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UB-ECM-PF 95/1

arXiv:hep-th/9501064v1 16 Jan 1995

Heat-kernel coe?cients of the Laplace operator on the 3-dimensional ball

M. Bordag Universit¨ at Leipzig, Institut f¨ ur Theoretische Physik, Augustusplatz 10, 04109 Leipzig, Germany K. Kirsten? Departament d’ECM, Facultat de F? ?sica Universitat de Barcelona, Av. Diagonal 647, 08028 Barcelona Spain January 10, 1995

Abstract We consider the heat-kernel expansion of the massive Laplace operator on the three dimensional ball with Dirichlet boundary conditions. Using this example, we illustrate a very e?ective scheme for the calculation of an (in principle) arbitrary number of heat-kernel coe?cients for the case where the basis functions are known. New results for the coe?cients B 5 , ..., B5 are presented.

2

?

Alexander von Humboldt foundation fellow, E-mail address: klaus@zeta.ecm.ub.es

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It is important to know the explicit form for the coe?cients in the short-time expansion of the heat-kernel, K (t), for some Laplacian-like operator on a d-dimensional manifold M. In mathematics the interest stems for example from connections between the heatequation and the Atiyah-Singer index theorem [1], whereas in physics the interest in this expansion lies for example in the domain of quantum ?eld theory where it is commonly known as the (integrated) Schwinger-De Witt proper-time expansion [2]. If the manifold M has a boundary ? M, the coe?cients Bn in the short time expansion have volume and boundary parts [3]. Thus K (t) ? (4πt)? 2 with Bk =

M

d

∞ k =0,1/2,1,...

Bk tk

(1)

dV bn +

?M

dS cn .

(2)

For the volume part e?ective systematic schemes have been developed (see for example [4]). The calculation of cn is in general more di?cult. Only relatively recently the coef?cient c2 for Dirichlet and Neumann boundary conditions have been found [5, 6]. When using the general formalism of ref. [5] for higher spin particles, Moss and Poletti [7] found a discrepancy with the direct calculations of D’Eath and Esposito [8] (see also [9]). The latter results have been con?rmed in [10] where a new systematic scheme for the calculation of c2 has been developed in the context of the Hartle-Hawking wave-function of the universe for the case when the full set of basis functions is known [10]. Finally, very recently the discrepancy have been resolved [11] and now the results found using the general algorithm [12] are in agreement with the direct calculations [8, 9, 10]. We will use a variant of the approach of ref. [10] in order to show that higher coe?cients cn may be calculated very e?ectively for the case that the basis functions are known. To illustrate the method in this letter we will concentrate on the calculation of the heat-kernel coe?cients of the elliptic operator

2 (?? + m2 )φn,l,m = (λ2 n,l,m + m )φn,l,m

(3)

in the three dimensional ball de?ned by BR = {x ∈ IR3 , |x| ≤ R}. We will treat explicitly Dirichlet boundary conditions, φn,l,m(|x| = R) = 0. As will be clear afterwards, other boundary conditions and higher dimensional balls may be treated in exactly the same way. Starting point is the equation Jl+ 1 (λn,l,m R) = 0

2

(4)

for the eigenvalues λn,l,m, which are (2l + 1)-times degenerated. We are especially interested in the calculation of the heat-kernel coe?cients de?ned in equation (1). Instead of

1

calculating the heat-kernel coe?cients itself, we will concentrate on the zeta function of the operator, eq. (3), and recover the heat-kernel coe?cients using the relations [13] Res ζ (s) = for s =

m m?1 +1 , 2 , ..., 1 ; ? 2l2 , 2 2

Bm ?s 2 m (4π ) 2 Γ(s)

(5)

for l ∈ IN0 , and ζ (?p) = (?1)p p! Bm +p 2 (4π ) 2

m

(6)

for p ∈ IN0 . Using relation (4) for the eigenvalues, one may write the zeta function in the form (for a similar procedure see [14]) ζ (s ) = =

