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8.1 Hadamard self-field regularization

In most practical computations we employ the Hadamard regularization [128199Jump To The Next Citation Point] (see Ref. [200] for an entry to the mathematical literature). Let us present here an account of this regularization, as well as a theory of generalized functions (or pseudo-functions) associated with it, following the investigations detailed in Refs. [36Jump To The Next Citation Point39Jump To The Next Citation Point].

Consider the class F of functions F (x) which are smooth (o o C) on 3 R except for the two points y1 and y2, around which they admit a power-like singular expansion of the type22

s um a n A n (- N, F (x) = r1 f1a(n1) + o(r1), (121) a0<a <n
and similarly for the other point 2. Here r1 = |x - y1|--> 0, and the coefficients 1fa of the various powers of r1 depend on the unit direction n1 = (x - y1)/r1 of approach to the singular point. The powers a of r1 are real, range in discrete steps (i.e. a (- (ai)i (- N), and are bounded from below (a0 < a). The coefficients 1fa (and 2fa) for which a < 0 can be referred to as the singular coefficients of F. If F and G belong to F so does the ordinary product F G, as well as the ordinary gradient @iF. We define the Hadamard partie finie of F at the location of the point 1 where it is singular as
integral d_O_1 (F )1 = ---- f0(n1), (122) 4p 1
where d_O_1 = d_O_(n1) denotes the solid angle element centered on y1 and of direction n1. Notice that because of the angular integration in Equation (122View Equation), the Hadamard partie finie is “non-distributive” in the sense that
(FG)1 /= (F)1(G)1 in general. (123)
The non-distributivity of Hadamard’s partie finie is the main source of the appearance of ambiguity parameters at the 3PN order, as discussed in Section 8.2.

The second notion of Hadamard partie finie (Pf) concerns that of the integral integral 3 d xF, which is generically divergent at the location of the two singular points y1 and y2 (we assume that the integral converges at infinity). It is defined by

{ } integral integral sum sa+3 ( F ) ( s ) ( ) Pfs1s2 d3xF = lim d3xF + 4p ------ -a- + 4pln -- r31F + 1 <--> 2 .(124) s-->0 S(s) a+3<0a + 3 r1 1 s1 1
The first term integrates over a domain S(s) defined as R3 from which the two spherical balls r < s 1 and r < s 2 of radius s and centered on the two singularities, denoted B(y ,s) 1 and B(y2, s), are excised: 3 S(s) =_ R \B(y1, s) U B(y2, s). The other terms, where the value of a function at point 1 takes the meaning (122View Equation) are such that they cancel out the divergent part of the first term in the limit where s --> 0 (the symbol 1 <--> 2 means the same terms but corresponding to the other point 2). The Hadamard partie-finie integral depends on two strictly positive constants s1 and s2, associated with the logarithms present in Equation (124View Equation). These constants will ultimately yield some gauge-type constants, denoted by ' r1 and ' r2, in the 3PN equations of motion and radiation field. See Ref. [36Jump To The Next Citation Point] for alternative expressions of the partie-finie integral.

We now come to a specific variant of Hadamard’s regularization called the extended Hadamard regularization and defined in Refs. [36Jump To The Next Citation Point39Jump To The Next Citation Point]. The basic idea is to associate to any F (- F a pseudo-function, called the partie finie pseudo-function Pf F, namely a linear form acting on functions G of F, and which is defined by the duality bracket

integral A G (- F , <Pf F,G > = Pf d3x F G. (125)
When restricted to the set D of smooth functions (i.e. o o 4 C (R )) with compact support (obviously we have D < F), the pseudo-function Pf F is a distribution in the sense of Schwartz [199Jump To The Next Citation Point]. The product of pseudo-functions coincides, by definition, with the ordinary pointwise product, namely Pf F.Pf G = Pf(F G). In practical computations, we use an interesting pseudo-function, constructed on the basis of the Riesz delta function [190], which plays a role analogous to the Dirac measure in distribution theory, d1(x) =_ d(x - y1). This is the so-called delta-pseudo-function Pf d1 defined by
integral 3 A F (- F, <Pf d1,F > = Pf d xd1F = (F )1, (126)
where (F)1 is the partie finie of F as given by Equation (122View Equation). From the product of Pf d1 with any Pf F we obtain the new pseudo-function Pf(F d1), that is such that
A G (- F , <Pf(F d1),G > = (F G)1. (127)
As a general rule, we are not allowed, in consequence of the “non-distributivity” of the Hadamard partie finie, Equation (123View Equation), to replace F within the pseudo-function Pf(F d1) by its regularized value: Pf(F d1) /= (F )1 Pf d1 in general. It should be noticed that the object Pf(F d1) has no equivalent in distribution theory.

