### 4.3 Explicit form of the Einstein-Langevin equation

We can write the Einstein–Langevin equation in a more explicit form by working out the expansion of up to linear order in the perturbation . From Equation (26), we see that this expansion can be easily obtained from Equation (30). The result is
Here we use a subscript on a given tensor to indicate that we are explicitly working in dimensions, as we use dimensional regularization, and we also use the superindex to generally indicate that the tensor is the first-order correction, linear in , in a perturbative expansion around the background .

Using the Klein–Gordon equation (2) and expression (4) for the stress-energy tensor and the corresponding operator, we can write

where is the differential operator
It is understood that indices are raised with the background inverse metric , and that all the covariant derivatives are associated with the metric .

Substituting Equation (38) into the -dimensional version of the Einstein–Langevin Equation (15), taking into account that satisfies the semiclassical Einstein equation (8), and substituting expression (39), we can write the Einstein–Langevin equation in dimensional regularization as

where the tensors , , and are computed from the semiclassical metric , and where we have omitted the functional dependence on and in , , , and to simplify the notation. The parameter is a mass scale, which relates the dimensions of the physical field with the dimensions of the corresponding field in dimensions, . Notice that, in Equation (41), all the ultraviolet divergences in the limit , which must be removed by renormalization of the coupling constants, are in and the symmetric part of the kernel , whereas the kernels and are free of ultraviolet divergences. If we introduce the bitensor defined by
where is defined by Equation (13), then the kernels and can be written as
where we have used

and the fact that the first term on the right-hand side of this identity is real, whereas the second one is purely imaginary. Once we perform the renormalization procedure in Equation (41), setting will yield the physical Einstein–Langevin equation. Due to the presence of the kernel , this equation will usually be non-local in the metric perturbation. In Section 6 we will carry out an explicit evaluation of the physical Einstein–Langevin equation, which will illustrate the procedure.

#### 4.3.1 The kernels for the vacuum state

When the expectation values in the Einstein–Langevin equation are taken in a vacuum state , such as, an “in” vacuum, we can be more explicit, since we can write the expectation values in terms of the Wightman and Feynman functions, defined as

These expressions for the kernels in the Einstein–Langevin equation will be very useful for explicit calculations. To simplify the notation, we omit the functional dependence on the semiclassical metric , which will be understood in all the expressions below.

From Equations (43), we see that the kernels and are the real and imaginary parts, respectively, of the bitensor . From expression (5) we see that the stress-energy operator can be written as a sum of terms of the form , where and are differential operators. It then follows that we can express the bitensor in terms of the Wightman function as

where is the differential operator (6). From this expression and relations (43), we get expressions for the kernels and in terms of the Wightman function .

Similarly, the kernel can be written in terms of the Feynman function as

where is the differential operator
Note that, in the vacuum state , the term in Equation (41) can also be written as .

Finally, the causality of the Einstein–Langevin equation (41) can be explicitly seen as follows. The non-local terms in that equation are due to the kernel , which is defined in Equation (29) as the sum of and . Now, when the points and are spacelike separated, and commute and, thus, , which is real. Hence, from the above expressions, we have that , and thus . This fact is expected since, from the causality of the expectation value of the stress-energy operator [359], we know that the non-local dependence on the metric perturbation in the Einstein–Langevin equation, see Equation (15), must be causal. See [208] for an alternative proof of the causal nature of the Einstein–Langevin equation.