8.1 Hadamard self-field regularization
In most practical computations we employ the Hadamard regularization [128, 199
] (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. [36
, 39
].
Consider the class
of functions
which are smooth (
) on
except for the
two points
and
, around which they admit a power-like singular expansion of the
type
and similarly for the other point 2. Here
, and the coefficients
of the various
powers of
depend on the unit direction
of approach to the singular point. The
powers
of
are real, range in discrete steps (i.e.
), and are bounded from below
(
). The coefficients
(and
) for which
can be referred to as the singular
coefficients of
. If
and
belong to
so does the ordinary product
, as well as the
ordinary gradient
. We define the Hadamard partie finie of
at the location of the point 1 where it
is singular as
where
denotes the solid angle element centered on
and of direction
. Notice that
because of the angular integration in Equation (122), the Hadamard partie finie is “non-distributive” in the
sense that
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 (
) concerns that of the integral
, which is
generically divergent at the location of the two singular points
and
(we assume that the integral
converges at infinity). It is defined by
The first term integrates over a domain
defined as
from which the two spherical balls
and
of radius
and centered on the two singularities, denoted
and
, are excised:
. The other terms, where the value of a
function at point 1 takes the meaning (122) are such that they cancel out the divergent part
of the first term in the limit where
(the symbol
means the same terms but
corresponding to the other point 2). The Hadamard partie-finie integral depends on two strictly
positive constants
and
, associated with the logarithms present in Equation (124). These
constants will ultimately yield some gauge-type constants, denoted by
and
, in the 3PN
equations of motion and radiation field. See Ref. [36
] 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. [36
, 39
]. The basic idea is to associate to any
a
pseudo-function, called the partie finie pseudo-function
, namely a linear form acting on functions
of
, and which is defined by the duality bracket
When restricted to the set
of smooth functions (i.e.
) with compact support (obviously we
have
), the pseudo-function
is a distribution in the sense of Schwartz [199
]. The
product of pseudo-functions coincides, by definition, with the ordinary pointwise product, namely
. 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,
. This is the so-called delta-pseudo-function
defined by
where
is the partie finie of
as given by Equation (122). From the product of
with any
we obtain the new pseudo-function
, that is such that
As a general rule, we are not allowed, in consequence of the “non-distributivity” of the Hadamard partie
finie, Equation (123), to replace
within the pseudo-function
by its regularized value:
in general. It should be noticed that the object
has no equivalent in
distribution theory.
Next, we treat the spatial derivative of a pseudo-function of the type
, namely
.
Essentially, we require (in Ref. [36
]) 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,
Furthermore, we assume that when all the singular coefficients of
vanish, the derivative of
reduces to the ordinary derivative, i.e.
. Then it is trivial to check that the rule (128)
contains as a particular case the standard definition of the distributional derivative [199
]. Notably, we see
that the integral of a gradient is always zero:
. 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
where
represents the “ordinary” derivative and
the distributional term. The following
solution of the basic relation (128) was obtained in Ref. [36
]:
where for simplicity we assume that the powers
in the expansion (121) of
are relative integers. The
distributional term (130) is of the form
(plus
). It is generated solely by the singular
coefficients of
(the sum over
in Equation (130) is always finite since there is a maximal order
of divergency in Equation (121)). The formula for the distributional term associated with the
th distributional derivative, i.e.
, where
, reads
We refer to Theorem 4 in Ref. [36
] for the definition of another derivative operator,
representing the most general derivative satisfying the same properties as the one defined by
Equation (130), and, in addition, the commutation of successive derivatives (or Schwarz
lemma).
The distributional derivative (129, 130, 131) does not satisfy the Leibniz rule for the derivation of a
product, in accordance with a general result of Schwartz [198
]. Rather, the investigation [36
] 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 (128), 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
is defined by Equation (122) in a preferred spatial hypersurface
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
is sufficient to get some unambiguous equations of motion which are
Lorentz invariant [42
]. To deal with the problem at 3PN order, a Lorentz-invariant variant
of the regularization, denoted
, was introduced in Ref. [39
]. 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
differs
from the simpler regularization
by relativistic corrections of order
at least. See
Ref. [39
] for the formulas defining this regularization in the form of some infinite power series in
. The regularization
plays a crucial role in obtaining the equations of motion at
the 3PN order in Refs. [37
, 38
]. In particular, the use of the Lorentz-invariant regularization
permits to obtain the value of the ambiguity parameter
in Equation (132)
below.