DIRECTOR AUDITOR Total de documentos con firma de autorización
2. Determinación del nivel de riesgo
geometrygab has a curvature far below the Planck scale. The partition function is Z1−loop=e−I[g]
Z
d[h](G.F.)e−I2[h;g], (3.6) where (G.F.) denotes all contributions induced by fixing the gauge in the path integral.
The second order action is given by I2[h;g] =− 1
16π Z
dDx√ g
−1
4¯h·Gh+1 2(δ¯h)2
, (3.7)
where·denotes the metric contraction of tensors. We have defined2
¯hab =hab−1
2gabhcc (3.8)
and
(Gh)ab=−∇c∇chab−2Ra bc dhcd. (3.9) We have also introduced the following operations on tensors T:
(δT)b...c=−∇aTab...c, (3.10a)
(αT)ab...c=∇(aTb...c), (3.10b)
following [131].
3.2. THE GRAVITATIONAL PARTITION FUNCTION 53 Consider the action for a conformal normalisable perturbation hab =ϕgab,
I2[ϕg;g] =−(D−1)(D−2) 64π
Z
dDx√
g(∂ϕ)2. (3.11)
This action is always negative, so that the path integral in (3.6) seems irremediably ill-defined. This is the perturbative manifestation of the fact that, if the path integral is taken without further prescription, there are geometries for which the action (3.2) can be made arbitrarily negative.
This challenge, known as the conformal factor problem, was addressed by Gibbons, Hawking and Perry [130], who proposed that the integration contour over the conformal direction in the space of metrics should be imaginary rather than real.3 We follow here the more detailed procedure of [131], straightforwardly extended to higher dimensions.
We will decompose the second order action, applying a standard gauge fixing procedure, and show that the conformal divergent modes are unphysical and do not contribute to the one-loop partition function.
The second order actionI2[h;g] is invariant for the diffeomorphism transformations hab→hab+∇aVb+∇bVa= (h+ 2αV)ab. (3.12) Following the Feynman-DeWitt-Faddeev-Popov gauge fixing method,
(G.F.) = (detC)δ(Ca[h]−wa). (3.13) We consider the linear class of gauges
Cb[h] =∇a
hab− 1 βgabhcc
, (3.14)
whereβ is an arbitrary constant, so that the Fadeev-Popov determinant (detC) is given
3This prescription is very suggestive because it corresponds to making the timelike direction in the Wheeler-DeWitt metric into a spacelike one, becoming a positive definite metric on the space of spacetime metrics. It is thus analogous to the imaginary time prescription in common Euclidean path integrals.
by the spectrum of the operator
(CV)a=−∇b∇bVa−RabVb+ 2
β −1
∇a∇bVb. (3.15)
To study the spectrum, let us consider the Hodge-de Rham decomposition4 of the gauge vectorV into harmonic (H), exact (E) and coexact (C) parts,
V =VH+VE+VC. (3.16)
This induces a decomposition of the action of C, which we denote by CH for harmonic vectors,CE for exact vectors and CC for coexact vectors.
The harmonic part satisfiesdVH= 0 and δVH= 0. We can check that CVH=− 4Λ
D−2VH. (3.17)
The spectrum is positive for Λ < 0 and zero for Λ = 0, with multiplicity given by the number of linearly independent harmonic vector fields. For Λ>0, the background solution satisfying (3.4) does not allow for harmonic vector fields if assumed to be compact and orientable [152]. Thus, the spectrum ofCH is never negative.
The exact part is such thatVE=dχ, whereχ is a scalar. We can show that specCE= spec
2
1 β −1
− 2Λ
D−2
, (3.18)
where the operator on the right-hand side acts on scalars, andis the Laplacian. For Λ<
0, the operator is positive forβ >1, being positive semi-definite for Λ = 0. For Λ>0, the Lichnerowicz-Obata theorem tells us that the spectrum of the Laplacian on a compact and orientable manifold satisfying (3.4) is bounded from above by−2DΛ(D−1)−1(D−2)−1, the saturation of the bound corresponding to the sphere [152]. This implies that, for Λ>0, the spectrum ofCE is positive forβ > D.
4Notice that, while the instanton background is not compact for asymptotically flat/AdS black holes, the off-shell perturbationshabin the path integral should be normalisable.
3.2. THE GRAVITATIONAL PARTITION FUNCTION 55 The coexact part is such thatδVC= 0. Hence
CVC= 2δαVC (3.19)
and the spectrum ofCC can be shown to be positive semi-definite, Z
dDx√
g[VC·CVC] = 2 Z
dDx√
g[αVC·αVC]≥0, (3.20) with equality for coexact Killing vectors.
