AÂµ + âÂµÇ«, leads to ghost modes having the form (hereafter Î½ â¡ +ân2 â 3 ... (n2 â 1) Ï2. + Î»n . (3.5). Following [3], we now try to dia...

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arXiv:gr-qc/9506053v1 26 Jun 1995

THE PRESENCE OF BOUNDARIES

Giampiero Esposito1,2

1

Istituto Nazionale di Fisica Nucleare, Sezione di Napoli, Mostra d’Oltremare Padiglione

20, 80125 Napoli, Italy; 2

Dipartimento di Scienze Fisiche, Mostra d’Oltremare Padiglione 19, 80125 Napoli, Italy.

Abstract. This paper describes recent progress in the analysis of relativistic gauge conditions for Euclidean Maxwell theory in the presence of boundaries. The corresponding quantum amplitudes are studied by using Faddeev-Popov formalism and zeta-function regularization, after expanding the electromagnetic potential in harmonics on the boundary 3-geometry. This leads to a semiclassical analysis of quantum amplitudes, involving transverse modes, ghost modes, coupled normal and longitudinal modes, and the decoupled normal mode of Maxwell theory. On imposing magnetic or electric boundary conditions, flat Euclidean space bounded by two concentric 3-spheres is found to give rise to gaugeinvariant one-loop amplitudes, at least in the cases considered so far. However, when flat Euclidean 4-space is bounded by only one 3-sphere, one-loop amplitudes are gaugedependent, and the agreement with the covariant formalism is only achieved on studying the Lorentz gauge. Moreover, the effects of gauge modes and ghost modes do not cancel 1

Euclidean Maxwell Theory in the Presence of Boundaries each other exactly for problems with boundaries. Remarkably, when combined with the contribution of physical (i.e. transverse) degrees of freedom, this lack of cancellation is exactly what one needs to achieve agreement with the results of the Schwinger-DeWitt technique. The most general form of coupled eigenvalue equations resulting from arbitrary gauge-averaging functions is now under investigation.

To appear in: Proceedings of the Conference on Heat-Kernel Techniques and Quantum Gravity, Winnipeg, August 1994.

2

Euclidean Maxwell Theory in the Presence of Boundaries 1. Introduction

The analysis of Euclidean Maxwell theory in the presence of boundaries can be seen as the first step in the quantization program for gauge fields and gravitation in the presence of boundaries [1-4]. This investigation enables one to get a better understanding of different quantization techniques of field theories with first-class constraints, i.e. reduction to physical degrees of freedom before quantization, or Faddeev-Popov Lagrangian formalism, or Batalin-Fradkin-Vilkovisky extended-phase-space formalism. Motivations also come from the quantization of closed cosmologies, and from perturbative properties of supergravity theories [1]. The main choices in order are the quantization technique, the background 4-geometry, the boundary 3-geometry, the boundary conditions respecting Becchi-Rouet-Stora-Tyutin invariance and local supersymmetry, the gauge condition and the regularization technique [5]. Here we are interested in the mode-by-mode analysis of BRST-covariant FaddeevPopov amplitudes, which relies on the expansion of the electromagnetic potential in harmonics on the boundary 3-geometry. By using zeta-function regularization and flat Euclidean backgrounds, the effects of relativistic gauges are as follows [1-4]. (i) In the Lorentz gauge, the mode-by-mode analysis of one-loop amplitudes agrees with the results of the Schwinger-DeWitt technique, both in the 1-boundary case (i.e. the disk) and in the 2-boundary case (i.e. the ring). (ii) In the presence of boundaries, the effects of gauge modes and ghost modes do not cancel each other. 3