∞ ∞ l 2s λ? n,l,m

n=0 l=0 m=?l ∞

(2l + 1)

l=0

γ

dk 2 ? (k + m2 )?s ln Jl+ 1 (kR), 2 2πi ?k

(7)

where the contour γ is counterclockwise enclosing all eigenvalues which are known to be 3 situated on the positive real axis. As it stands, the representation (7) is valid for ?s > 2 . Before considering in detail the l-summation, let us ?rst construct an analytic continuation of the k -integral in equation (7) alone and let us de?ne for that reason ζ ν (s ) =

γ

dk 2 ? (k + m2 )?s ln Jν (kR), 2πi ?k

(8)

with ν = l + 1/2. Deforming the contour to the imaginary axis, the analytic continuation, sin(πs) ζ ν (s ) = π

∞ mR

valid in the strip 1 < ?s < 1, may be found. A similar representation valid for m = 0 2 has been given in [15]. In order to continue, the idea is to make use of the uniform expansion of the Bessel function Iν (k ) for ν → ∞ as z = k/ν ?xed [16]. Actually we use the expansion ? 1 ln Iν (k ) ? √ ?k 1 ? t2 with t =

√ 1 . 1+z 2

k dk ? R

?

2

? m2 ?

??s

? ln Il+ 1 (k ), 2 ?k

(9)

dn (t) n n=0 ν

∞

(10)

Here the functions dn (t) ful?ll the recurrence relation (11)

? 1 n?1 1 dn (t) = t(1 ? t2 ) t ? 1 dn?1(t) ? dk (t)dn?k (t), 2 ?t 2 k=1 2

starting with d0 (t) = 1. In order to calculate up to B5 one needs the ?rst eleven coe?cients dn , which can be easily calculated using the recurrence. Adding and subtracting the leading terms of the asymptotic expansion for ν → ∞, eq. (9) may be split into two pieces,

N +1

ζν (s) = Nν (s) +

i=1

Ai ν (s ),

(12)

with sin(πs) Nν (s) = π and Ai ν (s ) sin(πs) = i?1 ν π

∞ mR/ν ∞ mR/ν

kν dk ? R

?

2

? m2 ?

?

? ?s

1 ? ln Il+ 1 (kν ) ? √ 2 ?k 1 ? t2

dn (t) n n=0 ν

N

(13)

kν dk ? R

2

?m

2?

??s

? di?1 (t). ?k

(14)

As it stands, the Ai ν , equation (14), are well de?ned (at least) in the strip 1/2 < ?s < 1. However, the analytic continuation in the parameter s to the whole complex plane in terms of known function may be provided. To explain afterwards some details of the calculation let us give explicitly only the Ai ν of the two leading terms in the asymptotics (10), A1 ν (s ) = m?2s sin(πs) 2π 2 ×

3

Rm

2

1 1 1 1 ν Γ(1 ? s) 2 F1 ? , s ? ; ; ? 2 2 2 2 mR ?2s m sin(πs) A2 ν (s ) = ? 4π ν 2 × Γ(s)Γ(1 ? s) 2 F1 1, s; 1; ? . mR Γ s?

,

(15)

(16)

Similar expressions for higher Ai ν can be calculated, e.g. using standard integration packages. As mentioned, we did explicitly calculate the coe?cients up to B5 and thus needed Ai ν for i = 1, ..., 11. Here 2 F1 (a, b, c; z ) denotes the hypergeometric function [17]. The representation (12) has the following very important properties. First of all, by considering the asymptotics of the integrand in equation (13) for k → mR/ν and k → ∞, it may be seen that N (s ) =

∞ l=0

(2l + 1)Nl+ 1 (s)

2

3

is analytic in the strip 1 ? N/2 < ?s < 1. For that reason it gives no contribution to the residue of ζ (s) in that strip. Furthermore, for s = ?k , k = 0, 1, 2, 3, we have N (s) = 0 and thus it does also not contribute to the values of the zeta function at those points. Together with eqs. (5), (6), this yields, that the heat-kernel coe?cients are only determined by the terms Ai (s) with Ai (s) =

∞ l=0

(2l + 1)Ai l + 1 (s ).

