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4.2 Influence action for stochastic gravity

In the spirit of the previous derivation of the Einstein–Langevin equation, we now seek a dynamical equation for a linear perturbation hab to the semiclassical metric gab, solution of Equation (7View Equation). Strictly speaking, if we use dimensional regularization we must consider the n-dimensional version of that equation. From the results just described, if such an equation were simply a linearized semiclassical Einstein equation, it could be obtained from an expansion of the effective action ± Seff[g + h ]. In particular, since, from Equation (18View Equation), we have that
± || ⟨ˆTab[g + h;x)⟩ = ∘--------2---------δSIF[g +-h-]||, (19 ) − det(g + h)(x) δh+ab(x ) | ± h =h
the expansion of ⟨ˆTab[g + h]⟩ to linear order in hab can be obtained from an expansion of the influence action SIF[g + h±] up to second order in h±ab.

To perform the expansion of the influence action, we have to compute the first and second order functional derivatives of ± SIF[g + h ] and then set + − hab = hab = hab. If we do so using the path integral representation (15View Equation), we can interpret these derivatives as expectation values of operators. The relevant second order derivatives are

| 4 δ2S [g + h ±]| ∘--------∘----------+IF----+---|| = − HabScd[g;x,y ) − Kabcd [g;x, y) + iN abcd[g;x,y ), − g (x ) − g(y)δh ab(x)δh cd(y)|h±=h | (20 ) --------4-----------δ2SIF[g-±]--|| abcd abcd ∘ ------∘ ------δh+ (x)δh −(y)|| = − H A [g;x,y ) − iN [g;x,y), − g (x ) − g(y) ab cd h±=h
where
N abcd[g;x,y) ≡ 1-⟨{ˆtab[g;x),ˆtcd[g;y)}⟩ , 2 ⟨ ( )⟩ Habcd[g;x,y) ≡ Im T∗ ˆTab[g;x)ˆT cd[g;y) , S abcd i ⟨[ ab cd ]⟩ H A [g;x,y) ≡ − 2- Tˆ [g;x ), ˆT [g;y) , ⟨ | ⟩ abcd -------−-4------- δ2Sm-[g-+-h,φ-]|| K [g;x,y) ≡ ∘ − g-(x-)∘ −-g(y) δhab(x )δhcd(y )| ˆ , φ= φ
with ˆtab defined in Equation (12View Equation); [ , ] denotes the commutator and { , } the anti-commutator. Here we use a Weyl ordering prescription for the operators. The symbol T ∗ denotes the following ordered operations: First, time order the field operators ˆ φ and then apply the derivative operators which appear in each term of the product ab cd T (x)T (y), where ab T is the functional (3View Equation). This ∗ T “time ordering” arises because we have path integrals containing products of derivatives of the field, which can be expressed as derivatives of the path integrals which do not contain such derivatives. Notice, from their definitions, that all the kernels which appear in expressions (20View Equation) are real and also Habcd A is free of ultraviolet divergences in the limit n → 4.

From Equation (18View Equation) and (20View Equation), since SIF[g,g] = 0 and SIF[g− ,g+ ] = − SI∗F[g+,g− ], we can write the expansion for the influence action SIF[g + h±] around a background metric gab in terms of the previous kernels. Taking into account that these kernels satisfy the symmetry relations

