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

where denotes an arbitrary length scale. The explicit expression of the gravitational source term involves some -dependent coefficients, and is given by When we recover Equation (15). In the following we assume, as usual in dimensional regularization, that the dimension of space is a complex number, , and prove many results by invoking complex analytic continuation in . We shall pose .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

which generalize Equations (116). As a result all the expressions of Section 7 acquire some explicit -dependent coefficients. For instance we find [30] Here means the retarded integral in space-time dimensions, which admits, though, no simple expression in physical space.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

The coefficients depend on , and the powers of involve the relative integers and whose values are limited by some , , and as indicated. Here we will be interested in functions which have no poles as (this will always be the case at 3PN order). Therefore, we can deduce from the fact that is continuous at the constraintFor 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,

where is a constant related to the usual Eulerian -function byNext 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

where and are the two masses. The terms corresponding to the -ambiguity in the acceleration of particle 1 read simply where the relative distance between particles is denoted (with being the unit vector pointing from particle 2 to particle 1). We start from the end result of Ref. [38] 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 where is a fully determined functional of the masses and , the relative distance , the coordinate velocities and , and also the gauge constants and . The only ambiguous term is the second one and is given by Equation (151).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. [30], 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 [199]. 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

where the ’s denote the extra terms following from the EHR prescriptions. The pHS-regularized acceleration (153) constitutes essentially the result of the first stage of the calculation of , as reported in Ref. [109].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

It is made of the sum of all the individual differences of Poisson or Poisson-like integrals as computed in Equation (149). The total difference (154) depends on the Hadamard regularization scales and (or equivalently on and , ), and on the parameters associated with dimensional regularization, namely and the characteristic length scale introduced in Equation (139). Finally, our main result is the explicit computation of the -expansion of the dimensional regularization (DR) acceleration as With this result we can prove two theorems [30]: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 [96]. 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. [38]), in the sense that

where 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
read^{30}:

http://www.livingreviews.org/lrr-2006-4 |
© Max Planck Society and the author(s)
Problems/comments to |