2

(17)

However, the Ai (s) may be given in terms of Hurwitz zeta functions and an explicit representation of Ai (s) showing the meromorphic structure in the whole complex plane may be given. The sum in (17) may be easily done by means of the Mellin-Barnes type integral representation of the hypergeometric function,

2 F1 (a, b; c; z )

=

1 Γ(c) Γ(a)Γ(b) 2πi

C

dt

Γ(a + t)Γ(b + t)Γ(?t) (?z )t , Γ(c + t)

(18)

where the contour C is such that the poles of Γ(a + t) and Γ(b + t) lie to the left of it and the poles of Γ(?t) to the right [17]. De?ning h(a, b, c; n) =

∞ l=0

1 l+ 2

n

1 l+ 2 ? F a, b ; c ; ? 2 1 ?

?

2

? ?

(19)

and closing the contour C to the left, we arrive at h(a, b, c; n) = (?1)k 2k ? × k! k =0 Γ(b ? a ? k )Γ(a + k ) 1 ζH ?n + 2a + 2k ; Γ(c ? a ? k ) 2 1 Γ(a ? b ? k )Γ(b + k ) ζH ?n + 2b + 2k ; +?2b Γ(c ? b ? k ) 2 ?2a

∞

(20)

,

which may be used for all summations we need for the calculation of the Ai ‘s. For example we obtain A1 = 1 1 1 1 Γ(1 ? s)h ? , s ? , ; 1 2 2 2 2 π 1 ∞ Γ l+s? 2 (?1)l 1 sin(πs) Γ(1 ? s ) , (mR)2l ζH 2l + 2s ? 2; = R2s 3 l! s+l 2 2π 2 l=0

3 2

m?2s sin(πs)

RmΓ s ?

A2 = ?

m?2s sin(πs) Γ(s)Γ(1 ? s)h(1, s, 1; 1) 2π ∞ sin(πs) 1 (?1)l , = ?R2s Γ(1 ? s) (mR)2l Γ (l + s) ζH 2l + 2s ? 1; 2π l! 2 l=0 4

and similar expressions for the other Ai ’s, i = 1, ..., 11. The sums appearing in the Ai (s) are convergent for |mR| < 1/2. Using this representation, equations (5), (6), together with ζH 1 + ?; 1 + O(?0 ), ? 1 (?1)n Γ(? ? n) = + O(?0 ), ? n! 1 2 =

the heat-kernel coe?cients may be easily determined. Summarizing, we ?nd the following new results for the coe?cients B 5 , ..., B5 ,

2

B5 = π 2

2

B3 = B7 =

2

B4 =

B9 =

2

B5 =

1 m2 ? ? m4 R2 6 120R2 64 16m2 4m4 R 2m6 R3 π ? + + ? 9009R3 315R 3 9 47 m2 m6 R2 m4 3 ? + + π2 ? 12 20160R4 120R2 3 2 4 202816 64m 8m π ? + ? 5 3 72747675R 9009R 315R 4m6 R m8 R3 + ? 9 18 6 3 m 521 47m2 π2 ? + 36 443520R6 20160R4 m8 R2 m4 ? ? 240R2 12 25426048 202816m2 32m4 π ? + ? 15058768725R7 72747675R5 9009R3 8m6 m8 R m10 R3 + + ? 945R 9 90

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For m = 0 the coe?cient B5/2 agrees with the one found by Kennedy [18]. As mentioned, other boundary conditions, higher dimensional balls, and even higher coe?cients may be found in exactly the same way without any additional complication. This and more details of the calculation will be given in a separate publication.

