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4.1 Influence action for semiclassical gravity

Let us formulate semiclassical gravity in this functional framework. Adopting the usual procedure of effective field theories [63, 95, 96, 97, 98, 366, 367], one has to take the effective action for the metric and the scalar field of the most general local form compatible with general covariance: S [g, φ] ≡ S [g ] + S [g, φ] + ... g m, where S [g] g and S [g, φ] m are given by Equations (9View Equation) and (1View Equation), respectively, and the dots stand for terms of order higher than two in the curvature and in the number of derivatives of the scalar field. Here, we shall neglect the higher-order terms as well as self-interaction terms for the scalar field. The second order terms are necessary to renormalize one-loop ultraviolet divergences of the scalar field stress-energy tensor, as we have already seen. Since ℳ is a globally hyperbolic manifold, we can foliate it by a family of t = constant Cauchy hypersurfaces Σt, and we will indicate the initial and final times by ti and tf, respectively.

The influence functional corresponding to the action (1View Equation) describing a scalar field in a spacetime (coupled to a metric field) may be introduced as a functional of two copies of the metric, denoted by g+ ab and g− ab, which coincide at some final time t = tf. Let us assume that, in the quantum effective theory, the state of the full system (the scalar and the metric fields) in the Schrödinger picture at the initial time t = ti can be described by a density operator, which can be written as the tensor product of two operators on the Hilbert spaces of the metric and of the scalar field. Let ρi(ti) ≡ ρi[φ+ (ti),φ− (ti)] be the matrix element of the density operator ˆρS(ti) describing the initial state of the scalar field. The Feynman–Vernon influence functional is defined as the following path integral over the two copies of the scalar field:

∫ ± i(Sm[g+,φ+]− Sm[g−,φ−]) ℱIF[g ] ≡ 𝒟 φ+ 𝒟 φ− ρi(ti)δ [φ+(tf) − φ − (tf)]e . (22 )
Alternatively, the above double path integral can be rewritten as a CTP integral, namely, as a single path integral in a complex time contour with two different time branches, one going forward in time from ti to tf and the other going backward in time from tf to ti (in practice one usually takes ti → − ∞). From this influence functional, the influence action S [g+,g− ] IF, or S [g ±] IF for short, defined by
ℱIF[g±] ≡ eiSIF[g± ], (23 )
carries all the information about the environment (the matter fields) relevant to the system (the gravitational field). Then we can define the CTP effective action for the gravitational field, Seff[g±], as
± + − ± Se ff[g ] ≡ Sg [g ] − Sg [g ] + SIF[g ]. (24 )
This is the effective action for the classical gravitational field in the CTP formalism. However, since the gravitational field is treated only at the tree level, this is also the effective classical action from which the classical equations of motion can be derived.

Expression (22View Equation) contains ultraviolet divergences and must be regularized. We shall assume that dimensional regularization can be applied, that is, it makes sense to dimensionally continue all the quantities that appear in Equation (22View Equation). For this we need to work with the n-dimensional actions corresponding to Sm in Equation (22View Equation) and Sg in Equation (9View Equation). For example, the parameters G, Λ, α, and β of Equation (9View Equation) are the bare parameters G B, Λ B, α B, and β B, and in S [g ] g, instead of the square of the Weyl tensor in Equation (9View Equation), one must use 2 abcd ab 3(RabcdR − RabR ), which by the Gauss–Bonnet theorem leads to the same equations of motion as the action (9View Equation) when n = 4. The form of Sg in n dimensions is suggested by the Schwinger–DeWitt analysis of the ultraviolet divergences in the matter stress-energy tensor using dimensional regularization. One can then write the Feynman–Vernon effective action ± Se ff[g ] in Equation (24View Equation) in a form suitable for dimensional regularization. Since both Sm and Sg contain second-order derivatives of the metric, one should also add some boundary terms [206Jump To The Next Citation Point, 361]. The effect of these terms is to cancel out the boundary terms, which appear when taking variations of Seff[g±], keeping the value of g+ ab and g − ab fixed at Σ ti and Σ tf. Alternatively, in order to obtain the equations of motion for the metric in the semiclassical regime, we can work with the action terms without boundary terms and neglect all boundary terms when taking variations with respect to ± gab. From now on, all the functional derivatives with respect to the metric will be understood in this sense.

The semiclassical Einstein equation (8View Equation) can now be derived. Using the definition of the stress-energy tensor ab √ --- T (x) = (2∕ − g)δSm ∕δgab and the definition of the influence functional, Equations (22View Equation) and (23View Equation), we see that

± || ⟨ˆTab[g;x)⟩ = ∘---2----δSIF[g--]|, (25 ) − g(x ) δg+ab(x)|| g±=g
where the expectation value is taken in the n-dimensional spacetime generalization of the state described by ρˆS (t ) i. Therefore, differentiating S [g ±] eff in Equation (24View Equation) with respect to g+ ab and then setting + − gab = gab = gab, we get the semiclassical Einstein equation in n dimensions. This equation is then renormalized by absorbing the divergences in the regularized ⟨Tˆab[g]⟩ into the bare parameters. Taking the limit n → 4 we obtain the physical semiclassical Einstein equation (8View Equation).
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