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6.2 The kernels in the Minkowski background

Since the two kernels (43View Equation) are free of ultraviolet divergences in the limit n → 4, we can deal directly with the F μναβ(x − y) ≡ limn →4 μ− 2(n−4)F μnναβ in Equation (42View Equation). The kernels N μναβ(x,y ) = Re F μναβ(x − y) and H μναβ (x, y) = Im F μναβ (x − y ) A are actually the components of the “physical” noise and dissipation kernels that will appear in the Einstein–Langevin equations once the renormalization procedure has been carried out. The bitensor μναβ F can be expressed in terms of the Wightman function in four spacetime dimensions, according to Equation (45View Equation). The different terms in this kernel can be easily computed using the integrals
∫ -d4k-- 2 2 0 2 2 0 0 I(p) ≡ (2π)4 δ(k + m )θ(− k )δ[(k − p) + m ]θ(k − p ) (97 )
and Iμ1...μr(p), which are defined as in Equation (97View Equation) by inserting the momenta kμ1 ...kμr with r = 1,...,4 into the integrand. All these integrals can be expressed in terms of I(p); see [259Jump To The Next Citation Point] for the explicit expressions. It is convenient to separate I(p) into its even and odd parts with respect to the variables μ p as
I(p) = IS(p) + IA (p ), (98 )
where IS(− p ) = IS(p) and IA(− p) = − IA (p). These two functions are explicitly given by
∘ --------- 1 m2 IS(p) = -------θ(− p2 − 4m2 ) 1 + 4---, 8(2π )3 p2 ∘ --------2 (99 ) I (p) = --−-1--sign (p0)θ(− p2 − 4m2 ) 1 + 4 m--. A 8(2π )3 p2
After some manipulations, we find
∫ ( ) μναβ π2 μναβ d4p − ipx m2 2 F (x ) = ---ℱx ----4 e 1 + 4 -2- I(p) 45 (2π∫) ( p ) 8π2 μν αβ d4p −ipx m2 2 + -9--ℱx ℱx (2π)4-e 3 Δξ + -p2 I(p), (100 )
where Δ ξ ≡ ξ − 16. The real and imaginary parts of the last expression, which yield the noise and dissipation kernels, are easily recognized as the terms containing IS(p) and IA(p), respectively. To write them explicitly, it is useful to introduce the new kernels
∘ --------- ∫ 4 2 ( 2)2 NA (x;m2 ) ≡ --1-- -d-p--eipxθ(− p2 − 4m2) 1 + 4m-- 1 + 4 m-- , 480 π (2π )4 p2 p2 ∘ --------- 1 ∫ d4p m2 ( m2 )2 NB (x;m2, Δ ξ) ≡ ---- -----4 eipx θ(− p2 − 4m2 ) 1 + 4--2 3Δ ξ + --2 , 72π (2π ) p p ∫ ∘ ---------( )2 (101 ) 2 -−-i- -d4p-- ipx 0 2 2 m2- m2- DA (x;m ) ≡ 480 π (2π )4 e sign (p )θ(− p − 4m ) 1 + 4 p2 1 + 4 p2 , ∫ 4 ∘ -------2-( 2)2 D (x;m2, Δ ξ) ≡ -−-i -d-p--eipx sign (p0)θ(− p2 − 4m2 ) 1 + 4m-- 3Δ ξ + m-- . B 72π (2π )4 p2 p2
Finally, we get
1 N μναβ(x,y) = -ℱ μxναβNA (x − y;m2 ) + ℱ μxνℱxαβNB (x − y;m2, Δ ξ), 6 (102 ) μναβ 1- μναβ 2 μν αβ 2 HA (x,y) = 6ℱ x DA (x − y; m ) + ℱ x ℱ x DB (x − y;m ,Δ ξ).
Notice that the noise and dissipation kernels defined in Equation (101View Equation) are actually real because, for the noise kernels, only the cospx terms of the exponentials ipx e contribute to the integrals, and, for the dissipation kernels, the only contribution of such exponentials comes from the isin px terms.