The Faddeev-Popov determinant contribution to the partition function is then det ˜C∼(det ˜CE)(det ˜CC), (3.21) the tilde denoting that the zero modes have been projected out. The harmonic contribution is not explicitly considered because, if it exists (Λ<0), it is a positive factor dependent only on Λ and on the dimension of the space of harmonic vector fields, as mentioned above; it will not be relevant to our discussion. The contribution from the exact part is fundamental since it will cancel the divergent conformal modes.
In order to make the results independent of the arbitrary vectorwin the gauge fixing (3.13), the ’t Hooft method of averaging over gauges is adopted. The arbitrariness is then expressed in terms of a constant γ introduced by the weighting factor of the averaging.
The final result will be independent ofγ, as required. The unconstrained effective action for the perturbations is given by
I2eff[h;g] =I2[h;g] + γ 32π
Z
dDx√
g Ca[h]Ca[h] =
− 1 16π
Z
dDx√ g
"
−1 4
¯h·Gh+1
2(1−γ)(δ¯h)2+γ 2
1− 2
β
δ¯h·dˆh−γ 8
1− 2
β 2
(dˆh)2
# , (3.22) where we denote ˆh≡hcc.
We now decompose the quantum field hab into a traceless-transverse (TT) part, a traceless-longitudinal (TL) part, built from a vectorη, and a trace (i.e. conformal) part,
hab =hT Tab +hT Lab + 1
Dgabˆh , (3.23)
with
hT Lab = 2(αη)ab+ 2
Dgabδη . (3.24)
The constantβ, unspecified in the gauge condition (3.14), can be chosen so that the trace ˆhand the longitudinal vector η decouple. This requires
β = 2
1−D−2 D
γ−1 γ
−1
. (3.25)
The effective action becomes I2eff[h;g] =− 1
16π Z
dDx√ g
"
− 1
4hT T ·GhT T −αη·α∆1η− 1
Dδηδη+
+ 2(1−γ)
δαη·δαη+ 1
D2αδη·αδη− 2
Dδαη·αδη
+
+ 4
D−2Λ
αη·αη− 1 D(δη)2
+1
2 ˆhFˆh
#
, (3.26)
where the operatorF is given by F =−D−2
4D
1 +D−2 D
γ−1 γ
− 1
DΛ. (3.27)
Recalling the choice of β (3.25), we find that the operator on the right-hand side of the expression (3.18) is given by 4DF/(D−2). The contribution of the ghosts (3.21) can be recast as
det ˜C∼(det ˜F)(det ˜CC). (3.28) For the vector η, as we did forV in the ghost part, we perform a Hodge-de Rham
3.2. THE GRAVITATIONAL PARTITION FUNCTION 57 decomposition into harmonic, coexact and exact parts,
η=ηH+ηC+ηE, (3.29)
respectively. UsingηE=dχ, the result for the effective action is then I2eff[h;g] =− 1
16π Z
dDx√ g
h
−1
4hT T ·GhT T +1 2
ˆhFˆh+
+ 4
D−2γΛαηH·αηH−γαηC·αCCηC− 4D
D−2γDχ· DF χi
, (3.30) where we defined the operator
Dab=∇a∇b− 1
Dgab. (3.31)
Notice that the Hodge-de Rham decomposition ofη in harmonic, coexact and exact parts gives, forhT Lab , a decomposition in 2αηH, 2αηC and 2Dχ, respectively.
Finally, we can evaluate the Gaussian integrals in the partition function. The pre-scription of [130] corresponds to taking the imaginary contour for ˆh. The dependence of the partition function is
Z1−loop∼(det ˜C)(det ˜G)−1/2(det ˜F)−1/2(det ˜CC)−1/2(det ˜F)−1/2
∼(det ˜G)−1/2(det ˜CC)1/2. (3.32)
Again, the tilde on the operators denotes that the zero modes have been projected out.5 The Gaussian integrals are subject toζ-function regularisation [131]. It is understood that the spectrum ofGhere is restricted to traceless-transverse normalisable modes.
Let us summarise the treatment of the conformal factor problem. Conformal modes make the action for perturbations negative. However, these modes do not contribute to the path integral. The two factors (detF)−1/2 arising from the Gaussian integrals in ˆh (taken through an imaginary contour) andχ cancel with the detF factor arising from the
5Zero modes can be dealt with by the standard collective coordinates method [153].
exact part of the Fadeev-Popov determinant. This makes the unphysical character of the divergence obvious, at least in perturbation theory.
The relevant operators are then CC and G. For a real metric, the operator CC is positive semi-definite, as we have shown above. Once its zero modes are projected out, it contributes a positive factor to the final result. Physical instabilities are only possible if there are negative eigenvalues of the operatorG,
(GhT T)ab=λhT Tab , (3.33)
in which case the eigenmodeshT Tab are called negative modes. Notice that, forλ6= 0, there is no gauge ambiguity (3.12) becauseGαV = 0.