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3.1 Semiclassical gravity

Semiclassical gravity describes the interaction of a classical gravitational field with quantum matter fields. This theory can be formally derived as the leading 1∕N approximation of quantum gravity interacting with N independent and identical free quantum fields [152Jump To The Next Citation Point, 175, 176, 348Jump To The Next Citation Point], which interact with gravity only. By keeping the value of N G finite, where G is Newton’s gravitational constant, one arrives at a theory in which formally the gravitational field can be treated as a c-number field (i.e., quantized at tree level) and matter fields are fully quantized. The semiclassical theory may be summarized as follows.

Let (ℳ, gab) be a globally hyperbolic four-dimensional spacetime manifold ℳ with metric gab and consider a real scalar quantum field φ of mass m propagating on that manifold; we assume a scalar field for simplicity. The classical action S m for this matter field is given by the functional

∫ 1- 4 √--- [ ab ( 2 ) 2] Sm [g,φ] = − 2 d x − g g ∇a φ ∇bφ + m + ξR φ , (1 )
where ∇a is the covariant derivative associated with the metric gab, ξ is a coupling parameter between the field and the scalar curvature of the underlying spacetime R and g = detgab.

The field may be quantized in the manifold using the standard canonical quantization formalism [34Jump To The Next Citation Point, 121Jump To The Next Citation Point, 362Jump To The Next Citation Point]. The field operator in the Heisenberg representation ˆ φ is an operator-valued distribution solution of the Klein–Gordon equation,

(□ − m2 − ξR)ˆφ = 0, (2 )
where □ = ∇ ∇a a. We may write the field operator as φˆ[g; x) to indicate that it is a functional of the metric gab and a function of the spacetime point x. This notation will be used also for other operators and tensors.

The classical stress-energy tensor is obtained by functional derivation of this action in the usual way:

T ab(x) = √2---δSm-, (3 ) − g δgab
leading to
ab a b 1- ab( c 2 2) T [g,φ] = ∇ φ∇ φ − 2 g ∇ φ∇c φ + m φ + ξ(gab□ − ∇a ∇b + Gab) φ2, (4 )
where Gab is the Einstein tensor. With the notation T ab[g,φ] we explicitly indicate that the stress-energy tensor is a functional of the metric gab and the field φ.

The next step is to define a stress-energy tensor operator ˆab T [g;x). Naively, one would replace the classical field φ[g;x) in the above functional with the quantum operator φˆ[g; x), but this procedure involves taking the product of two distributions at the same spacetime point; this is ill-defined and we need a regularization procedure. There are several regularization methods, which one may use. One is the point-splitting or point-separation regularization method [83Jump To The Next Citation Point, 84Jump To The Next Citation Point], in which one introduces a point y in the neighborhood of the point x and then uses as the regulator the vector tangent at point x of the geodesic joining x and y; this method is discussed in [303Jump To The Next Citation Point, 304Jump To The Next Citation Point, 305Jump To The Next Citation Point] and in Section 5. Another well-known method is dimensional regularization, in which one works in n dimensions, where n is not necessarily an integer, and then uses as the regulator the parameter ε = n − 4; this method is implicitly used in this section. The regularized stress-energy operator using the Weyl ordering prescription, i.e., symmetrical ordering, can be written as

ˆab 1- aˆ bˆ ab ˆ2 T [g] = 2 {∇ φ[g],∇ φ[g]} + 𝒟 [g]φ [g], (5 )
where ab 𝒟 [g] is the differential operator
( ) 𝒟ab ≡ (ξ − 1∕4)gab□ + ξ Rab − ∇a ∇b . (6 )
Note that if dimensional regularization is used, the field operator ˆ φ [g;x ) propagates in an n-dimensional spacetime. Once the regularization prescription has been introduced, a regularized and renormalized stress-energy operator ˆTRab[g; x) may be defined as
R C Tˆab[g;x) = ˆTab[g;x) + Fab[g; x)ˆI, (7 )
which differs from the regularized ˆTab[g; x) by the identity operator times some tensor counter-terms C F ab[g;x), which depend on the regulator and are local functionals of the metric; see [258Jump To The Next Citation Point] for details. The field states can be chosen in such a way that for any pair of physically acceptable states, i.e., Hadamard states in the sense of [362Jump To The Next Citation Point], |ψ ⟩ and |ϕ⟩, the matrix element ⟨ψ|TaRb|ϕ⟩, defined as the limit when the regulator takes the physical value, is finite and satisfies Wald’s axioms [121Jump To The Next Citation Point, 359Jump To The Next Citation Point]. These counter-terms can be extracted from the singular part of a Schwinger–DeWitt series [44, 83Jump To The Next Citation Point, 84Jump To The Next Citation Point, 121Jump To The Next Citation Point]. The choice of these counter-terms is not unique, but this ambiguity can be absorbed into the renormalized coupling constants, which appear in the equations of motion for the gravitational field.

