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Spectral Boundary Conditions in One-Loop Quantum Cosmology

For fermionic fields on a compact Riemannian manifold with boundary one has a choice between local and non-local (spectral) boundary conditions. The one-loop prefactor in the Hartle-Hawking amplitude in quantum cosmology can then be studied using the generalized Riemann zeta-function formed from the squared eigenvalues of the four-dimensional fermionic operators. For a massless Majorana spin-1/2 field, the spectral conditions involve setting to zero half of the fermionic field on the boundary, corresponding to harmonics of the intrinsic three-dimensional Dirac operator on the boundary with positive eigenvalues. Remarkably, a detailed calculation for the case of a flat background bounded by a three-sphere yields the same value zeta(0)=11/360 as was found previously by the authors using local boundary conditions. A similar calculation for a spin-3/2 field, working only with physical degrees of freedom (and, hence, excluding gauge and ghost modes, which contribute to the full Becchi-Rouet-Stora-Tyutin-invariant amplitude), again gives a value zeta(0)=-289/360 equal to that for the natural local boundary conditions.

of freedom (and, hence, excluding gauge and ghost modes, which contribute to the full Becchi-Rouet-Stora-Tyutin-invariant amplitude), again gives a value ζ(0) = − 289 360 equal to that for the natural local boundary conditions.

I. INTRODUCTION
In recent work by the authors, 1,2 one-loop effects in quantum cosmology for fermionic fields have been studied using local boundary conditions. In the case of a massless spin-1 2 field ψ A , ψ A ′ in a Riemannian background, which we shall refer to loosely as a Majorana spin-1 2 field, 2 the simplest natural local boundary conditions are (using two-component spinors) where e n AA ′ is the spinor version of the unit Euclidean normal e n µ to the boundary and ǫ = ±1. Note that the primed field ψ A ′ is taken to be independent of ψ A , not related by any conjugation operation. A first-order differential operator for this Riemannian boundary-value problem exists which is symmetric and has self-adjoint extensions. One Because of the first-order nature of the fermionic operators, one has a choice between local boundary conditions such as Eq. (1.1) and non-local (spectral) boundary conditions.
While spectral boundary conditions are not in any obvious way related to supersymmetry, they are nevertheless of considerable mathematical interest, and are the subject of this paper. Their mathematical foundations lie in the theory of elliptic equations and in the index theory for the Dirac operator. 6 To illustrate these boundary conditions, consider again the case of a massless Majorana spin-1 2 field ψ A , ψ A ′ in the region of flat Euclidean four-space bounded by a three-sphere of radius a. Denote by τ the Euclidean distance from the centre of the sphere. Then the field ψ A , ψ A ′ may be expanded in terms of harmonics on the family of spheres centred on the origin, 2,7 as In the summations, n runs from 0 to ∞, p and q from 1 to (n + 1)(n + 2). The α pq n are a collection of matrices introduced for convenience, where, for each n, α pq n is block-diagonal in the indices pq, with blocks 1 1 1 −1 . The harmonics ρ nqA have positive eigenvalues 1 2 (n + 3 2 ) of the intrinsic three-dimensional Dirac operator e n AA ′ e BA ′ j (3) D j on the threesphere, while the harmonics σ nqA have negative eigenvalues − 1 2 (n + 3 2 ). Here e BA ′ j is the spinor version of the orthonormal spatial triad on the three-sphere, and (3) D j is the three-dimensional covariant derivative (j = 1, 2, 3). 7 Similarly, the harmonics σ nqA ′ have positive eigenvalues 1 2 (n + 3 2 ) of the corresponding three-dimensional operator on primed spinors, and the harmonics ρ nqA ′ have negative eigenvalues − 1 2 (n + 3 2 ). This expansion can be summarized more simply as where the (±) parts correspond to positive and negative eigenvalues, respectively, for the intrinsic three-dimensional Dirac operator.
In studying the classical boundary-value problem for the massless Dirac equation, one finds that classical solutions corresponding to boundary data with a non-zero coefficient gives a boundary contribution to the index of the Dirac operator for the manifold with boundary. 6 [In the generic case, the index will be non-zero, and the classical boundaryvalue problem will not be well-posed. 6 ] In this paper we are however not concerned with the signature, but rather, as explained above, with the zeta-function ζ(s) formed from the eigenvalues of the four-dimensional Dirac operator, subject now to boundary conditions in which ψ A (+) and ψ A ′ (+) are specified on the boundary in our flat-space example.
Thus, just as one has a well-posed classical problem with these boundary data, one similarly expects that the analogous quantum amplitude, the Hartle-Hawking path integral for the fermions, is naturally studied by taking spectral boundary conditions in which ψ A (+) and ψ A ′ (+) , or equivalently the m np (a) and r np (a) in our example, are specified on the boundary. Here the Euclidean action is (1.7) The fermionic fields are taken to be anti-commuting, and Berezin integration is being used. 7 With our conventions, the Infeld-van der Waerden connection symbols σ AA ′ a are taken to obtained is again identical to that found using the natural local boundary conditions for the physical degrees of freedom. 1 Some comments are included in Sec. VI.