Euclidean Maxwell Theory in the Presence of Boundaries (iii) When combined with the contribution of physical degrees of freedom, i.e. the transverse part of the potential, this lack of cancellation is exactly what one needs to achieve agreement with the results of the Schwinger-DeWitt technique. (iv) Thus, physical degrees of freedom are, by themselves, insufficient to recover the full information about one-loop amplitudes. (v) Even on taking into account physical, non-physical and ghost modes, the analysis of relativistic gauges different from the Lorentz gauge yields gauge-invariant amplitudes only in the 2-boundary case. (vi) Gauge modes obey a coupled set of second-order eigenvalue equations. For some choices of gauge conditions it is possible to decouple such a set of differential equations, by means of two functional matrices which diagonalize the original operator matrix. (vii) For arbitrary choices of relativistic gauges, gauge modes remain coupled. The explicit proof of gauge invariance of quantum amplitudes becomes a problem in homotopy theory. Hence there seems to be a deep relation between the Atiyah-Patodi-Singer theory of Riemannian 4-manifolds with boundary [6], the zeta-function, and the BKKM function (section 5). Denoting by Φ(A) the gauge-averaging function appearing in the Faddeev-Popov action, and by ǫ the ghost field [1-2], magnetic boundary conditions take the form

Φ(A)|∂M = 0

,

Ak |∂M = 0

4

,

ǫ|∂M = 0

,

(1.1)

Euclidean Maxwell Theory in the Presence of Boundaries while electric boundary conditions are

A0 |∂M = 0

,

∂ǫ |∂M = 0 ∂n

,

∂Ak |∂M = 0 ∂τ

.

(1.2)

Following [1-5], the boundary 3-geometries are taken to be 3-spheres. The normal and tangential components of the electromagnetic potential on a family of 3-spheres are given by [1-5] A0 (x, τ ) =

∞ X

Rn (τ )Q(n) (x)

,

(1.3)

n=1

∞ X (n) (n) Ak (x, τ ) = fn (τ )Sk (x) + gn (τ )Pk (x)

,

(1.4)

n=2

(n)

(n)

where Q(n) (x), Sk (x), Pk (x) are scalar, transverse and longitudinal vector harmonics on S 3 respectively. Section 2 is a brief summary of my early work on relativistic gauge conditions for Euclidean Maxwell theory [1-2]. Section 3, following [3], solves the technical problems of section 2, i.e. how to decouple gauge modes and how to evaluate the full ζ(0). Section 4, relying on [4], studies coupled eigenvalue equations for arbitrary gauge-averaging functions. Concluding remarks are presented in section 5.

5

Euclidean Maxwell Theory in the Presence of Boundaries 2. Relativistic gauge conditions for Euclidean Maxwell theory

In my early work on Euclidean Maxwell theory [1-2], I studied a gauge-averaging function defined as (K being the extrinsic-curvature tensor of the boundary) ΦE (A) ≡ =

∂A0 (3) i + ∇ Ai = (4) ∇µ Aµ − A0 Tr K ∂τ ∞ X

n=1

R˙ n (τ )Q(n) (x) − τ −2

∞ X

gn (τ )Q(n) (x)

,

(2.1)

n=2

since I wanted to obtain the 1-dimensional Laplace operator acting on the decoupled mode R1 , and I was interested in relativistic gauges different from the Lorentz gauge. After integration by parts one then finds that, ∀n ≥ 2, on defining the operators 1 d (n2 − 1) d2 b + An (τ ) ≡ − 2 − dτ τ dτ ατ 2 bn (τ ) ≡ 1 B α

,

d2 (n2 − 1) 3 d − 2− + dτ τ dτ τ2

(2.2)

,

(2.3)

the part of the Euclidean action quadratic in coupled gauge modes becomes [1-2] Z Z 1 1 3 1 1 τ gn (n) b bn Rn dτ An gn dτ + τ Rn B IE (g, R) = 2 0 (n2 − 1) 2 0

1 + 1− α

Z

1

τ gn R˙ n dτ +

0

Z

1

gn Rn dτ

,

(2.4)

0

after setting to 1 the 3-sphere radius in the 1-boundary problem. This leads to the coupled eigenvalue equations [1-2] g˙ n λn (n2 − 1) 1 τ −¨ gn − τ R˙ n + Rn = 2 + gn + 1 − τ gn 2 2 (n − 1) τ ατ α (n − 1) 6

,

(2.5)

Euclidean Maxwell Theory in the Presence of Boundaries 3 ˙ (n2 − 1) gn 1 3 1 ¨ − Rn − Rn + + Rn − τ g˙ n 1 − = λn τ 3 Rn τ 2 α τ τ α α

.

