. The position and the order of the poles is known, and also the residue of the rightmost one, which can be determined by using either the Dixmier trace or the Wodzicki residue of the principal symbol of the operator. Here, we obtain the residua of all the remaining poles and illustrate their very simple calculation through such powerful functional analytical tools, by means of two fundamental examples with physical application [4]. The additional determination that is carried out of the regular part of the analytic continuation completes the analysis of the meromorphic structure of the zeta functions. A pseudodifferential operator A of order m on a manifold Mn is defined through its symbol a(x, ξ), which is a function belonging to the space S m (Rn × Rn ) of C∞ functions such that for any α, β there exists a constant Cα,β so that ∂ξα ∂xβ a(x, ξ) ≤ Cα,β (1 + |ξ|)m−|α| . The definition of A is given (in the distribution sense) by Af (x) = (2π)−n

Z

ei a(x, ξ)fˆ(ξ) dξ,

(1)

where f is a smooth function (f ∈ S) and fˆ its Fourier transform. When a(x, ξ) is a polynomial in ξ one gets a differential operator but, in general, the order m can be even complex. Pseudodifferential operators are useful tools, both in mathematics and in physics. They were crucial for the proof of the uniqueness of the Cauchy problem [5] and also for the proof of the Atiyah-Singer index formula [6]. In quantum field theory they appear in any analytic continuation process (as complex powers of differential operators, like the Laplacian) [7]. They constitute nowadays the basic starting point of any rigorous formulation of quantum field theory through microlocalization [8], a concept that is considered to be the most important step towards the understanding of linear partial differential equations since the appearance of distributions. For A a positive-definite elliptic ΨDO of positive order m ∈ R, acting on the space of smooth sections of an n-dimensional vector bundle E over a closed, n-dimensional manifold M, the zeta function is defined as ζA (s) = tr A−s =

X

λ−s j ,

j

Re s >

n ≡ s0 . m

(2)

Here s0 is called the abscissa of convergence of ζA (s), which is proven to have a meromorphic continuation to the whole complex plane C (regular at s0 ), provided that A admits a spectral cut: Lθ = {λ ∈ C; Arg λ = θ, θ1 < θ < θ2 }, Spec A ∩ Lθ = ∅ (the Agmon-Nirenberg condition). Strictly speaking, the definition of ζA (s) depends on the position of the cut Lθ , not so that of the determinant [9] detζ A = exp[−A′ (0)], which only depends on the homotopy class 2

of the cut. The precise structure of the analytical continuation is well known [10]: it has at most simple poles at sk = (n − k)/m, k = 0, 1, 2, . . . , n − 1, n + 1, . . .. The applications of this zeta-function definition of a determinant in physics are important [11, 12]. A zeta function with the same meromorphic structure in the complex s-plane and extending the ordinary definition to operators of complex order m ∈ C\Z, has been recently obtained in Ref. [3]. (It is clear that operators of complex order do not admit spectral cuts.) The construction in [3] starts from the definition of a trace, obtained as the integral over the manifold of the trace density of the difference between the Schwartz kernel of A and the Fourier transform of a number of first homogeneous terms (in ξ) of the usual decomposition of the symbol of A: a(x, ξ) = am (x, ξ) + am−1 (x, ξ) + · · · + am−N (x, ξ) + · · · In order to write down an action in operator language one needs a functional that replaces integration. For the Yang-Mills theory this is the Dixmier trace, which constitutes the unique extension of the usual trace to the ideal L(1,∞) of the compact operators T such that the partial sums of its spectrum diverge logarithmically as the number of terms in the sum, i.e. σN (T ) ≡

N −1 X j=0

µj = O(log N),

µ0 ≥ µ1 ≥ · · ·

(3)

The definition of the Dixmier trace of T , Dtr T [13], is then a refinement of the limit limN →∞ log1N σN (T ). (It is directly given by this limit when the Cesaro means M(ρ) of the sequence in N are convergent as ρ → ∞.) As observed by Connes [14], the Hardy-Littlewood theorem can be stated in a way that connects the Dixmier trace with the residue of the zeta function of the operator T −1 at s = 1: Dtr T = lims→1+ (s − 1)ζT −1 (s). The Wodzicki (or noncommutative) residue [15] is the only extension of the Dixmier trace to ΨDOs which are not in L(1,∞) . Even more, it is the only trace at all one can define in the algebra of ΨDOs up to a multiplicative constant. It is given by the integral res A =

