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Mixed Boundary Conditions in Euclidean Quantum Gravity

This paper studies a new set of mixed boundary conditions in Euclidean quantum gravity. These involve, in particular, Robin boundary conditions on the perturbed 3-metric and hence lead, by gauge invariance, to Robin conditions on the whole ghost 1-form. The corresponding trace anomaly is evaluated in the case of flat Euclidean 4-space bounded by a 3-sphere. In general, this anomaly differs from the ones resulting from other local or non-local boundary conditions studied in the recent literature.


Introduction
Since the main problem in quantum cosmology is to find a suitable set of boundary conditions which completely determine the path integral for the quantum state of the universe [1][2], the search for such boundary conditions is of crucial importance in Euclidean quantum gravity [3]. For this purpose, it has been found in [4] that, following earlier work in [5], a set of gauge-invariant boundary conditions for Euclidean quantum gravity can be written as (1.5) With our notation [4], g is the flat background 4-metric, h is its perturbation, τ is the Euclidean-time coordinate, which becomes a radial coordinate if flat 4-space is bounded by a 3-sphere (see section 3). Moreover, ϕ µ is the ghost 1-form, with normal component ϕ 0 and tangential components ϕ i , and the stroke denotes covariant differentiation tangentially with respect to the 3-dimensional Levi-Civita connection of the boundary. We are interested in the Faddeev-Popov formalism for quantum amplitudes. Thus, we deal with a gaugeaveraging term in the Euclidean action, i.e.
where Φ ν is taken to be the de Donder functional (∇ being the 4-dimensional Levi-Civita connection of the background) [4,6] Φ dD ν (h) ≡ ∇ µ h µν − 1 2 g µν g ρσ h ρσ (1.7) and we consider a ghost η µ and an anti-ghost η µ . These ghosts reflect the invariance of the classical theory under gauge transformations of metric perturbations of the form The boundary conditions for ϕ µ are the same as those for η µ and η µ [7]. This property should not be surprising, since already in the simpler case of Euclidean Maxwell theory the ghost and anti-ghost obey the same boundary conditions imposed on the scalar field occurring in the gauge transformations of the potential (see Appendix A). It is therefore sufficient to consider ϕ µ , and then multiply by −2 the resulting contribution to the amplitudes, since the ghosts are anti-commuting and complex-valued.
The boundary conditions (1.1) reflect a natural choice in the classical variational problem in general relativity, where one often fixes at the boundary the intrinsic 3-geometry [2,8]. Since, denoting by K ij the unperturbed extrinsic-curvature tensor of the boundary where c ij is the metric on a unit 3-sphere), one has the invariance of (1.  (i) In classical general relativity one can also fix at the boundary the trace of the extrinsiccurvature tensor (rather than the induced 3-metric) [8]. This choice of boundary conditions is also relevant for the theory of the quantum state of the universe [1,9].
(ii) In one-loop quantum cosmology, one can also consider Hawking's magnetic boundary conditions for quantum gravity, which set to zero at the boundary the linearized magnetic curvature, and hence the first derivatives with respect to τ of the perturbed 3-metric (see section 7.3 of [2]).
are local but not entirely gauge-invariant, since (1.14) makes it impossible to preserve (1.10) under the action of (1.9). It therefore appears that neither locality nor complete gauge invariance of the boundary conditions are always respected in Euclidean quantum gravity. The minimal requirement is instead that the boundary conditions on the ghost 1form ϕ µ should lead to gauge invariance of at least a subset of the boundary conditions on metric perturbations. The latter choice reflects a careful analysis of well-posed variational problems in classical general relativity, which usually fix at the boundary the 3-metric, or its normal derivatives, or the trace of the extrinsic-curvature tensor [8].

Mixed boundary conditions
For a given choice of quantization technique and background 4-geometry, the amplitudes of quantum gravity may depend on the boundary 3-geometry and on the boundary conditions, and hence a large number of interesting problems can be studied for various gaugeaveraging terms in the Euclidean action. For the reasons described in the introduction, we are interested in the most general class of Robin boundary conditions on perturbations of the 3-geometry. Thus, denoting by λ a real dimensionless parameter, we begin by requiring The invariance of the boundary conditions (2.1) under the transformations (1.9) is ensured by the following set of Robin conditions on the ghost 1-form: Note that (2.3) is a local equation which ensures the validity of a more complicated condition, involving its 3-dimensional covariant derivative with respect to the Levi-Civita connection of the boundary.
At this stage, the remaining set of boundary conditions on metric perturbations can be obtained by bearing in mind that, in the case of flat Euclidean 4-space with a 3-sphere boundary, one has from (1.8) Note that, since such a ζ(0) has a cubic dependence on λ, a real value of λ (compatible with reality of the eigenvalues E n resulting from self-adjointness) always exists such that our trace anomaly (3.9) can agree with the values found in [4] in the case of Luckock- in [15,16]. Moreover, it appears necessary to understand whether any set of local and gauge-invariant boundary conditions can be found in Euclidean quantum gravity when a 4-dimensional background geometry is analyzed. In the two-dimensional case, the author of [17] has found that for R 2 + T 2 gravity with an independent spin-connection, local and

Appendix B
Since the calculations leading to (3.1)-(3.8) are not straightforward, we give a brief outline of the most difficult, i.e. the contribution of scalar-type perturbations to the trace anomaly.
To avoid repeating ourselves, we refer the reader to [6,13,14] for the meaning of our notation and for the technique used in our investigation.
As the eigenvalues tend to ∞ and the order n of basis functions tends to ∞, the n-dependent coefficient in the equation obeyed by the eigenvalues by virtue of boundary conditions takes the form (cf [4]) ρ ∞ (n) = 12 n(n 2 − 1) (n 2 − 4) . (B.1) Thus, since the degeneracy of coupled scalar modes is n 2 for all n ≥ 3, one has to consider 1 2 n 2 log(ρ ∞ (n)), whose expansion as n → ∞ does not have terms proportional to 1 n .