(2.6)

The boundary conditions are regularity at the origin, i.e. gn (0) = Rn (0) = 0 ∀n ≥ 2, and magnetic conditions on S 3 : gn (1) = R˙ n (1) = 0 ∀n ≥ 2, or electric conditions on S 3 : g˙ n (1) = Rn (1) = 0 ∀n ≥ 2. I could then find power-series solutions in the form

gn (τ ) = τ

µ

∞ X

an,k (n, k, λn )τ k

,

(2.7)

k=0

Rn (τ ) = τ

µ−1

∞ X

bn,k (n, k, λn )τ k

,

(2.8)

k=0

where regular solutions are obtained for µ = q + n2 − (1)

3 4

(1) µ+

q = + n2 −

3 4

+

1 2

(2)

=

(1)

=

or µ = µ+

− 21 , while singular solutions (here discarded) correspond to µ = µ− (2)

(2)

−µ+ , µ = µ− = −µ+ . The decoupled mode R1 was found to give the contributions − 41 and − 43 to ζ(0) in the magnetic and electric cases respectively, while the contribution of ghost modes was obtained by applying the zeta-function at large x [1-2]:

ζ(s, x2 ) ≡

∞ ∞ X X

n=n0 m=m0

λn,m + x2

−s

.

By virtue of the gauge choice (2.1), the gauge transformation on the potential:

(2.9)

ǫ

Aµ ≡

q ǫn (τ ) = Aµ + ∇µ ǫ, leads to ghost modes having the form (hereafter ν ≡ + n2 − 43 ) e √

√ τ Jν ( E τ ). This is proved after evaluating the difference ΦE (A) − Φ(ǫ A) which leads to 7

Euclidean Maxwell Theory in the Presence of Boundaries a second-order operator whose eigenfunctions are proportional to Bessel functions of noninteger order. Referring the reader to [1-2] and to the appendix for a detailed treatment of how to evaluate ζ(0) out of the zeta-function at large x, we just state that, in our case, after defining αν (x) ≡

√

ν 2 + x2 , the contribution to ζ(0) resulting from the ghost can be

obtained as -2 times half the coefficient of x−6 in the asymptotic expansion Γ(3)ζ(3, x2 ) ∼ σ1 + σ2

,

(2.10)

where [1-2] σ1 ∼

∞ X

n

2

n=0

h

1 2 −4 −3 − νx−6 + ν 2 x−6 α−1 αν ν + ν x 2

1 −6 3 −5 i 3 − α + αν + ν 2 x−2 α−5 ν 8 2 ν 8

σ2 ∼ −

×

∞ X l X l=1 r=0

∞ X

alr

l r+ 2

,

(2.11)

l l r+ +1 r+ +2 2 2

n2 ν 2r αν−(l+2r+6)

.

(2.12)

n=0

By using suitable contour formulae, and re-expressing αν (x) in terms of αn (x) ≡

√

n 2 + x2 ,

I was able to evaluate all ghost contributions to ζ(0), but the one resulting from the first term on the right-hand side of (2.11). It was therefore necessary to express coupled gauge modes in a more convenient form after decoupling them, and to complete the calculation for the ghost field. For this purpose, I started a collaboration with Dr. Kamenshchik and our students (section 3).