Z

S∗M

tr an (x, ξ) dξ,

(4)

with S ∗ M ⊂ T ∗ M the co-sphere bundle on M (some authors put a coefficient in front of the integral). If dim M = n = − ord A (M compact Riemann, A elliptic, n ∈ N) it coincides with the Dixmier trace, and one has [15] Ress=1 ζA (s) =

1 res A−1 . n

(5)

However, the Wodzicki residue continues to make sense for ΨDOs of arbitrary order and, even if the symbols aj (x, ξ), j < m, are not invariant under coordinate choice, the integral in (4) is, and defines a trace. In particular, the residua of the poles of the extended definition of zeta function to operators of complex order are also given by the noncommutative residue. Moreover, an interesting connection of the Wodzicki residue with the second coefficient of the heat-kernel expansion of the Laplacian has been found recently [16, 17]. 3

A complete determination of the meromorphic structure of the zeta function in the complex plane is obtained as follows. Relying on the above results, what is missing for the description of the singularities are the residua of all the remaining poles. As for the regular part of the analytic continuation, specific methods have to be used and the results are non-trivial: asymptotic series, and not convergent ones, appear most often [18]. Proposition 1. Under the conditions of existence of the zeta function of A, given above, and asuming that the symbol a(x, ξ) of the operator A is analytic in ξ −1 at ξ −1 = 0, the formula for the determination of the residue of the rightmost pole (by means of the Wodzicki residue) can be generalized to calculate all the residua of the zeta function poles, in the way: 1 Z 1 −sk −sk res A = tr a−n (x, ξ) dn−1ξ. Ress=sk ζA (s) = ∗ m m S M

(6)

Proof. One just has to notice that the homogeneous component of degree −n of the corresponding power of the principal symbol of A is obtained by taking the appropriate derivative of a power of the symbol with respect to ξ −1 at ξ −1 = 0, namely: −sk a−n (x, ξ) =

∂ ∂ξ −1

!k

i ξ n−k a(k−n)/m (x, ξ)

h

ξ −n .

(7)

ξ −1 =0

The proof then follows by simple algebraic manipulation. Corollary. As a consequence of the additivity property of the trace, it is now clear that, given two operators A1 and A2 satisfying the conditions above, the singularity structure of the zeta function of the operator A1 + A2 can be determined from those of the zeta functions of A1 and A2 . In fact, for all k, Ress=sk ζA1 +A2 (s) = Ress=sk ζA1 (s) + Ress=sk ζA2 (s).

(8)

These results will now be illustrated with two examples. Aside from the Riemann and Hurwitz zeta functions, the Epstein ones [19] and generalizations thereof are most basic tools in the zeta function regularization method [18]. They appear in the calculation of the vacuum energy or effective potentials of quantum physical systems involving toroidal compactification, finite temperature, massive particles, or a chemical potential. Consider a spacetime with topology R×T 2 [20] and a general metric on T 2 : ds2 = hab dxa dxb , with 



1 τ1  1 , hab =  τ2 τ1 |τ |2

(9)

(τ1 , τ2 ) being the Teichm¨ uller parameters, τ = τ1 +iτ2 , τ2 > 0. The Laplace-Beltrami operator 2 1 2 2 (|τ |2 n21 − 2τ1 n1 n2 + n22 ). In is: L = − τ2 (|τ | ∂1 − 2τ1 ∂1 ∂2 + ∂22 ) and its eigenvalues λn1 ,n2 = 4π τ2 the massive case the spectrum runs over n1 , n2 ∈ Z. If m = 0 the zero-mode n1 = n2 = 0 has to be excluded. Under different boundary conditions (Dirichlet and Neumann, for instance) one gets a restriction of the indices to non-negative values and in many situations —as is 4

the case of spherical compactification— a one-dimensional variant of the Laplacian appears [18]. This leads us to consider two families of such operators —plus boundary conditions in general. The corresponding zeta functions belong to the family of generalized Epstein zeta functions: X ′ (am2 + bmn + cn2 + q)−s , Re s > 1 (10) ζE (s; a, b, c; q) ≡ m,n∈Z (where q is the mass, chemical potential or finite-temperature contribution) or to the more simple version [21]: ζG (s; a, c; q) ≡

∞ h X

a(n + c)2 + q

n=−∞

i−s

,

Re s > 1/2.