Habcd (x, y) = Hcdab(y,x), Habcd(x,y) = − Hcdab(y,x), Kabcd(x,y) = Kcdab(y, x), (21 ) S S A A
and introducing the new kernel
Habcd(x,y ) ≡ HabScd (x,y) + HaAbcd(x,y), (22 )
the expansion of SIF can be finally written as
∫ ∘ ------ SIF [g + h± ] = 1- d4x − g(x)⟨ˆTab[g;x)⟩[hab(x)] 2 ∫ 1 4 4 ∘ ------∘ ------ ( abcd abcd ) − -- d xd y − g(x) − g (y )[hab(x )] H [g;x, y) + K [g; x,y) {hcd(y)} 8 ∫ ∘ ------∘ ------ + -i d4xd4y − g(x) − g (y )[h (x )]N abcd[g;x, y)[h (y )] + 𝒪 (h3), (23 ) 8 ab cd
where we have used the notation
+ − + − [hab] ≡ hab − hab, {hab} ≡ h ab + hab. (24 )
From Equations (23View Equation) and (19View Equation) it is clear that the imaginary part of the influence action does not contribute to the perturbed semiclassical Einstein equation (the expectation value of the stress-energy tensor is real), however, as it depends on the noise kernel, it contains information on the fluctuations of the operator ab Tˆ [g].

We are now in a position to carry out the derivation of the semiclassical Einstein–Langevin equation. The procedure is well known [44Jump To The Next Citation Point167Jump To The Next Citation Point58Jump To The Next Citation Point11026296246]: It consists of deriving a new “stochastic” effective action from the observation that the effect of the imaginary part of the influence action (23View Equation) on the corresponding influence functional is equivalent to the averaged effect of the stochastic source ab ξ coupled linearly to the perturbations h±ab. This observation follows from the identity first invoked by Feynman and Vernon for such purpose:

( ∫ ) 1- 4 4 ∘ ------∘ ------ abcd exp − 8 d x d y − g(x ) − g (y)[hab(x )]N (x,y)[hcd(y)] = ∫ ( ∫ ∘ ------ ) 𝒟ξ 𝒫[ξ]exp i- d4x − g(x)ξab(x )[hab(x )] , (25 ) 2
where 𝒫[ξ] is the probability distribution functional of a Gaussian stochastic tensor ab ξ characterized by the correlators (13View Equation) with abcd N given by Equation (11View Equation), and where the path integration measure is assumed to be a scalar under diffeomorphisms of (ℳ, gab). The above identity follows from the identification of the right-hand side of Equation (25View Equation) with the characteristic functional for the stochastic field ξab. The probability distribution functional for ξab is explicitly given by
[ ∫ ------ ------ ] 𝒫 [ξ] = det (2πN )−1∕2exp − 1- d4xd4y∘ − g(x )∘ − g(y)ξab(x)N −1 (x,y)ξcd(y) . (26 ) 2 abcd

We may now introduce the stochastic effective action as

Sseff[g + h±,ξ ] ≡ Sg [g + h+ ] − Sg[g + h− ] + SsIF[g + h±,ξ], (27 )
where the “stochastic” influence action is defined as
∫ s ± ± 1- 4 ∘ ------ ab 3 SIF[g + h ,ξ] ≡ Re SIF[g + h ] + 2 d x − g(x) ξ (x) [hab(x)] + 𝒪 (h ). (28 )
Note that, in fact, the influence functional can now be written as a statistical average over ξab:
ℱ [g + h±] = ⟨exp (iSs [g + h ±,ξ])⟩ . IF IF s

The stochastic equation of motion for hab reads

s ± || δS-eff[g-+-h--,ξ]| = 0, (29 ) δh+ab(x) |h±=h
which is the Einstein–Langevin equation (14View Equation); notice that only the real part of SIF contributes to the expectation value (19View Equation). To be precise, we get first the regularized n-dimensional equations with the bare parameters, with the tensor Aab replaced by 23Dab, where
∫ ( ) Dab ≡ √-1---δ-- dnx √ −-g RcdefRcdef − RcdRcd − gδgab gab( ) = --- RcdefRcdef− RcdRcd + □R − 2RacdeRbcde− 2RacbdRcd + 4RacRcb − 3□Rab + ∇a ∇bR. 2 (30 )
Of course, when n = 4 these tensors are related, Aab = 2Dab 3. After that we renormalize and take the limit n → 4 to obtain the Einstein–Langevin equations in the physical spacetime.
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