8

Euclidean Maxwell Theory in the Presence of Boundaries 3. Decoupling gauge modes and evaluating ζ(0)

The system (2.5)-(2.6) is more conveniently re-expressed in the form (we choose α = 1 in this section) Abn gn + Bbn Rn = 0

,

(3.1)

b n Rn = 0 Cbn gn + D

,

(3.2)

where 1 d (n2 − 1) d2 − + λn Abn ≡ 2 + dτ τ dτ τ2 (n2 − 1) Bbn ≡ − τ

,

1 Cbn ≡ − 3 τ

2 2 bn ≡ d + 3 d − (n − 1) + λn D dτ 2 τ dτ τ2

,

(3.3)

,

(3.4)

.

(3.5)

Following [3], we now try to diagonalize the system (3.1)-(3.2) by introducing the operator matrix (n) Oij

≡ =

1 Wn

ˆ + V Cˆ + V Dβ ˆ Aˆ + Bβ ˆ + Cˆ + Dβ ˆ W Aˆ + W Bβ

Vn 1

Abn Cbn

Bbn bn D

1 βn

αn 1

ˆ +B ˆ + V Cα ˆ +VD ˆ Aα ˆ + WB ˆ + Cα ˆ +D ˆ W Aα

.

(3.6)

n

The basic idea is that the functions αn and βn should create the linear combinations of decoupled modes, while the functions Vn and Wn should select decoupled equations.

9

Euclidean Maxwell Theory in the Presence of Boundaries (n)

Setting to zero the off-diagonal matrix elements of Oij one finds the equation for αn dαn αn Vn d d2 αn 1 dαn d2 + +3 + + (αn + Vn ) 2 + 2 dτ dτ τ τ dτ dτ 2 τ dτ (n2 − 1) (n2 − 1) αn Vn − (αn + Vn ) + λn (αn + Vn ) − − =0 τ2 τ τ3

,

(3.7)

solved by [3] 1 αn (τ ) = − ± ν τ = −Vn (τ ) 2

,

(3.8)

and an equation for βn solved by 1 βn (τ ) = (ν + 1/2)(ν − 1/2)

1 ±ν 2

1 = −Wn (τ ) τ

.

(3.9)

Choosing the opposite signs in the round brackets of (3.8)-(3.9), the corresponding diagonal matrix elements are Bessel operators multiplied by 2ν/(ν + 1/2). Thus, in the 2-boundary problem, where both I- and K-functions are admissible solutions, one finds decoupled modes in the form [3] gn (τ ) = C1 Iν− 12 (M τ ) + C2

+ C3 Kν− 12 (M τ ) + C4

1 Rn (τ ) = τ

C1

1 ν− 2

1 ν− 2

Iν+ 21 (M τ )

Kν+ 21 (M τ )

,

(3.10)

−1 I 1 (M τ ) + C2 Iν+ 21 (M τ ) (ν + 1/2) ν− 2

−1 K 1 (M τ ) + C4 Kν+ 12 (M τ ) +C3 (ν + 1/2) ν− 2

10

,

(3.11)

Euclidean Maxwell Theory in the Presence of Boundaries since the diagonal matrix elements are

(n) O11

(n) O22

2ν = (ν + 1/2)

2ν = (ν + 1/2)

d2 1 d (ν − 1/2)2 + − + λn dτ 2 τ dτ τ2

d2 3 d ((ν + 1/2)2 − 1) + − + λn dτ 2 τ dτ τ2

,

(3.12)

.

