Natural Science Forum / Physics / Research / December 2007

You are right in the saying that the equations of motion of one or
several particles coupled to a field are inconsistent as usually
written, which for the example of a massless scalar field is as
follows:

 del^2 phi(x) = int ds delta(x-z(s)),             (1)
 m z''(s) = - grad_perp phi(z(s)),                (2)

where del^2 is the D'Alambertian, phi(x) is a scalar field, z(s) is the
particle world line parametrized by proper time s, and grad_perp
corresponds to the space-time gradient projected into the subspace
orthogonal to the world line. Because of a quirk of the usual way of
coupling the scalar field to a particle, the factor m actually
represents an "effective mass", which depends on phi(z(s)), but that's
not too important at the moment.  There could also be more than one
particle, hence more than one world-line, and more than one z. In that
case, the RHS of (1) is additive.

The inconsistency comes from the fact that any solution to (1) gives a
phi(x) that diverges on the particle's world line. Therefore the value,
let alone the gradient, of phi(x) is not defined at z(s), which makes
the RHS of (2) undefined. This has been known since before the time of
Abraham and Lorentz, who were the first to suggest a resolution to
problem.

Having looked at the monograph of Trump and Schieve, according to the
authors, this problem has been one of the major motivations for work on
alternative formulations of relativistic N-body dynamics. Their approach
is decides to get rid of (1) and (2) completely, and start with
something new. But more on that later.

However, one can also try to change (2) so that the RHS is well
defined, while the rest of the equations is unmodified. The motivation
is two fold: the divergence of phi(x) at the world line is due to the
way the world line comes into the RHS of (1) (hence, removing this
self-field may eliminate the divergence alltogether), and if phi(x) is
produced by other means (distant particles, for example), then we know
that the equation of motion for the particle is definitely (2).

The first modification of these equations of motion is due to Abraham,
Lorentz and Dirac, who described a way to subtract the self-field
contribution from the RHS of (2) (along with theoretical reasons for
doing so), leaving behind a finite and well defined force. Their result
was consistent with the well known and tested Larmor formula for the
total radiated power from an accelerating charge. Unfortunately, the
regime where the Abraham-Lorentz-Dirac equation becomes important in
the dynamics of a moving charge, is not easily accessible
experimentally. So, for a long time (and arguably still) the question
of validity of the ALD equation has been largely academic.

Fortunately, this situation is rapidly changing. In recent years, the
question of gravitational radiation of moving massive compact objects
has come to the attention of gravitational wave astronomers (LIGO and
LISA experiments). The theoretical predictions of the emitted wave
forms can be compared to observed data, once the signal to noise ratio
of the gravitational wave detectors gets high enough. Subsequently,
there has been quite a bit of theoretical work on the subject. A great
review of this recent work can be found in an article by Eric Poisson
(2004, http://www.livingreviews.org/lrr-2004-6). Further, the journal
of Classical and Quantum Gravity had a special issue in 2005 dedicated
almost entirely to this problem (volume 22, issue 15,
http://www.iop.org/EJ/toc/0264-9381/22/15).

For those who are curious, the approach described by Poisson consists
of generalizing the arguments used to obtain the ALD equation to curved
spaces and arbitrary fields, including the gravitational field. It also
fixes the most unsavory feature of the ALD equation; it is converted to
second order in time, as opposed to third order. Others have also
generalized the approach to point particles with a little bit of
internal structure (spinning particles, for instance).

The moral of this story is that yes, equations (1) and (2) as written
are not consistent and hence cannot describe the motion of point
particles coupled to fields. On the other hand, there is a well defined
way of modifying the RHS of equation (2) such that the results agrees
with all known observations and can produce predictions that can very
soon be testable. This question of what the correct modification of
equations (1) and (2) will soon no longer be academic.

Now, let me come back to the monograph by Trump and Schieve.  They
propose an alternative formulation of N-body dynamics, without fields
and using ideas very similar to interaction potentials in
non-relativistic Newtonian mechanics. In my understanding, it can be
summarized as follows. For a system of N particles, take R^(4N) as the
configuration space and throw in another copy of R^(4N) for canonically
conjugate momenta, the result is an 8N-dimensional phase space. The
configuration space is seen as an N-fold product of Minkowski
space-times, each endowed with an appropriate metric. The configuration
space is now also endowed with a metric, but this metric is now of the
signature of N minuses and 3N plusses. A corresponding metric is
induced on the space of canonical momenta. The dynamics are specified
by giving a Hamiltonian function and imposing the canonical equations
of motion. The Hamiltonian is required to be invariant under the
Poincare group, as it acts on each copy of Minkowski space separately
and simultaneously. There is an external time parameter, as in the
usual formulation of Hamiltonian classical mechanics. Once an initial
condition is specified, the dynamics yield a curve in the
8N-dimensional phase space. The curve can be first projected into the
4N-dimensional configuration space, and then represented as N curves in
one copy of 4-dimensional Minkowski space. These N curves are
interpreted as the world-lines of N interacting particles.

Unfortunately, there are problems with this approach. Here are some.

The "synchronization" of the initial conditions is arbitrary, but not
irrelevant. To specify initial conditions, on needs to give an event (4
coordinates) from the world line of each particle, and a 4-velocity
tangent to the world line at each of these events. Now, suppose I've
got a set of world lines, obtained by a specific set of initial
conditions. I can choose a different set of initial conditions (events
and 4-velocities) consistent with the same set of world lines and plug
them into the dynamical equations. Generically, the result of solving
the canonical evolution equations with the second set of initial
conditions will yield a different set of world lines! In other words,
in this theory, a set of world lines is not enough to determine the
state of N particles (the coresponding point in phase space), one also
needs to specify a synchronization convention. This is definitely more
information that is justified, say, by the non-relativistic limit.

The causal structure of one trajectory in N copies of Minkowski
space-time is not the same as the causal structure of N trajectories in
the same Minkowski space-time. The result is the appearance of
so-called "exotic" world lines as possible solutions. These are world
lines that fail to remain time-like at all times. And, in my opinion,
it is not sufficient to invoke particle creation or annihilation to
explain this away. The analogy with Feynman diagrams is false, since
the lines in Feynman graphs are not particle trajectories.

In their monograph, Trump and Schieve study "Coulomb" scattering in
this formalism. However, in the attempt to eliminate fields, they have
also eliminated a reason for using the Coulomb potential away from the
non-relativistic regime. Recall that the Coulomb potential is obtained
by solving for the electromagnetic *field* of a stationary charge. In
addition, there appears no way to introduce radiation losses into this
Hamiltonian formalism, which is a well known physical effect (cf.
cyclotron physics).

Lastly, an appendix in Trump and Schieve's book attempts to invalidate
the result of the famous relativistic interaction no-go theorem due to
Currie, Jordan, and Sudarshan (see Rev.Mod.Phys. v.35 p.350).
Unfortunately, a careful reading of this appendix reveals that Trump
and Schieve simply managed to get lost in derivatives and
reparametrizations while interpreting the assumptions of the CJK paper.
The argument given in this appendix has no bearing on the result of the
CJK no-go theorem.

Igor