(11)

The restriction of these series to non-negative values of the indices will be denoted by ζEt and ζGt , respectively. The parenthesis in (10) is an inhomogeneous quadratic form, Q(x, y) + q, restricted to the integers. We assume that a, c > 0 and ∆ = 4ac − b2 > 0 [22]. The starting point for the derivation of the formulas is Jacobi’s theta function identity +∞ X

−(n+z)2 t

e

n=−∞

=

r

∞ X π 2 2 e−π n /t cos(2πnz) , 1+ t n=1

#

"

z, t ∈ C, Re t > 0.

(12)

Example 1. Consider first the second function, which is more simple. Making use of the Jacobi identity we get the analytical continuation ζG (s; a, c; q) =

r

·

π Γ(s − 1/2) 1/2−s 4π s −1/4−s/2 1/4−s/2 q + a q a Γ(s) Γ(s)

∞ X

q

ns−1/2 cos(2πnc)Ks−1/2 (2πn q/a),

n=1

(13)

where Kν is the modified Bessel function of the second kind. Associated with the above zeta functions, but considerably more difficult to treat, are the corresponding truncated sums, with indices running from 0 to ∞. In this case the Jacobi identity is of no use. By means of specific techniques of analytic continuation of zeta functions [18], we obtain ∞ 1 q −s X (−1)m Γ(m + s) q −m ζGt (s; a, c; q) ∼ − c q −s + ζH (−2m, c) (14) 2 Γ(s) m=1 m! a r ∞ q π Γ(s − 1/2) 1/2−s 2π s −1/4−s/2 1/4−s/2 X ns−1/2 cos(2πnc)Ks−1/2 (2πn q/a). q + a q + a 2Γ(s) Γ(s) n=1



 



The first series is asymptotic [23, 24]. From the expressions above one can calculate the determinants of Klein-Gordon and Dirac operators on compact spaces as, for instance, Ncubes, cylinders and spheres S N (whenever the spectrum λn is a polynomial in n). The meromorpic structure of these zeta functions is described by the general theory. According to it, in principle, poles at the positions s = −1, −2, −3, . . . could also be possible. 5

They just have zero residue, as we shall now prove. The residue of the rightmost pole at s = 1/2 can be obtained from the Dixmier trace: −1 h i−1/2 1 NX 1 a(j + c)2 + q =√ , N →∞ log N a j=0

Dtr G−1/2 = lim

(15)

which is in fact the value of the residue of the pole at s = 1/2 in Eq. (13), thus Ress=1/2 ζG (s) = Dtr G−1/2 . Now to the second step: the residue of this pole at s = 1/2 can also be obtained from the Wodzicki residue. In fact, we have: res G

−1/2

=

Z

S∗R

2 −1/2 tr g−1 (ξ) dξ = √ a

(16)

(note that g −1/2 (ξ) = ξ −1 + O(ξ −2 ) and that the 0−dimensional sphere is reduced to two points, namely S ∗ R = S 0 = {−1, 1}). Thus, Ress=1/2 ζG (s) = 21 res G−1/2 . Having dealt with the rightmost pole in all possible ways, we now analyze the others by means of the generalized Wodzicki residue. We will prove that Ress=1/2−k ζG (s) =

1 res Gk−1/2 , 2

Ress=−k ζG (s) = 0,

k = 0, 1, 2, . . .

(17)

The decomposition of the corresponding symbol into homogeneous parts yields (k − 1/2)(k − 3/2) · · · 1/2 q k k − 1/2 q + · · · + +··· g k−1/2 (ξ) = ξ 2k−1 1 + 1! ξ 2 k! ξ 2k (2k − 1)!! q k = ξ 2k−1 + · · · + + ···, k = 0, 1, 2, . . . , k! 2k ξ

#

"

g k (ξ) =



ξ2 + q

k

= ξ 2k + · · · + q k ,

(18)

k = 0, 1, 2, . . . ,

therefore, k−1/2

tr g−1

(ξ) =

(2k − 1)!! q k , k! 2k

k tr g−1 (ξ) = 0,

k = 0, 1, 2, . . . .