(3.13)

In the case of magnetic boundary conditions at two 3-spheres of radii τ− and τ+ respectively, the gauge-averaging function (2.1) leads to (see (1.1)) gn (τ− ) = gn (τ+ ) = 0, R˙ n (τ− ) = R˙ n (τ+ ) = 0, ∀n ≥ 2. The Barvinsky-Kamenshchik-Karmazin-Mishakov formalism, described by Dr. Kamenshchik in this same volume, can be now applied. For coupled gauge modes, the Ilog value vanishes, while the Ipole (∞) value is the coefficient of expansion of cient of

1 n

n2 2

log

h

4ν 2 (ν+1/2)2

i

in the expansion of

1 n

in the

as n → ∞. The Ipole (0) value is instead given by the coeffin2 2

log

h

(ν−1/2) (ν+1/2)

i

as n → ∞. Remarkably, in the 2-boundary

problem one finds Ipole (∞) = Ipole (0) = − 11 48 , which implies that coupled gauge modes give a vanishing contribution to the full ζ(0). Along the same lines, one finds that, for ghost modes, Ilog = Ipole (∞) = Ipole (0) = 0. For transverse modes, Ipole (∞) = Ipole (0) = 0, while Ilog =

∞ X 2(n2 − 1) 1 (−1) = ζR (0) − ζR (−2) = − 2 2 n=2

.

(3.14)

Last, but not least, the decoupled normal mode R1 (τ ) = C1 τ1 I1 (M τ ) + C2 τ1 K1 (M τ ), contributes

1 2

to ζ(0). Hence the full ζ(0) vanishes in the 2-boundary problem about flat

Euclidean backgrounds: 1 1 ζ(0) = − + = 0 2 2 11

.

(3.15)

Euclidean Maxwell Theory in the Presence of Boundaries 4. Coupled eigenvalue equations for arbitrary gauge-averaging functions

Within the Faddeev-Popov formalism, the study of arbitrary gauge conditions is equivalent to the introduction of a gauge-averaging function in the form (the boundary being given by 3-spheres) Φ(A) ≡ γ1 (4) ∇0 A0 + =

γ2 A0 Tr K − γ3 (3) ∇i Ai 3

R1 γ1 R˙ 1 + γ2 τ

Q(1) (x)

∞ X g R n n + γ3 2 Q(n) (x) + γ1 R˙ n + γ2 τ τ n=2

.

(4.1)

This Φ(A) should be inserted in the Faddeev-Popov Euclidean action [2,4]

IeE ≡ Igh +

Z

M

1 [Φ(A)]2 p µν Fµν F + det g d4 x 4 2α

,

(4.2)

and one may distinguish 7 different cases [4]. We here focus on the most general choice for Φ(A), when the dimensionless parameters γ1 , γ2 , γ3 are all different from zero. Thus, defining ρ≡1+

γ1 γ3 α

,

(4.3)

µ≡1+

γ2 γ3 α

,

(4.4)

the operators appearing in a system of the kind (3.1)-(3.2) now take the form [4] 1 d γ 2 (n2 − 1) d2 − 3 + λn Abn ≡ 2 + dτ τ dτ α τ2 12

,

(4.5)

Euclidean Maxwell Theory in the Presence of Boundaries (n2 − 1) d −µ Bbn ≡ −ρ(n2 − 1) dτ τ 1 γ3 ρ d γ1 − γ2 3 + Cbn ≡ 2 τ dτ α τ

,

(4.6)

,

(4.7)

i 1 hγ γ12 d2 3γ12 1 d 2 2 b Dn ≡ 2γ1 − γ2 − (n − 1) 2 + λn + + α dτ 2 α τ dτ α τ

.

(4.8)

(n)

If one now tries to set to zero the off-diagonal matrix elements of Oij (cf. (3.6)), one finds the following systems of equations (hereafter γ1 = 1 for simplicity [4]):

Vn + αn = 0

,

(4.9)

αn + α3 Vn dαn 1 dVn 2 +2 1− + − ρ(n2 − 1) = 0 dτ α dτ α d2 αn + dτ 2 +

1 ρVn + 2 τ τ

,

(4.10)

,

(4.11)

dαn γ 2 (n2 − 1) µ − 3 αn − (n2 − 1) 2 dτ α τ τ

hγ iV γ3 1 2 n (1 − γ2 )Vn αn 3 + (2 − γ2 ) − (n2 − 1) 2 = 0 α τ α τ Wn + βn = 0

,

Wn + α3 βn dβn ρ 2 + + 2 =0 dτ τ τ 1 d2 βn + α dτ 2

31 − ρ(n2 − 1)Wn ατ

(4.12)

,

(4.13)

1 dβn − µ(n2 − 1)Wn βn dτ τ

γ32 2 1 γ2 2 2 − γ2 − (n − 1) βn − (n − 1)Wn 2 α α τ

+

+

1 γ3 (1 − γ2 ) 3 = 0 α τ

.