Acknowledgments

It is a pleasure to thank S. Leseduarte, E. Elizalde, P. Gilkey, S. Dowker and G. Esposito for interesting discussions and helpful comments. K.K. thanks the Department ECM of the University of Barcelona for their warm hospitality. Furthermore, K.K. acknowledges ?nancial support from the Alexander von Humboldt Foundation (Germany). 5

References

[1] P.B. Gilkey, Invariance theory, the heat-equation and the Atiyah-Singer index theorem, (Publish or Perish, Wilmington, 1984). [2] N. Birrell and P.C.W. Davies, Quantum ?elds in curved space, (Cambridge University Press, Cambridge, 1982). B.S. De Witt, Relativity, groups and topology, (Gordon and Breach, New York, 1965). [3] P. Greiner, Arch. Rat. Mech. Anal. 41 (1971) 163. P.B. Gilkey, J. Di?. Geom. 10 (1976) 601. [4] I.G. Avramidi, Nucl. Phys. B 355 (1991) 712. S.A. Fulling and G. Kennedy, Transac. Amer. Math. Soc. 310 (1988) 583. P. Amsterdamski, A.L. Berkin and D.J. O’Connor, Class. Quantum Grav. 6 (1989) 1981. [5] T.P. Branson and P.B. Gilkey, Commun. Part. Di?. Eqs. 15 (1990) 245. [6] I.G. Moss and J.S. Dowker, Phys. Lett. B 229 (1989) 261. D.M. Mc Avity and H. Osborn, Class. Quantum Grav. 8 (1991) 603. A. Dettki and A. Wipf, Nucl. Phys. B 377 (1992) 252. J.S. Dowker and J.P. Scho?eld, J. Math. Phys. 31 (1990) 808. G. Cognola, L. Vanzo and S. Zerbini, Phys. Lett. B 241 (1990) 381. [7] I.G. Moss and S. Poletti, Nucl. Phys. B 341 (1990) 155. I.G. Moss and S. Poletti, Phys. Lett. B 245 (1990) 355. [8] P.D. D’Eath and G. Esposito, Phys. Rev. D 43 (1991) 3234. [9] G. Esposito, Class. Quantum Grav. 11 (1994) 905. G. Esposito, A.Yu. Kamenshchik, I.V. Mishakov and G. Pollifrone, Class. Quantum Grav. 11 (1994) 2939. G.Esposito, A.Yu. Kamenshchik, I.V. Mishakov and G. Pollifrone, Phys. Rev. D 50 (1994) 6329. G. Esposito, Quantum gravity, quantum cosmology and Lorentzian geometries, Second corrected and enlarged edition, (Lecture Notes in Physics, New Series m: Monographs, Vol. m12, Springer-Verlag, Berlin, 1994). [10] A.Yu. Kamenshchik and I.V. Mishakov, Int. J. Mod. Phys. A 7 (1992) 3713. A.O. Barvinsky, A.Yu. Kamenshchik and I.P. Karmazin, Ann. Phys. 219 (1992) 201. [11] D.V. Vassilevich, Vector ?elds on a disk with mixed boundary conditions, (St. Petersburg preprint SPBU-94-6, gr-qc/9404052, to appear in J. Math. Phys. [12] I.G. Moss and S. Poletti, Phys. Lett. B 333 (1994) 326.

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[13] A. Voros, Commun. Math. 110 (1987) 439. [14] M. Bordag, Vacuum energy in smooth background ?elds, Preprint NTZ 12-94, Leipzig University, J. Phys. A: Math. Gen., to appear. [15] E. Elizalde, S. Leseduarte and A. Romeo, J. Phys. A: Math. Gen. 26 (1993) 2409. S. Leseduarte and A. Romeo, J. Phys. A: Math. Gen. 27 (1994) 2483. [16] M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions (Natl. Bur. Stand. Appl. Math. Ser. 55) (Washington, DC: US GPO) (1972 reprinted by New York: Dover). [17] I.S. Gradshteyn and I.M. Ryzhik, Tables of Integrals, Series and Products, Academic Press, New York, (1965). [18] G. Kennedy, J. Phys. A: Math. Gen. 11 (1978) L173.

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