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6.2 The kernels in the Minkowski background

Since the two kernels (36View Equation) are free of ultraviolet divergences in the limit n --> 4, we can deal directly with the F mnab(x - y) =_ limn -->4 m- 2(n-4)F mnnab in Equation (35View Equation). The kernels N mnab(x,y) = Re F mnab(x - y) and Hmnab (x,y) = Im F mnab(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 bi-tensor mnab F can be expressed in terms of the Wightman function in four spacetime dimensions, according to Equation (38View Equation). The different terms in this kernel can be easily computed using the integrals
integral -d4k-- 2 2 0 2 2 0 0 I(p) =_ (2p)4 d(k + m )h(- k )d[(k - p) + m ]h(k - p ), (90)
and Im1...mr(p), which are defined as the previous one by inserting the momenta km1 ...kmr with r = 1,...,4 in the integrand. All these integrals can be expressed in terms of I(p); see [209Jump 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 m p as
I(p) = IS(p) + IA(p), (91)
where IS(-p) = IS(p) and IA(- p) = - IA(p). These two functions are explicitly given by
V~ --------- 1 m2 IS(p) = -------h(- p2- 4m2) 1 + 4---, 8(2p)3 p2 V~ -------2- (92) I (p) = ---1---sign p0h(- p2 - 4m2) 1 + 4 m-. A 8(2p)3 p2
After some manipulations, we find
integral ( ) mnab p2 mnab d4p - ipx m2 2 F (x) = ---Fx ----4 e 1 + 4 -2- I(p) 45 (2p integral ) ( p ) 8p2 mn ab d4p -ipx m2 2 + -9--Fx Fx (2p)4-e 3Dq + -p2 I(p), (93)
where Dq =_ q- 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
V~ --------- integral 4 2 ( 2 )2 NA(x; m2) =_ --1-- -d-p--eipx h(- p2- 4m2) 1 + 4m-- 1 + 4m-- , 480p (2p)4 p2 p2 V~ --------- 1 integral d4p m2 ( m2 )2 NB(x; m2, Dq) =_ ---- ----4-eipx h(-p2 - 4m2) 1 + 4--2 3Dq + --2 , 72p (2p) p p integral V~ ---------( )2 (94) 2 ---i- -d4p-- ipx 0 2 2 m2- m2- DA(x; m ) =_ 480p (2p)4 e sign p h(- p - 4m ) 1 + 4 p2 1 + 4 p2 , integral 4 V ~ -------2-( 2 )2 D (x;m2, Dq) =_ --i- -d-p--eipx sign p0h(- p2 - 4m2) 1 + 4m-- 3Dq + m-- , B 72p (2p)4 p2 p2
and we finally get
1 N mnab(x,y) = -F mxnabNA(x - y;m2) + F mxnFxabNB(x - y;m2, Dq), 6 (95) mnab 1- mnab 2 mn ab 2 HA (x,y) = 6F x DA(x - y; m ) + F x F x DB(x - y;m ,Dq).
Notice that the noise and dissipation kernels defined in Equation (94View 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 isinpx terms.

The evaluation of the kernel mnab 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 (39View Equation), this kernel can be written in terms of the Feynman propagator (81View Equation) as

-(n-4) mnab mnab m H Sn (x,y) = Im K (x - y), (96)
where
mnab -(n-4){ m (a b) n mn ( a b ) K (x) =_ -m 2@ @ DFn(x) @ @ DFn(x) + 2D @ DFn(x)@ DFn(x) ab m n mn ab ( 2 ) +2D[ (@ DFn(x) @ DFn(x)) + 2D D D Fn(x) + jmn@(aD (x)@b) + jab@(mD (x)@n) + D (0)(jmnDab + jabDmn) Fn Fn Fn 1- mn ab ( 2 )] n } + 4j j DFn(x)[] - m DFn(0) d (x) . (97)
Let us define the integrals
integral dnk 1 Jn(p) =_ m-(n-4) ----------------------------------------, (98) (2p)n (k2 + m2 - ie)[(k - p)2 + m2 - ie]
and Jm1...mr(p) n obtained by inserting the momenta km1 ...kmr into the previous integral, together with
integral n I0 =_ m -(n- 4) -d--k- ------1-------, (99) n (2p)n (k2 + m2 - ie)
and m1...mr I0n which are also obtained by inserting momenta in the integrand. Then, the different terms in Equation (97View Equation) can be computed; these integrals are explicitly given in [209Jump To The Next Citation Point]. It is found that m I0n = 0, and the remaining integrals can be written in terms of I0n and Jn(p). It is useful to introduce the projector P mn orthogonal to pm and the tensor P mnab as
2 mn mn 2 m n mnab m(a b)n mn ab p P =_ j p - p p , P =_ 3P P - P P . (100)
Then the action of the operator F mn x is simply written as F mn integral dnp eipx f(p) = - integral dnp eipxf (p)p2P mn x, where f(p) is an arbitrary function of m p.

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

{ [ mnab i 1 mnab n 2 mn ab n K (x) = ----2- kn --F x d (x) + 4Dq F x F x d (x) (4p) 90 2---m2--- mn ab m(a b)n m(a b) n + 3 (n - 2) (j j []x - j j []x + j @x @x n(a b) m mn a b ab m n n +j @ x @ x- j @x@x - j @x@x)d (x) 4m4 m(a b)n mn ab n ] + n(n----2) (2j j - j j )d (x) integral ( ) -1-- mnab dnp---ipx m2- 2 2 + 180F x (2p)n e 1 + 4 p2 f(p ) integral ( )2 2- mn ab -dnp-- ipx m2- 2 + 9F x F x (2p)n e 3Dq + p2 f(p ) [ ] - --4-F mnab+ -1--(60q - 11)F mnF ab dn(x) 675 x 270 x x [2 1 ] } - m2 ----F mxnab+ ---FmxnF axb Dn(x) + O(n - 4), (101) 135 27
where kn has been defined in Equation (86View Equation), and f(p2) and Dn(x) are given by
V ~ --------- integral 1 ( 2 ) 2 f(p2) =_ da ln 1 + p--a(1 - a) - ie = -iph(- p2 - 4m2) 1 + 4m-- + f(p2), (102) 0 m2 p2 integral dnp 1 Dn(x) =_ ----n-eipx -2, (103) (2p) p
where
integral 1 | | 2 || p2- || f(p ) =_ 0 da ln |1 + m2 a(1 - a)|.

The imaginary part of Equation (101View Equation) gives the kernel components -(n-4) mnab m H Sn (x, y), according to Equation (96View Equation). It can be easily obtained multiplying this expression by - i and retaining only the real part f(p2) of the function f(p2).


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