(4.14)

13

Euclidean Maxwell Theory in the Presence of Boundaries Remarkably, Eqs. (4.9)-(4.10) are solved by

αn (τ ) =

α ρ(n2 − 1)τ + α0,n τ (3−α)/2 (α − 1)

,

(4.15)

and (4.15) is also a solution of (4.11), at least in the limit α → ∞, which yields αn (τ ) ∼ (n2 − 1)τ

.

(4.16)

By contrast, (4.12)-(4.13) are solved by

βn (τ ) =

3 1 α ρ + β0,n τ 2 (1− α ) (α − 1) 3τ

,

(4.17)

but (4.17) is not a solution of (4.14), not even in the limit α → ∞, which yields βn (τ ) ∼

√ 1 + β0,n τ 3τ

.

(4.18)

These limiting properties reflect the impossibility to find solutions for both αn (τ ) and βn (τ ) for arbitrary gauge parameters γ1 , γ2 , γ3 and α. Hence gauge modes cannot be decoupled for arbitrary choices of gauge-averaging functions [4].

5. Concluding remarks

The main open problem seems to be the explicit proof of gauge invariance of one-loop amplitudes for relativistic gauges, in the case of flat Euclidean space bounded by two concentric 3-spheres. For this purpose, one may have to show that, for coupled gauge modes, Ilog and the difference Ipole (∞) − Ipole(0) are not affected by a change in the gauge 14

Euclidean Maxwell Theory in the Presence of Boundaries parameters γ1 , γ2 , γ3 , α (section 4). Although this is what happens in the particular cases studied so far [3-4], at least 3 technical achievements are necessary to obtain a rigorous proof, i.e. (1) To relate the regularization at large x of section 2 to the BKKM regularization, based on the BKKM function [3-5]:

2

I(M , s) ≡

∞ X

n=n0

h i d(n) n−2s log fn (M 2 )

,

(5.1)

where d(n) is the degeneracy of the eigenvalues parametrized by the integer n, and fn (M 2 ) is the function occurring in the equation obeyed by the eigenvalues by virtue of boundary conditions, after taking out fake roots. (2) To evaluate Ilog from an asymptotic analysis of coupled eigenvalue equations. (3) To evaluate Ipole (∞) − Ipole (0) by relating the analytic continuation to the whole complex-s plane of the difference I(∞, s) − I(0, s) (see (5.1)) to the analytic continuation of the zeta-function. If this last step can be performed, it may involve a non-local, integral transform relating the BKKM function (5.1) to the zeta-function, and a non-trivial application of the Atiyah-Patodi-Singer theory of Riemannian 4-manifolds with boundary [6]. In other words, one might have to prove that, in the 2-boundary problem only, Ipole (∞) − Ipole (0) resulting from coupled gauge modes is the residue of a meromorphic function, invariant under a smooth variation (in γ1 , γ2 , γ3 , α) of the matrix of elliptic self-adjoint operators appearing in (4.5)-(4.8). 15

