The motion of a charged particle interacting with its own
electromagnetic field is an area of research that has a long history;
this problem has never ceased to fascinate its investigators. On the
one hand the theory ought to be straightforward to formulate: one has
Maxwell's equations that tell the field how to behave (given the
motion of the particle), and one has the Lorentz-force law that tells
the particle how to move (given the field).
On the other hand the theory is fundamentally ambiguous because of the field singularities
that necessarily come with a point particle. While each separate
sub-problem can easily be solved, to couple the field to the particle
in a self-consistent treatment turns out to be tricky. I believe it is
this dilemma (the theory is straightforward but tricky) that has been
the main source of the endless fascination.
For readers of Classical and Quantum Gravity, the fascination does not end there. For them it
is also rooted in the fact that the electromagnetic self-force problem
is deeply analogous to the gravitational self-force problem, which is
of direct relevance to future gravitational wave observations. The
motion of point particles in curved spacetime has been the topic of a
recent Topical Review [1], and it was the focus of a recent Special
Issue [2].
It is surprising to me that radiation reaction is a subject that
continues to be poorly covered in the standard textbooks, including
Jackson's bible [3]. Exceptions are Rohrlich's excellent text [4],
which makes a very useful introduction to radiation reaction, and
the Landau and Lifshitz classic [5], which contains what is probably
the most perfect summary of the foundational ideas (presented in
characteristic terseness).
It is therefore with some trepidation that
I received Herbert Spohn's book, which covers both the classical and
quantum theories of a charged particle coupled to its own field (the
presentation is limited to flat spacetime). Is this the text that
graduate students and researchers should turn to in order to get a
complete and accessible education in radiation reaction?
My answer is that while the book does indeed contain a lot of useful
material, it is not a very accessible source of information, and it is
certainly not a student-friendly textbook. Instead, the book presents
a technical account of the author's personal take on the theory, and
represents a culminating summary of the author's research
contributions over more than a decade. The book is written in a fairly
mathematical style (the author is Professor of Mathematical Physics at
the Technische Universitat in Munich), and it very much emphasises
mathematical rigour. This makes the book less accessible than I would
wish it to be, but this is perhaps less a criticism than a statement
about my taste, expectation, and attitude.
The presentation of the classical theory begins with a point particle,
but Spohn immediately smears the charge distribution to eliminate the
vexing singularities of the retarded field. He considers both the
nonrelativistic Abraham model (in which the extended particle is
spherically symmetric in the laboratory frame) and the relativistic
Lorentz model (in which the particle is spherical in its rest
frame).
In Spohn's work, the smearing of the charge distribution is
entirely a mathematical procedure, and I would have wished for a more
physical discussion. A physically extended body, held together against
electrostatic repulsion by cohesive forces (sometimes called Poincaré
stresses) would make a sound starting point for a classical theory of
charged particles, and would have nicely (and physically) motivated
the smearing operation adopted in the book.
Spohn goes on to derive energy–momentum relations for the extended
objects, and to obtain their equations of motion. A compelling aspect
of his presentation is that he formally introduces the 'adiabatic
limit', the idea that the external fields acting on the charged body
should have length and time scales that are long compared with the
particle's internal scales (respectively the electrostatic classical
radius and its associated time scale).
As a consequence, the equations
of motion do not involve a differentiated acceleration vector (as is
the case for the Abraham–Lorentz–Dirac equations) but are proper
second-order differential equations for the position vector. In
effect, the correct equations of motion are obtained from the
Abraham–Lorentz–Dirac equations by a reduction-of-order procedure that
was first proposed (as far as I know) by Landau and Lifshitz [5]. In
Spohn's work this procedure is not {\it ad hoc}, but a natural consequence
of the adiabatic approximation.
An aspect of the classical portion of the book that got me
particularly excited is Spohn's proposal for an experimental test of
the predictions of the Landau–Lifshitz equations. His proposed
experiment involves a Penning trap, a device that uses a uniform
magnetic field and a quadrupole electric field to trap an electron for
very long times. Without radiation reaction, the motion of an electron
in the trap is an epicycle that consists of a rapid (and small)
cyclotron orbit superposed onto a slow (and large) magnetron
orbit. Spohn shows that according to the Landau–Lifshitz equations,
the radiation reaction produces a damping of the cyclotron motion. For
reasonable laboratory situations this damping occurs over a time scale
of the order of 0.1 second. This experiment might well be within
technological reach.
The presentation of the quantum theory is based on the nonrelativistic
Abraham model, which upon quantization leads to the well-known
Pauli-Fierz Hamiltonian of nonrelativistic quantum
electrodynamics. This theory, an approximation to the fully
relativistic version of QED, has a wide domain of validity that
includes many aspects of quantum optics and laser-matter
interactions. As I am not an expert in this field, my ability to
review this portion of Spohn's book is limited, and I will indeed
restrict myself to a few remarks.
I first admit that I found Spohn's presentation to be tough
going. Unlike the pair of delightful books by Cohen-Tannoudji,
Dupont-Roc, and Grynberg [6, 7], this is not a gentle introduction to
the quantum theory of a charged particle coupled to its own
electromagnetic field. Instead, Spohn proceeds rather quickly through
the formulation of the theory (defining the Hamiltonian and the
Hilbert space) and then presents some applications (for example, he
constructs the ground states of the theory, he examines radiation
processes, and he explores finite-temperature aspects).
There is a lot of material in the eight chapters devoted to the quantum theory, but
my insufficient preparation and the advanced nature of Spohn's
presentation were significant obstacles; I was not able to draw much
appreciation for this material.
One of the most useful resources in Spohn's book are the historical
notes and literature reviews that are inserted at the end of each
chapter. I discovered a wealth of interesting articles by reading
these, and I am grateful that the author made the effort to collect
this information for the benefit of his readers.
References
[1] Poisson E 2004 Radiation reaction of point particles in curved
spacetime Class. Quantum Grav21 R153–R232
[2] Lousto C O 2005 Special issue: Gravitational Radiation from Binary
Black Holes: Advances in the Perturbative Approach, Class. Quantum
Grav22 S543–S868
[3] Jackson J D 1999 Classical Electrodynamics Third Edition (New
York: Wiley)
[4] Rohrlich F 1990 Classical Charged Particles (Redwood City, CA: Addison–Wesley)
[5] Landau L D and Lifshitz E M 2000 The Classical Theory of FieldsFourth Edition (Oxford: Butterworth–Heinemann)
[6] Cohen-Tannoudji C Dupont-Roc J and Grynberg G 1997 Photons and
Atoms - Introduction to Quantum Electrodynamics (New York: Wiley-Interscience)
[7] Cohen-Tannoudji C, Dupont-Roc J and G Grynberg G 1998 Atom–Photon
Interactions: Basic Processes and Applications (New York: Wiley-Interscience)