Next, we treat the spatial derivative of a pseudo-function of the type Pf F, namely @ (Pf F ) i. Essentially, we require (in Ref. [36Jump To The Next Citation Point]) that the so-called rule of integration by parts holds. By this we mean that we are allowed to freely operate by parts any duality bracket, with the all-integrated (“surface”) terms always zero, as in the case of non-singular functions. This requirement is motivated by our will that a computation involving singular functions be as much as possible the same as if we were dealing with regular functions. Thus, by definition,

A F, G (- F , <@i(Pf F ),G > = - <@i(Pf G), F >. (128)
Furthermore, we assume that when all the singular coefficients of F vanish, the derivative of Pf F reduces to the ordinary derivative, i.e. @i(Pf F ) = Pf(@iF). Then it is trivial to check that the rule (128View Equation) contains as a particular case the standard definition of the distributional derivative [199Jump To The Next Citation Point]. Notably, we see that the integral of a gradient is always zero: <@i(Pf F ),1> = 0. This should certainly be the case if we want to compute a quantity (e.g., a Hamiltonian density) which is defined only modulo a total divergence. We pose
@i(Pf F) = Pf(@iF ) + Di[F], (129)
where Pf(@iF ) represents the “ordinary” derivative and Di[F ] the distributional term. The following solution of the basic relation (128View Equation) was obtained in Ref. [36Jump To The Next Citation Point]:
( ) [ sum ] Di[F] = 4p Pf ni1 1-r1 f -1 + -1- f-2-k d1 + 1 <--> 2, (130) 2 1 k>0rk1 1
where for simplicity we assume that the powers a in the expansion (121View Equation) of F are relative integers. The distributional term (130View Equation) is of the form Pf(Gd1) (plus 1 <--> 2). It is generated solely by the singular coefficients of F (the sum over k in Equation (130View Equation) is always finite since there is a maximal order a0 of divergency in Equation (121View Equation)). The formula for the distributional term associated with the lth distributional derivative, i.e. DL[F ] = @L Pf F - Pf @LF, where L = i1i2...il, reads
sum l DL[F ] = @i1...ik-1Dik[@ik+1...ilF ]. (131) k=1
We refer to Theorem 4 in Ref. [36Jump To The Next Citation Point] for the definition of another derivative operator, representing the most general derivative satisfying the same properties as the one defined by Equation (130View Equation), and, in addition, the commutation of successive derivatives (or Schwarz lemma)23.

The distributional derivative (129View Equation, 130View Equation, 131View Equation) does not satisfy the Leibniz rule for the derivation of a product, in accordance with a general result of Schwartz [198Jump To The Next Citation Point]. Rather, the investigation [36Jump To The Next Citation Point] suggests that, in order to construct a consistent theory (using the “ordinary” product for pseudo-functions), the Leibniz rule should be weakened, and replaced by the rule of integration by part, Equation (128View Equation), which is in fact nothing but an “integrated” version of the Leibniz rule. However, the loss of the Leibniz rule stricto sensu constitutes one of the reasons for the appearance of the ambiguity parameters at 3PN order.

The Hadamard regularization (F )1 is defined by Equation (122View Equation) in a preferred spatial hypersurface t = const of a coordinate system, and consequently is not a priori compatible with the Lorentz invariance. Thus we expect that the equations of motion in harmonic coordinates (which manifestly preserve the global Lorentz invariance) should exhibit at some stage a violation of the Lorentz invariance due to the latter regularization. In fact this occurs exactly at the 3PN order. Up to the 2.5PN level, the use of the regularization (F )1 is sufficient to get some unambiguous equations of motion which are Lorentz invariant [42Jump To The Next Citation Point]. To deal with the problem at 3PN order, a Lorentz-invariant variant of the regularization, denoted [F ]1, was introduced in Ref. [39Jump To The Next Citation Point]. It consists of performing the Hadamard regularization within the spatial hypersurface that is geometrically orthogonal (in a Minkowskian sense) to the four-velocity of the particle. The regularization [F ]1 differs from the simpler regularization (F )1 by relativistic corrections of order 1/c2 at least. See Ref. [39Jump To The Next Citation Point] for the formulas defining this regularization in the form of some infinite power series in 1/c2. The regularization [F ]1 plays a crucial role in obtaining the equations of motion at the 3PN order in Refs. [37Jump To The Next Citation Point38Jump To The Next Citation Point]. In particular, the use of the Lorentz-invariant regularization [F ]1 permits to obtain the value of the ambiguity parameter wkinetic in Equation (132View Equation) below.

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