Semiclassical gravity

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 [142 , 143 , 128

, 277

] 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) while 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 just assume a scalar field for simplicity. The classical action $S m$ for this matter field is given by the functional

where $∇a$ is the covariant derivative associated to the metric $gab$ , $ξ$ is a coupling parameter between the field and the scalar curvature of the underlying spacetime $R$ , and $g = det gab$ .

The field may be quantized in the manifold using the standard canonical quantization formalism [25 , 100 , 285 ]. The field operator in the Heisenberg representation $ˆ φ$ is an operator-valued distribution solution of the Klein–Gordon equation, the field equation derived from Equation (1 ),

We may write the field operator as $ˆφ[g;x)$ to indicate that it is a functional of the metric $g ab$ 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, $T ab(x ) = (2 ∕√ − g) δSm ∕δgab$ , leading to

where $□ = ∇a ∇a$ , and $Gab$ 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 $Tˆab[g;x)$ . Naively one would replace the classical field $φ[g;x)$ in the above functional by the quantum operator $ˆφ[g;x)$ , 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 [67 , 68 ], in which one introduces a point $y$ in a neighborhood of the point $x$ and then uses as the regulator the vector tangent at the point $x$ of the geodesic joining $x$ and $y$ ; this method is discussed for instance in [242 , 243 , 245 ] and in Section 5. Another well known method is dimensional regularization in which one works in arbitrary $n$ 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

where $ab 𝒟 [g]$ is the differential operator

Note that if dimensional regularization is used, the field operator $φˆ[g;x )$ propagates in an $n$ -dimensional spacetime. Once the regularization prescription has been introduced, a regularized and renormalized stress-energy operator $ˆR Tab[g; x)$ may be defined as

which differs from the regularized $ˆTab[g; x)$ by the identity operator times some tensor counterterms $F C[g;x) ab$ , which depend on the regulator and are local functionals of the metric (see [208

] 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 [285 ]), $|ψ ⟩$ and $|ϕ⟩$ , the matrix element $⟨ψ|TRab|ϕ ⟩$ , defined as the limit when the regulator takes the physical value is finite and satisfies Wald’s axioms [100

, 282

]. These counterterms can be extracted from the singular part of a Schwinger–DeWitt series [100

, 67

, 68

, 31 ]. The choice of these counterterms 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

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 respectively the renormalized coupling constants, 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 (7

) may be derived from the gravitational action

where $Cabcd$ is the Weyl tensor. The tensors $Aab$ and $Bab$ come from the functional derivatives with respect to the metric of the terms quadratic in the curvature in Equation (8

); they are explicitly given by

ab 1 δ ∫ 4√ --- cdef A = √-------- d − gCcdefC − g δgab = 1gabC Ccdef− 2RacdeRb + 4RacR b− 2RRab − 2□Rab + 2∇a ∇bR + 1gab□R, (9 ) 2 cdef cde c 3 3 3 ab 1 δ ∫ 4√ --- 2 B = √-------- d − gR − g δgab = 1gabR2 − 2RRab + 2∇a ∇bR − 2gab□R, (10 ) 2

where $R abcd$ and $R ab$ are the Riemann and Ricci tensors, respectively. These two tensors are, like the Einstein and metric tensors, symmetric and divergenceless: $a a ∇ Aab = 0 = ∇ Bab$ .

A solution of semiclassical gravity consists of a spacetime ( $ℳ, gab$ ), a quantum field operator $ˆφ [g]$ which satisfies the evolution equation (2 ), and a physically acceptable state $|ψ[g]⟩$ for this field, such that Equation (7 ) 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 seems 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 should 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 [92 ]. A criterion based on the ratio of the fluctuations to the mean was proposed by Kuo and Ford [194 ] (see also work via zeta-function methods [241 , 69 ]). This proposal was questioned by Phillips and Hu [163 , 242 , 243 ] 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 bi-tensor quantities, equivalent to a stipulation of the resolution level of measurements; see also the response by Ford [93 , 95 ]. A different criterion is recently suggested by Anderson et al. [10 , 9 ] based on linear response theory. A partial summary of this issue can be found in our Erice Lectures [168 ].