The evaluation of the kernel μναβ H Sn (x,y) is a more involved task. Since this kernel contains divergences in the limit n → 4, we use dimensional regularization. Using Equation (46View Equation), this kernel can be written in terms of the Feynman propagator (88View Equation) as

−(n−4) μναβ μναβ μ H Sn (x,y) = Im K (x − y), (103 )
where
{ ( ) K μναβ(x ) ≡ − μ −(n− 4) 2 ∂μ∂(αΔFn (x)∂ β)∂ νΔFn (x) + 2 𝒟μν ∂αΔFn (x)∂β ΔFn (x ) +2 𝒟 αβ (∂μΔ (x)∂ νΔ (x )) + 2𝒟 μν𝒟 αβ (Δ2 (x )) [ Fn Fn Fn μν (α β) αβ (μ ν) ( μν αβ αβ μν) + η ∂ ΔFn (x )∂ + η ∂ ΔFn (x)∂ + ΔFn (0) η 𝒟 + η 𝒟 ( ) ] } + 1η μνηαβ ΔFn (x)□ − m2 ΔFn (0) δn(x) . (104 ) 4
Let us define the integrals
∫ dnk 1 Jn (p) ≡ μ−(n−4) ----------------------------------------, (105 ) (2π)n (k2 + m2 − iε)[(k − p)2 + m2 − iε]
and Jμ1...μr(p) n obtained by inserting the momenta kμ1 ...k μr into Equation (105View Equation), together with
∫ dnk 1 I0n ≡ μ −(n− 4) -----n --2----2------, (106 ) (2π ) (k + m − iε)
and Iμ1...μr 0n, which are also obtained by inserting momenta into the integrand. Then the different terms in Equation (104View Equation) can be computed; these integrals are explicitly given in [259Jump To The Next Citation Point]. It is found that μ I0n = 0, and the remaining integrals can be written in terms of I0n and Jn(p). It is useful to introduce the projector P μν orthogonal to pμ and the tensor P μναβ as
p2P μν ≡ ημνp2 − pμpν, P μναβ ≡ 3P μ(αP β)ν − P μνPαβ. (107 )
Then the action of the operator ℱ μxν is simply written as ∫ ∫ ℱμxν dnpeipxf (p ) = − dnp eipxf(p) p2Pμν, where f(p) is an arbitrary function of pμ.

Finally, after a rather long but straightforward calculation, and after expanding around n = 4, we get

{ i [ 1 K μναβ(x) = ------ κn --ℱ μxναβδn(x ) + 4Δ ξ2ℱ μxνℱ αxβδn(x) (4π)2 90 2 m2 ( + --------- η μνηαβ□x − ημ(αη β)ν□x + ημ(α∂βx)∂νx 3 (n − 2) +η ν(α∂ β)∂ μ− ημν∂α∂β − ηα β∂μ∂ν)δn(x ) 4 x x x x ] x x + --4m-----(2ημ(αηβ)ν − ημνηαβ)δn (x ) n (n − 2) ∫ n ( 2)2 + -1-ℱ μναβ -d--p-eipx 1 + 4m-- ¯φ(p2) 180 x (2π )n p2 ∫ n ( 2)2 + 2ℱ μνℱ αβ -d-p--eipx 3Δ ξ + m-- φ¯(p2) 9 x x (2π )n p2 [ 4 1 ] − ----ℱ μxναβ+ ----(60ξ − 11)ℱ μxνℱ αxβ δn(x) 675 270 } [ 2 1 ] − m2 ----ℱxμναβ+ ---ℱμxνℱ αxβ Δn (x) + 𝒪(n − 4 ), (108 ) 135 27
where κn has been defined in Equation (93View Equation), and ¯ 2 φ (p ) and Δn (x) are given by
∘ --------- ∫ 1 ( p2 ) m2 ¯φ(p2) ≡ d α ln 1 + --2α(1 − α ) − iε = − iπθ (− p2 − 4m2 ) 1 + 4--2 + ϕ (p2), (109 ) ∫0 m p dnp ipx 1 Δn (x) ≡ (2π)n-e p2, (110 )
where
| | 2 ∫ 1 | p2 | ϕ(p ) ≡ d α ln ||1 + --2 α (1 − α)||. 0 m

The imaginary part of Equation (108View Equation) gives the kernel components μναβ μ−(n−4)H Sn (x,y), according to Equation (103View Equation). It can be easily obtained by multiplying this expression by − i and retaining only the real part ϕ (p2) of the function φ¯(p2).


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