II. EIGENVALUES FOR SPECTRAL BOUNDARY CONDITIONS ON S 3
In the case of a region of four-dimensional flat Euclidean space bounded by a threesphere of radius a, we decompose the massless spin-1 2 field ψ A , ψ A ′ as in Eqs.
where Because of the degeneracy (n + 1)(n + 2) in the label p, and because of the splitting of the action I E in Eq. (2.3) into identical pieces involving (m np , m np ) and (r np , r np ), the complete path integral (1.6) splits into a product of Berezin integrals : . (2.5) The boundary conditions in each integration are then naturally arising from variation of the action (1.7), subject to the boundary conditions are clearly found by separation of variables, being of the form Here the pair x nk (τ ), x nk (τ ) is an eigenvector corresponding to variation of the action . Here the constant µ with dimensions of mass has been introduced in order to make the product dimensionless. 2 As shown below, the λ n,k are purely imaginary, or equivalently the E nk are real. Further, the E nk occur in equal and opposite pairs, since if the pair x nk (τ ), x nk (τ ) corresponds to an eigenvalue E nk in Eq. (2.10), then clearly the pair x nk (τ ), − x nk (τ ) corresponds to the eigenvalue −E nk . Hence the path integral can also be written as n,k . The Berezin integration rules imply that one should only include those values of k, say k = 1, 2, ..., which correspond to positive values of E nk . This formal expression must then be regularized using zeta-function methods.
The coupled first-order equations (2.10,11) lead to the second-order equation together with a corresponding equation for x nk (τ ). The solutions x nk (τ ), x nk (τ ) obeying the boundary conditions (2.6) are subject to the eigenvalue condition The quantities E 2 nk are clearly real and positive since they are eigenvalues for the selfadjoint problem (2.12), subject to boundary conditions x nk (0) = x nk (a) = 0, and since Eq. (2.12) involves a positive operator. For a given n, the coefficients A nk in Eqs. (2.13,14) can be chosen such that the eigenfunctions x nk (τ ) are orthonormal in the inner product where the E nk (k = 1, 2, ...) are the positive eigenvalues obeying Eq. (2.15).
Following the standard procedure, as, for example, in Ref. 2, the formally divergent infinite product (2.16) is regularized by studying the zeta-function for the squared eigenvalues Here the degeneracy d k (n) = 2(n + 1)(n + 2) is in fact independent of k. The series (2.17) converges for Re(s) > 2, and can be analytically continued to a meromorphic function with poles only at s = 1 2 , 1, 3 2 , 2. The formal expression log(K HH ), with K HH given by Eq. (2.16), is then evaluated as − 1 2 ζ ′ (0) − ζ(0) log µ.