(19)

Thus, we obtain (2k − 1)!! q k 1 1 k−1/2 = , Ress=−k ζG (s) = res Gk = 0, k = 0, 1, 2, . . . , Ress=1/2−k ζG (s) = res G k 2 k! 2 2 (20) which coincide in fact with the residues of the poles of ζG (s) at s = sk , as we wanted to see. The fact that some of the would-be poles are actually not present follows from the decomposition (18) showing clearly that their residua are zero. We see in this example how the Wodzicki residue under its more general form allows us to calculate all the residua of the poles of the zeta function. The meromorphic structure of the analytical continuation of the zeta function is absolutely specified through the Dixmier trace and the Wodzicki residue in its general form. However, we have also shown through our explicit calculation that 6

what remains can in no way be considered as a trivial analytic part. It may be given by a convergent series but, possibly, by an asymptotic one. Example 2. It is more involved. In the homogeneous case the analytic continuation of this Epstein zeta function is given by the Chowla-Selberg formula [22] √ 22s π as−1 −s ζE (s; a, b, c; 0) = 2ζ(2s) a + Γ(s − 1/2)ζ(2s − 1) Γ(s)∆s−1/2   ∞ X 2s+5/2 π s πn √ s−1/2 √ n σ1−2s (n) cos(πnb/a) Ks−1/2 + ∆ . (21) Γ(s) ∆s/2−1/4 a n=1 a where σs (n) ≡ d|n ds , sum over the s-powers of the divisors of n. (There is a missprint in the transcription of formula (21) in Ref. [25]). We observe that the rhs of (21) exhibits a simple pole at s = 1, with residue Ress=1 ζE (s; a, b, c; 0) = √2π∆ . In the general case, we have obtained the meromorphic continuation: P

ζE (s; a, b, c; q) = −q

+

s

−s

!s

∞ X q q 1/4 π 4 2πq 1−s √ + k s−1/2 Ks−1/2 2πk + √ qa k=1 a (s − 1) ∆ Γ(s) a !s ∞ s r r   X q a aq 2π (22) + k s−1 Ks−1 4πk a q∆ k=1 ∆

" 

∞ X X 4aq 2 (2π)s k s−1/2 cos(πkb/a) d1−2s ∆ + 2 a d k=1 d|k



1/4−s/2





s

r 



πk 4aq Ks−1/2  ∆ + 2  . a d

This is a fundamental result. It looks rather different from the Chowla-Selberg formula (21), but it can actually be viewed as its natural extension to the case q 6= 0. It also shares all the good properties of (21). As in example 1, the only pole of this zeta function can be obtained by using either the Dixmier trace or the Wodzicki residue. In fact, N X 1 2π (am2 + bmn + cn2 + q)−1 = √ , 2 N →∞ log(N ) ∆ m,n=−N

Dtr E −1 = lim

(23)

which is the value of the residue of the pole at s = 1 , thus Ress=1 ζE (s) = Dtr E −1 . Moreover, this value can also be obtained from the Wodzicki residue: res E −1

1 −1 2 = 2 tr e−2 (ξ) d ξ = a S∗R Z

Z

0

2π



b  tan θ + 2a

!2

−1

∆ + 2 4a

4π d(tan θ) = √ . ∆

(24)

Here the integral is performed over the unit circumference (S ∗ R2 = S 1 , |ξ| = 1). Thus, Ress=1 ζE (s) = 12 res E −1 . We have thus shown again how the rightmost pole of the zeta function can be obtained either from the Dixmier trace or from the Wodzicki residue. The fact that this is the only pole of our zeta function also follows from the calculation of the generalized Wodzicki residua. According to the general theory, the other possible poles would 7

be at s = sk = 1 − k/2, k = 1, 3, 4, . . .. We must obtain the homogeneous component of degree −2 of the principal symbol of the operator E k/2−1 : 

ek/2−1 (ξ1 , ξ2 ) = aξ12 + bξ1 ξ2 + cξ22 + q

k/2−1

.

(25)

But it is clear that neither for k odd nor for k even is there any component of this principal symbol of degree −2. All corresponding residua are zero and none of these poles exists. The most difficult case in the family of Epstein-like zeta functions corresponds to having a truncated range. This comes about when one imposes boundary conditions of the usual Dirichlet or Neumann type [18]. Jacobi’s theta function identity is then useless and no expression in terms of a convergent series for the analytical continuation to values of Re s below the abscissa of convergence can be obtained. The best one gets is an asymptotic series expression. However, the issue of extending the Chowla-Selberg formula or, better still, the more general one we have obtained before for inhomogeneous Epstein zeta functions in two indices, is not simple and had never been adressed in the literature. In order to obtain the analytic continuation to Re s ≤ 1 of the truncated inhomogeneous Epstein zeta function in P 2 2 −s two dimensions, ζEt (s; a, b, c; q) ≡ ∞ m,n=0 (am + bmn + cn + q) , we can proceed in two ways: either by a direct calculation that leads to the generalized Chowla-Selberg formula [18] or by using the formulas for the Epstein zeta function in one dimension (example 1) recurrently. In both cases the result is ζEt (s; a, b, c; q) ≡ s

(4a) ∼ Γ(s)

∞ X

∞ X

(am2 + bmn + cn2 + q)−s

m,n=0

bn (−1)m Γ(m + s) (2a)2m (∆n2 + 4aq)−m−s ζH −2m; m! 2a m,n=1

∞ X (−1)n Γ(n + s − 1)Bn b q 1−s − (s − 1)∆Γ(s − 1) n=0 n!



