Dimensional regularization was invented as a means to preserve the gauge symmetry of perturbative quantum field theories [202, 51, 57, 73]. Our basic problem here is to respect the gauge symmetry associated with the diffeomorphism invariance of the classical general relativistic description of interacting point masses. Hence, we use dimensional regularization not merely as a trick to compute some particular integrals which would otherwise be divergent, but as a powerful tool for solving in a consistent way the Einstein field equations with singular point-mass sources, while preserving its crucial symmetries. In particular, we shall prove that dimensional regularization determines the kinetic ambiguity parameter (and its radiation-field analogue ), and is therefore able to correctly keep track of the global Lorentz-Poincaré invariance of the gravitational field of isolated systems.
The Einstein field equations in space-time dimensions, relaxed by the condition of harmonic coordinates , take exactly the same form as given in Equations (9, 14). In particular denotes the flat space-time d’Alembertian operator in dimensions. The gravitational constant is related to the usual three-dimensional Newton’s constant by
We parametrize the 3PN metric in dimensions by means of straightforward -dimensional generalizations of the retarded potentials , , , , and of Section 7. Those are obtained by post-Newtonian iteration of the -dimensional field equations, starting from the following definitions of matter source densities
As reviewed in Section 8.1, the generic functions we have to deal with in 3 dimensions, say , are smooth on except at and , around which they admit singular Laurent-type expansions in powers and inverse powers of and , given by Equation (121). In spatial dimensions, there is an analogue of the function , which results from the post-Newtonian iteration process performed in dimensions as we just outlined. Let us call this function , where . When the function admits a singular expansion which is a little bit more complicated than in 3 dimensions, as it reads
For the problem at hand, we essentially have to deal with the regularization of Poisson integrals, or iterated Poisson integrals (and their gradients needed in the equations of motion), of the generic function . The Poisson integral of , in dimensions, is given by the Green’s function for the Laplace operator,29 [96, 30]. The main technical step of our strategy consists of computing, in the limit , the difference between the -dimensional Poisson potential (147), and its Hadamard 3-dimensional counterpart given by , where the Hadamard partie finie is defined by Equation (122). Actually, we must be very precise when defining the Hadamard partie finie of a Poisson integral. Indeed, the definition (122) stricto sensu is applicable when the expansion of the function , when , does not involve logarithms of ; see Equation (121). However, the Poisson integral of will typically involve such logarithms at the 3PN order, namely some where formally tends to zero (hence is formally infinite). The proper way to define the Hadamard partie finie in this case is to include the into its definition, so we arrive at  [37, 38]. On the other hand, the constant remaining in the result (148) is the source for the appearance of the physical ambiguity parameter , as it will be related to it by Equation (150). Denoting the difference between the dimensional and Hadamard regularizations by means of the script letter , we pose (for the result concerning the point 1) add to the Hadamard-regularization result in order to get the -dimensional result. However, we shall only compute the first two terms of the Laurent expansion of when , say . This is the information we need to clear up the ambiguity parameter. We insist that the difference comes exclusively from the contribution of terms developing some poles in the -dimensional calculation.
Next we outline the way we obtain, starting from the computation of the “difference”, the 3PN equations of motion in dimensional regularization, and show how the ambiguity parameter is determined. By contrast to and which are pure gauge, is a genuine physical ambiguity, introduced in Refs. [36, 38] as the single unknown numerical constant parametrizing the ratio between and (where is the constant left in Equation (148)) as for the 3PN harmonic coordinates acceleration in Hadamard’s regularization, abbreviated as HR. Since the result was obtained by means of the specific extended variant of Hadamard’s regularization (in short EHR, see Section 8.1) we write it as
Our strategy is to express both the dimensional and Hadamard regularizations in terms of their common “core” part, obtained by applying the so-called “pure-Hadamard-Schwartz” (pHS) regularization. Following the definition of Ref. , the pHS regularization is a specific, minimal Hadamard-type regularization of integrals, based on the partie finie integral (124), together with a minimal treatment of “contact” terms, in which the definition (124) is applied separately to each of the elementary potentials (and gradients) that enter the post-Newtonian metric in the form given in Section 7. Furthermore, the regularization of a product of these potentials is assumed to be distributive, i.e. in the case where and are given by such elementary potentials (this is in contrast with Equation (123)). The pHS regularization also assumes the use of standard Schwartz distributional derivatives . The interest of the pHS regularization is that the dimensional regularization is equal to it plus the “difference”; see Equation (155).
To obtain the pHS-regularized acceleration we need to substract from the EHR result a series of contributions, which are specific consequences of the use of EHR [36, 39]. For instance, one of these contributions corresponds to the fact that in the EHR the distributional derivative is given by Equations (129, 130) which differs from the Schwartz distributional derivative in the pHS regularization. Hence we define.
The next step consists of evaluating the Laurent expansion, in powers of , of the difference between the dimensional regularization and the pHS (3-dimensional) computation. As we reviewed above, this difference makes a contribution only when a term generates a pole , in which case the dimensional regularization adds an extra contribution, made of the pole and the finite part associated with the pole (we consistently neglect all terms ). One must then be especially wary of combinations of terms whose pole parts finally cancel (“cancelled poles”) but whose dimensionally regularized finite parts generally do not, and must be evaluated with care. We denote the above defined difference by:
Theorem 8 The pole part of the DR acceleration (155) can be re-absorbed (i.e. renormalized)
into some shifts of the two “bare” world-lines: and , with, say,
, so that the result, expressed in terms of the “dressed” quantities, is finite when .
The situation in harmonic coordinates is to be contrasted with the calculation in ADM-type coordinates within the Hamiltonian formalism, where it was shown that all pole parts directly cancel out in the total 3PN Hamiltonian: No renormalization of the world-lines is needed . A central result is then as follows:
Theorem 9 The renormalized (finite) DR acceleration is physically equivalent to the Hadamard-regularized (HR) acceleration (end result of Ref. ), in the sense thatwhere denotes the effect of the shifts on the acceleration, if and only if the HR ambiguity parameter entering the harmonic-coordinates equations of motion takes the unique value (135).
The precise shifts and needed in Theorem 9 involve not only a pole contribution (which would define a renormalization by minimal subtraction (MS)), but also a finite contribution when . Their explicit expressions read30:
© Max Planck Society and the author(s)