The semiclassical Einstein equation for the metric gab can then be written as

G [g] + Λg − 2 (αA + βB )[g] = 8πG ⟨ˆT R[g]⟩, (8 ) ab ab ab ab ab
where ⟨ˆTRab[g]⟩ is the expectation value of the operator ˆTaRb[g,x) after the regulator takes the physical value in some physically acceptable state of the field on (ℳ, gab). Note that both the stress tensor and the quantum state are functionals of the metric, hence the notation. The parameters G, Λ, α and β are the renormalized coupling constants, respectively the gravitational constant, the cosmological constant and two dimensionless coupling constants, which are zero in the classical Einstein equation. These constants must be understood as the result of “dressing” the bare constants, which appear in the classical action before renormalization. The values of these constants must be determined by experiment. The left-hand side of Equation (8View Equation) may be derived from the gravitational action
1 ∫ √ ---[1 ] Sg[g] = ----- d4x − g --R − Λ + αCabcdCabcd + βR2 , (9 ) 8πG 2
where C abcd is the Weyl tensor. The tensors A ab and B ab come from functional derivatives with respect to the metric of the terms quadratic in the curvature in Equation (9View Equation); they are explicitly given by
∫ ab --1----δ- 4√--- cdef A = √ −-gδg d − gCcdefC ab = 1gabCcdefCcdef − 2RacdeRb + 4RacR b− 2-RRab 2 cde c 3 ab 2- a b 1- ab − 2□R + 3∇ ∇ R + 3 g □R, (10 ) 1 δ ∫ √--- Bab = √-------- d4 − gR2 − gδgab 1- ab 2 ab a b ab = 2g R − 2RR + 2∇ ∇ R − 2g □R, (11 )
where Rabcd and Rab are the Riemann and Ricci tensors, respectively. These two tensors are, like the Einstein and metric tensors, symmetric and divergence-less: a a ∇ Aab = 0 = ∇ Bab.

A solution of semiclassical gravity consists of a spacetime (ℳ, gab), a quantum field operator ˆφ[g], which satisfies the evolution Equation (2View Equation) and a physically acceptable state |ψ [g]⟩ for this field, such that Equation (8View Equation) is satisfied when the expectation value of the renormalized stress-energy operator is evaluated in this state.

For a free quantum field this theory is robust in the sense that it is self-consistent and fairly well understood. As long as the gravitational field is assumed to be described by a classical metric, the above semiclassical Einstein equations seem to be the only plausible dynamical equation for this metric: the metric couples to matter fields via the stress-energy tensor and for a given quantum state the only physically observable c-number stress-energy tensor that one can construct is the above renormalized expectation value. However, lacking a full quantum-gravity theory, the scope and limits of the theory are not so well understood. It is assumed that the semiclassical theory will break down at Planck scales, which is when simple order-of-magnitude estimates suggest that the quantum effects of gravity should not be ignored, because the energy of a quantum fluctuation in a Planck-size region, as determined by the Heisenberg uncertainty principle, is comparable to the gravitational energy of that fluctuation.

The theory is expected to break down when the fluctuations of the stress-energy operator are large [111Jump To The Next Citation Point]. A criterion based on the ratio of the fluctuations to the mean was proposed by Kuo and Ford [237Jump To The Next Citation Point] (see also work over zeta-function methods [86Jump To The Next Citation Point, 302Jump To The Next Citation Point]). This proposal was questioned by Phillips and Hu [198Jump To The Next Citation Point, 303Jump To The Next Citation Point, 304Jump To The Next Citation Point] because it does not contain a scale at which the theory is probed or how accurately the theory can be resolved. They suggested the use of a smearing scale or point-separation distance for integrating over the bitensor quantities, which is equivalent to a stipulation of the resolution level of measurements; see also the response by Ford [113Jump To The Next Citation Point, 115Jump To The Next Citation Point]. A different criterion was recently suggested by Anderson et al. [10Jump To The Next Citation Point, 11Jump To The Next Citation Point] based on linear-response theory. A partial summary of this issue can be found in our Erice Lectures [207Jump To The Next Citation Point].

More recently, in collaboration with A. Roura [203Jump To The Next Citation Point, 204Jump To The Next Citation Point], we have proposed a criterion for the validity of semiclassical gravity, which is based on the stability of the solutions of the semiclassical Einstein equations with respect to quantum metric fluctuations. The two-point correlations for the metric perturbations can be described in the framework of stochastic gravity, which is closely related to the quantum theory of gravity interacting with N matter fields, to leading order in a 1∕N expansion. We will describe these developments in the following sections.


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