Euclidean Maxwell Theory in the Presence of Boundaries Other problems are the mode-by-mode analysis of curved backgrounds, and a deeper understanding of why, in the 1-boundary case, one-loop amplitudes are gauge-dependent [3-4]. So far, this undesirable property seems to hold since relativistic gauges different from the Lorentz gauge involve explicitly the trace of the extrinsic-curvature tensor of the boundary, and hence are ill-defined at the origin of flat Euclidean 4-space, where a smooth vector field matching the normal at the boundary cannot be defined. It should be emphasized that the mode-by-mode analysis appearing in [3] has led to the (first) correct calculation of the conformal anomaly for spin-1 fields in the Lorentz , as confirmed in gauge about flat Euclidean 4-space bounded by a 3-sphere, i.e. ζ(0) = − 31 90 [7], where the same ζ(0) value has been obtained by using the Schwinger-DeWitt technique and the recent results appearing in [8]. Our ζ(0) values in the 2-boundary case all coincide with the Schwinger-DeWitt value, as well [3-4]. Even more recently, the mode-by-mode analysis of non-relativistic gauges has been initiated by myself and Dr. Kamenshchik [5]. In that case, boundary conditions are quite different from (1.1)-(1.2), since the modes for the normal component A0 of the potential are not subject to any boundary condition [5]. Still, the resulting ζ(0) value agrees with the prediction of the relativistic analysis, at least in the 2-boundary problem about flat Euclidean backgrounds. The results and open problems presented so far seem to strengthen the evidence in favour of the field-theory quantization program for manifolds with boundary being able to shed new light on the consistency or the limits of modern quantum field theories. Its

16

Euclidean Maxwell Theory in the Presence of Boundaries ultimate consequence might also be a better understanding of the boundary conditions relevant for quantum cosmology [1].

Acknowledgments

I am indebted to A. Yu. Kamenshchik, I. V. Mishakov and G. Pollifrone for collaboration on many topics described in my contribution to this volume.

Appendix

The zeta-function regularization at large x used in [1-2] relies on the following properties. For problems with boundaries, the eigenfunctions are usually expressed in terms of Bessel functions. By virtue of the boundary conditions, a linear (or non-linear) combination of Bessel functions is set to zero. Denoting by Fp the function occurring in this eigenvalue condition, and using the zeta-function at large x defined in (2.9), one has the identity

2

Γ(3)ζ(3, x ) =

∞ X

Np

p=0

1 d 2x dx

3

−p log (ix) Fp (ix)

,

(A.1)

where Np is the corresponding degeneracy. On the other hand, by virtue of the asymptotic expansion G(t) ∼

∞ X

n

Bn t 2 −2

n=0

17

t → 0+

(A.2)

Euclidean Maxwell Theory in the Presence of Boundaries of the integrated heat kernel G(t), one finds

2

Γ(3)ζ(3, x ) =

Z

∞

0

2 −x2 t

t e

G(t) dt ∼

n −n−2 x Bn Γ 1 + 2 n=0 ∞ X

.

(A.3)

Thus, by comparison, one finds that ζ(0) = B4 is half the coefficient of x−6 in the uniform asymptotic expansion of the right-hand side of (A.1).

References

[1] Esposito G. (1994) Quantum Gravity, Quantum Cosmology and Lorentzian Geometries, Lecture Notes in Physics, New Series m: Monographs, Vol. m12, second corrected and enlarged edition (Berlin: Springer-Verlag). [2] Esposito G. (1994) Class. Quantum Grav. 11, 905. [3] Esposito G., Kamenshchik A. Yu., Mishakov I. V. and Pollifrone G. (1994) Class. Quantum Grav. 11, 2939. [4] Esposito G., Kamenshchik A. Yu., Mishakov I. V. and Pollifrone G. Relativistic Gauge Conditions in Quantum Cosmology (DSF preprint 95/8, to appear in Phys. Rev. D). [5] Esposito G. and Kamenshchik A. Yu. (1994) Phys. Lett. B 336, 324. [6] Atiyah M. F., Patodi V. K. and Singer I. M. (1976) Math. Proc. Camb. Phil. Soc. 79, 71. [7] Moss I. G. and Poletti S. J. (1994) Phys. Lett. B 333, 326. [8] Vassilevich D. V. (1995) J. Math. Phys. 36, 3174.

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