4aq ∆

−n

q −s π π Γ(s − 1/2) 1/2−s πq 1−s 1 √ + + + + q 4 a c Γ(s) 2(s − 1) ∆ 4 "   !s ∞ r   X q 1/4 π q 1 s−1/2 2 + k K 2πk √ s−1/2 Γ(s) a qa k=1 a r

aq + ∆ 

+

1/4 s

s

a π q∆

∞ X

!s

∞ X

k=1

k

s−1/2

Ks−1/2

!

r 



aq + 2πk ∆ r



r

s

q a 2π a q∆

X 2 4aq (2π)s k s−1/2 cos(πkb/a) d1−2s ∆ + 2 a d k=1 d|k 

(26)

1/4−s/2

!s

∞ X

k

s−1

Ks−1 4πk

k=1





s

r

aq ∆





πk 4aq Ks−1/2  ∆ + 2  . a d

This imposing formula is new too. The first series is in general asymptotic, but it converges for a wide range of values of the parameters. The second series is always asymptotic and contributes to the pole at s = 1. As in the case of the fundamental formula, Eq. (22), the pole structure is here explicitly, although much more elaborate. Apart from the pole at 8

s = 1, there is here a sequence of poles at s = ±1/2, −3/2, −5/2, . . .. Calculations similar to the ones above lead to the determination of the residua of the poles in this case by means of the same expressions as before. Acknowledgments. The author is indebted with Klaus Fredenhagen and Romeo Brunetti for enlightening discussions and with all the members of the II. Institut f¨ ur Theoretische Physik der Universit¨at Hamburg for the very kind hospitality. This investigation has been supported by the Alexander-von-Humboldt Foundation, DFG-CSIC, DGICYT and CIRIT.

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[10] P.G. Gilkey, Invariance theory, the heat equation and the Atiyah-Singer index theorem, Math. lecture series 11 (Publish or Perish Inc., Boston, Ma., 1984). [11] A. Schwarz, Commun. Math. Phys. 67, 1 (1979); A. Schwarz and Yu. Tyupkin, Nucl. Phys. B242, 436 (1984); A. Schwarz, Abstracts (Part II), Baku International Topological Conference Baku (1987); A. Schwarz, Lett. Math. Phys. 2, 247 (1978). [12] E. Witten, Commun. Math. Phys. 121, 351 (1989). [13] J. Dixmier, Existence de traces non normales, C.R. Acad. Sci. Paris 262, 1107 (1966). [14] A. Connes, Non-commutative geometry (Academic Press, New York, 1994). [15] M. Wodzicki, Invent. Math. 75, 143 (1984); Non-commutative residue. Chap. I. Fundamentals, in K-theory, arithmetic and geometry, Lect. Notes in Math. 1289, 320 (Springer, Berlin, 1987). [16] W. Kalau and M. Walze, Gravity, noncommutative geometry and the Wodzicki residue, gr-qc/9312031, to appear in J. Geom. and Phys. [17] T. Ackermann, A note on the Wodzicki residue, funct-an/9506006. [18] E. Elizalde, Ten physical applications of spectral zeta functions (Springer, Berlin, 1995). [19] P. Epstein, Math. Ann. 56, 615 (1903); 65, 205 (1907). [20] K. Kirsten and E. Elizalde, Phys. Lett. B365, 72 (1995). [21] E. Elizalde, J. Math. Phys. 35, 6100 (1994). [22] S. Chowla and A. Selberg, Proc. Nat. Acad. Sci. U.S.A. 35, 317 (1949). [23] E. Elizalde, J. Phys. A22, 931 (1989); E. Elizalde and A. Romeo, Phys. Rev. D40, 436 (1989); E. Elizalde, J. Math. Phys. 31, 170 (1990). [24] E. Elizalde, J. Math. Phys. 35, 3308 (1994). [25] S. Iyanaga and Y. Kawada, Eds., Encyclopedic dictionary of mathematics, Vol. II (The MIT press, Cambridge, 1977), p. 1372 ff.

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