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6.1 Perturbations around Minkowski spacetime

The Minkowski metric ηab in a manifold ℳ, which is topologically 4 IR, together with the usual Minkowski vacuum, denoted as |0⟩, is the simplest solution to the semiclassical Einstein equation (8View Equation), the so-called trivial solution of semiclassical gravity [110Jump To The Next Citation Point]. It constitutes the ground state of semiclassical gravity. In fact, we can always choose a renormalization scheme in which the renormalized expectation value ⟨0| ˆT ab [η]|0⟩ = 0 R. Thus, the Minkowski spacetime 4 (IR ,ηab) plus the vacuum state |0⟩ is a solution to the semiclassical Einstein equation with renormalized cosmological constant Λ = 0. The fact that the vacuum expectation value of the renormalized stress-energy operator in Minkowski spacetime should vanish was originally proposed by Wald [359], and it may be understood as a renormalization convention [121, 135]. Note that other possible solutions of semiclassical gravity with zero vacuum expectation value of the stress-energy tensor are the exact gravitational plane waves, since they are known to be vacuum solutions of Einstein equations, which induce neither particle creation nor vacuum polarization [91, 125, 128].

As we have already mentioned, the vacuum |0⟩ is an eigenstate of the total four-momentum operator in Minkowski spacetime, but not an eigenstate of ˆR Tab[η]. Hence, even in the Minkowski background there are quantum fluctuations in the stress-energy tensor and, as a result, the noise kernel does not vanish. This fact leads us to consider stochastic corrections to this class of trivial solutions of semiclassical gravity. Since in this case the Wightman and Feynman functions (44View Equation), their values in the two-point coincidence limit and the products of derivatives of two of such functions appearing in expressions (45View Equation) and (46View Equation) are known in dimensional regularization, we can compute the Einstein–Langevin equation using the methods outlined in Sections 3 and 4.

To perform explicit calculations it is convenient to work in a global inertial coordinate system {xμ} and in the associated basis, in which the components of the flat metric are simply ημν = diag (− 1,1, ...,1). In Minkowski spacetime, the components of the classical stress-energy tensor (4View Equation) reduce to

T μν[η, φ] = ∂ μφ∂νφ − 1ημν∂ ρφ∂ρφ − 1-ημνm2 φ2 + ξ(ημν□ − ∂μ∂ν) φ2, (85 ) 2 2
where □ ≡ ∂μ∂μ and the formal expression for the components of the corresponding “operator” in dimensional regularization, see Equation (5View Equation), is
{ } ˆTμnν[η] = 1- ∂ μˆφn,∂νφˆn + 𝒟 μνˆφ2n, (86 ) 2
where 𝒟 μν is the differential operator (6View Equation), with gμν = ημν, R μν = 0, and ∇ μ = ∂μ. The field ˆ φn (x) is the field operator in the Heisenberg representation in an n-dimensional Minkowski spacetime, which satisfies the Klein–Gordon equation (2View Equation). We use here a stress-energy tensor, which differs from the canonical one that corresponds to ξ = 0; both tensors, however, define the same total momentum.

The Wightman and Feynman functions (44View Equation) for g = η μν μν are well known:

G+ (x,y ) = iΔ+ (x − y ), G (x,y) = Δ (x − y), (87 ) n n Fn Fn
∫ + -dnk-- ikx 2 2 0 Δ n(x ) = − 2πi (2π)n e δ(k + m )θ (k ), ∫ n ikx (88 ) Δ (x ) = − d--k------e-------- for ε → 0+, Fn (2 π)n k2 + m2 − iε
where 2 μ ν k ≡ ημνk k and μ ν kx ≡ ημνk x. Note that the derivatives of these functions satisfy ∂xμ Δ+n (x − y) = ∂μΔ+n(x − y) and ∂yμΔ+n(x − y) = − ∂μΔ+n(x − y), and similarly for the Feynman propagator ΔFn (x − y).

To write down the semiclassical Einstein equation (8View Equation) in n dimensions for this case, we need to compute the vacuum expectation value of the stress-energy operator components (86View Equation). Since, from (87View Equation), we have that ˆ2 + ⟨0|φn(x)|0⟩ = iΔFn (0) = iΔ n (0 ), which is a constant (independent of x), we have simply

( ) ⟨ || μν || ⟩ ∫ dnk kμkν ημν m2 n∕2 ( n ) 0|Tˆn [η ]|0 = − i ----n--2-----2-----= ---- --- Γ − -- , (89 ) (2π) k + m − iε 2 4 π 2
where the integrals in dimensional regularization have been computed in the standard way [259Jump To The Next Citation Point], and where Γ (z) is Euler’s gamma function. The semiclassical Einstein equation (8View Equation) in n dimensions before renormalization reduces now to
⟨ | | ⟩ -ΛB---ημν = μ −(n−4) 0 ||Tˆμν[η]|| 0 . (90 ) 8πGB n
Thus, this equation simply sets the value of the bare coupling constant Λ ∕G B B. Note from Equation (89View Equation) that in order to have ˆμν ⟨0 |TR |0⟩[η] = 0, the renormalized and regularized stress-energy tensor “operator” for a scalar field in Minkowski spacetime, see Equation (7View Equation), has to be defined as
( )n−4- μν −(n−4) μν ημν m4 m2 2 ( n) ˆTR [η] = μ Tˆn [η ] −-2--(4π-)2 4πμ2- Γ − 2- , (91 )
which corresponds to a renormalization of the cosmological constant
4 ΛB--= Λ-− 2----m-----κn + 𝒪 (n − 4), (92 ) GB G π n(n − 2)
( )n−4- ( ) --1--- eγm2- 2 --1--- 1- eγm2- κn ≡ n − 4 4πμ2 = n − 4 + 2 ln 4πμ2 + 𝒪 (n − 4), (93 )
with γ being Euler’s constant. In the case of a massless scalar field, 2 m = 0, one simply has ΛB ∕GB = Λ ∕G. Introducing this renormalized coupling constant into Equation (90View Equation), we can take the limit n → 4. We find that, for (IR4, ηab,|0⟩) to satisfy the semiclassical Einstein equation, we must take Λ = 0.

We can now write down the Einstein–Langevin equations for the components hμν of the stochastic metric perturbation in dimensional regularization. In our case, using ⟨0|ˆφ2n(x)|0⟩ = iΔFn (0) and the explicit expression of Equation (41View Equation), we obtain

[ ( )] --1--- G (1)μν + ΛB h μν − 1-ημνh (x) − 4αBD (1)μν(x) − 2βBB (1)μν(x) 8πGB 2 3 ∫ − ξG (1)μν(x )μ −(n−4)iΔFn (0) + 1- dny μ −(n− 4)H μnναβ(x,y )h αβ(y) = ξμν(x). (94 ) 2
The indices in hμν are raised with the Minkowski metric and h ≡ hρ ρ; here a superindex (1) denotes the components of a tensor linearized around the flat metric. Note that in n dimensions the two-point correlation functions for the field μν ξ is written as
⟨ μν αβ ⟩ −2(n− 4) μναβ ξ (x)ξ (y )s = μ N n (x,y). (95 )

Explicit expressions for (1)μν D and (1)μν B are given by

(1)μν 1 μναβ (1)μν μν αβ D (x) = --ℱx hαβ(x), B (x) = 2ℱ x ℱ x h αβ(x), (96 ) 2
with the differential operators μν μν μ ν ℱ x ≡ η □x − ∂x∂x and μναβ μ(α β)ν μν αβ ℱ x ≡ 3ℱ x ℱx − ℱx ℱx.
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