III. GREEN'S FUNCTIONS AND THE HEAT KERNEL
The quantity ζ(0), which gives the divergence and scaling properties of the one-loop amplitude, is evaluated by studying the heat kernel, defined for T > 0 by This is related to the zeta-function ζ(s) of Eq. (2.17) by for Re(s) > 2. The heat kernel G(T ) has the standard asymptotic expansion One proceeds by considering, for each n = 0, 1, 2, ..., the Green's function defined for  By setting τ = τ ′ and integrating, one recovers the contribution to the heat kernel The Laplace transform of the Green's function, obeys the differential equation Following Eq. (3.6),Ĝ n (τ, τ ′ , σ 2 ) is zero whenever either τ or τ ′ is 0 or a. It can be found explicitly in terms of modified Bessel functions (cf Refs. 8-11), aŝ where τ > (τ < ) is the larger (smaller) of τ and τ ′ . This gives a splitting of the Laplace transformĜ(τ, τ ′ , σ 2 ) of the function G(τ, τ ′ , T ), defined as G(τ, τ ′ , T ) = ∞ n=0 2(n + 1)(n + 2)G n (τ, τ ′ , T ) , (3.12) in the formĜ is the "free contribution", which corresponds to the boundary conditions of vanishing at the origin and at infinity. The "interacting contribution" iŝ   The problem of computing G F (T ) is thus reduced to that of computing the right-hand side of Eq. (4.6). Using again the identity I ′ 0 (y) = I 1 (y), together with I ′ 1 (y) = I 0 (y)−y −1 I 1 (y), one has the relations among indefinite integrals : ye −y I 0 (y) dy = ye −y (I 0 + I 1 ) − y 2 e −y (I 0 + I 1 ) + y 2 e −y I 1 + I 0 − I 1 y dy = y 2 e −y (I 0 + I 1 ) − ye −y (I 0 + I 1 ) dy , (4.7) giving 3 ye −y I 0 (y) dy = y 2 e −y (I 0 + I 1 ) − ye −y I 0 + I 0 e −y dy . (4.8) In addition, Hence the integral on the right-hand side of Eq. (4.6) is found to be e −y (2y − 1)I 0 (y) dy = e −y 2 3 y 2 I 0 + (y 2 + y)I 1 − y(I 0 + I 1 ) . (4.10) The relations (4.1)-(4.6) and (4.10) imply that, as In deriving Eq. (4.11), we have used the following asymptotic relations 11 valid as z → ∞ : The Laplace transform of the kernel of the interacting part is given by nf (n; σa) , (4.14) where f (n; σa) = 1 + n 2 σ 2 a 2 I n (σa)K n (σa) − I ′ n (σa)K ′ n (σa) − I ′ n (σa) σaI n (σa) (4.15) is the function defined in Eq. for large σ 2 , using the asymptotic expansion of f (n; σa) valid uniformly 8−10 with respect to n at large σa, and then taking the inverse Laplace transform before taking the limit N → ∞. The first series in Eq. (4.14) has already been studied in the case of scalar fields. 9 In the case of the second series, the Watson transform used in Refs. 8-9 is a source of complications, because nf (n; σa) is not an even function of n. Instead, we take the inverse Laplace transform of the large-σ 2 expansion of nf (n; σa), and compute the sum (as an asymptotic series valid as T → 0 + ) with the help of the Euler-Maclaurin formula. 13 Setting r = n √ n 2 +σ 2 a 2 , one has the asymptotic series 8−9 nf (n; σa) ∼ n √ n 2 + σ 2 a 2 σ 2 a 2 (4.16) valid as σ → ∞, uniformly in n. This is derived from the uniform asymptotic expansions of I ν (z), K ν (z), I ′ ν (z), and K ′ ν (z) described in Refs. 10,11. Denoting by L I the inverse Laplace transform, one thus has the asymptotic series In order to compute sums of the type ∞ n=1 n (2m + 1) e − n 2 T a 2 where m = 0,1,2,..., we can use the Euler-Maclaurin formula 13 for the function F (y) = ye − y 2 T a 2 . In Eq. (4.18), the B i denote the Bernoulli numbers.
Thus we get The other sums arising from Eq. and so on.
Combining the resulting contribution to the asymptotic expansion of G I (T ) as T → 0 + with the other piece, arising from the first term in Eq. (4.14), given in Ref. 9, as well as the expansion (4.11) of G F (T ), one finds the asymptotic expansion of the heat kernel : The spin-3 2 field, as appearing, e.g., in N = 1 supergravity, is described by a potential ψ A µ , ψ A ′ µ in the Euclidean regime (µ = 0, 1, 2, 3). In a Hamiltonian treatment, 15 the quantities ψ A i , ψ A ′ i are the dynamical variables (i = 1, 2, 3), while ψ A 0 , ψ A ′ 0 appear as Lagrange multipliers. The gravitational field is correspondingly described by the tetrad e AA ′ µ . Taking the geometry to be flat, and x 0 = τ to be the radial distance from the origin, while x i (i = 1, 2, 3) are coordinates on the 3-sphere, we impose the gauge conditions In the summations, n runs from 0 to ∞, and p and q now run from 1 to (n + 1)(n + 4).
For each n, α pq n is a matrix again block-diagonal in the indices pq, with blocks Here where n AA ′ = i e n AA ′ is the Lorentzian normal, 2,7 and the harmonic ρ nq(ABC) , symmetric in its three unprimed indices, has positive eigenvalue 1 2 (n + 5 2 ) of the appropriate 3-dimensional Dirac operator. Similarly where the harmonic σ nq(A ′ B ′ C ′ ) , symmetric in its three primed indices, also has positive eigenvalue 1 2 (n + 5 2 ). The harmonics µ nqABB ′ and β : The resulting Berezin integration for the Hartle-Hawking amplitude K HH gives the formal .

ACKNOWLEDGMENTS
We are much indebted to Michael Atiyah, Gary Gibbons, Stephen Hawking and Jorma Louko for several enlightening discussions.