As we have already mentioned the vacuum is an eigenstate of the total four-momentum operator in Minkowski spacetime, but not an eigenstate of . 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 to consider the stochastic corrections to this class of trivial solutions of semiclassical gravity. Since, in this case, the Wightman and Feynman functions (37), their values in the two-point coincidence limit, and the products of derivatives of two of such functions appearing in expressions (38) and (39) 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 and in the associated basis, in which the components of the flat metric are simply . In Minkowski spacetime, the components of the classical stress-energy tensor (3) reduce to

where , and the formal expression for the components of the corresponding “operator” in dimensional regularization, see Equation (4), is where is the differential operator (5), with , , and . The field is the field operator in the Heisenberg representation in an -dimensional Minkowski spacetime, which satisfies the Klein–Gordon equation (2). We use here a stress-energy tensor which differs from the canonical one, which corresponds to ; both tensors, however, define the same total momentum.The Wightman and Feynman functions (37) for are well known:

with where and . Note that the derivatives of these functions satisfy and , and similarly for the Feynman propagator .To write down the semiclassical Einstein equation (7) in dimensions for this case, we need to compute the vacuum expectation value of the stress-energy operator components (79). Since, from (80), we have that , which is a constant (independent of ), we have simply

where the integrals in dimensional regularization have been computed in the standard way [209], and where is Euler’s gamma function. The semiclassical Einstein equation (7) in dimensions before renormalization reduces now to This equation, thus, simply sets the value of the bare coupling constant . Note, from Equation (82), that in order to have , the renormalized and regularized stress-energy tensor “operator” for a scalar field in Minkowski spacetime, see Equation (6), has to be defined as which corresponds to a renormalization of the cosmological constant where with being Euler’s constant. In the case of a massless scalar field, , one simply has . Introducing this renormalized coupling constant into Equation (83), we can take the limit . We find that, for to satisfy the semiclassical Einstein equation, we must take .We can now write down the Einstein–Langevin equations for the components of the stochastic metric perturbation in dimensional regularization. In our case, using and the explicit expression of Equation (34), we obtain

The indices in are raised with the Minkowski metric, and ; here a superindex denotes the components of a tensor linearized around the flat metric. Note that in dimensions the two-point correlation functions for the field is written asExplicit expressions for and are given by

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