Natural Science Forum / Physics / Research / January 2005

Electrodynamics can be (should be?) treated on the same level of
abstraction as thermodynamics - see for example A. Sommerfeld,
"Lectures on Theoretical Physics - Vol IV - Electrodynamics" (highly
recommended book).

-drl

Check out F. W. Hehl's papers at arXiv

Once you have Maxwell's equations, you are pretty much set and
experimental input is only necessary for comparing with predictions. Of
course you'll also have to make assumptions about how various sorts of
matter behave in E&M fields, for instance macroscopic polarization and
magnetization. Or do you want to derive Maxwell's equations from some more
basic assumption?

Jackson itself has a very nice bibliography. Some titles that spring to
mind are

_Classical Electricity and Magnetism_ by Panofsky and Phillips
_Classical Electrodynamics_ by Julian Seymour Schwinger
_Course of Theoretical Physics_ Vols. 2 and 8 by Landau and Lifshitz.

See vol. 2 of the latter for a derivation of electrodynamics purely from
an action principle.

Hope this helps.

Igor

One approach:
The Wigner classification of the representations of the Poincare' group
can be applied both to classical and quantum theory, not just quantum
theory.

Another approach:
The classical theory, in fact, factors into a "classical" classical
part (which is the macroscopic field equations which are a subset of
those listed in Maxwell), plus a "non-classical" classical part: the
constitutive relations (which are different that those posed by
Maxwell, in as far as Maxwell posed any).

The former part is quite general, given its huge invariance group and
can be considered separately from the rest.  Then you do, indeed, have
something quite analogous to what you see in thermodynamics.  In fact
more than an analogy: the differential form
theta = -E.dD - H.dB
suddenly comes into prominence.  Using the relations D = E + P, B = H +
M (treating epsilon_0 = mu_0 = 1), this yields
theta = d(-1/2 (E^2 + H^2)) - E.dP - H.dM,
which up to a total differential (and signs and constant factors) is
just the quantity of heat exchange dQ.

The integrability of this form gives you a stress tensor; otherwise you
have a force law given in part by a stress tensor, plus an irreversible
part expressed in terms of theta.  So, one might adopt as an axiom,
which provides a generalizing envelope for any set of constitutive
relations, that theta be an exact differential.  This then implies all
sorts of variant formulations, e.g., a quasi-Lagrangian:
L = L(D,B); dL/dD = -E; dL/dB = -H
and quasi-potentials derived from it by Legendre transforms, e.g., a
quasi-Hamiltonian:
h = h(E,H): dh/dE = D; dh/dH = B; h = D.E + B.H - L.
These play the analogous role of thermodynamic potentials.

So, this gives you (ironically) a set of Maxwell relations, like you
see in thermodynamics, for instance:
(dD/dH)_E = (dB/dE)_H
(del_E x D)_H = 0; (del_H x B)_E = 0.
(Out of this, one finds the constitutive matrices epsilon^{ij} =
dD^j/dE_i; and mu^{ij} = dB^j/dH_i are both symmetric, for instance).

The second part of the factoring then focuses on the constitutive
relations, themselves.  Absent these, the considerations above apply
generally to all sorts of 4-D spacetimes (Galilean, Minkowski, even
Euclidean), since the field equations:
div D = rho; div B = sigma
curl D - dH/dt = J; curl B + dE/dt = -K
sigma = 0, K = 0
and force/power laws
F = rho E + J x B + sigma H - K x D
P = J.E + K.H
are S(GL(4) x GL(2)) invariant (allowing for non-zero sigma, K) with a
subgroup thereof (strictly larger than GL(4), I believe) being the
little group for sigma = 0, K = 0.

So, all the information about spacetime structure is actually locked
into the constitutive relations themselves.  For instance, the set
D = epsilon_0 (E - v x B)
B = mu_0 (D + v x H)
is Galilean invariant, (provided v transforms in the obvious way under
Galilean transformations), and essentially postulates the existence of
an 'ether frame'.  The set
D = epsilon_0 E; B = mu_0 H
in contrast gives you a Minkowski spacetime (up to conformal
invariance).

The whole purpose of the enterprise, one may suppose you're seeking, is
to get a foundation that's consistent and that either removes or
explains away the classical singularity in the force law and stress
tensor.

That's precisely the advantage of this kind of framework.  Because, now
that the duality structure (D* = E, B* = H) or (equivalently) the
constitutive relations (D = epsilon_0 E, B = mu_0 H) have been
separated out, you have more room for movement to address the general
issues, isolate the problem(s) and resolve them.

There's a lot that's non-trivial that needs to be said with this
factoring.  A key theorem would be the condition that results from
requiring that the expressions for force and power remain regular.  In
particular, you can ask what combinations of point-like singularities
are possible that do not result in singularity multiplied by
singularity.  For instance, tracing out the assumption that rho is
singular at a point, you find from (div D = rho) that D will be too.
Then from the law (F = rho E + ...) you find that E must NOT be
singular at that point.  Therefore, the relation (D = epsilon_0 E)
cannot hold in the neighborhood of that point -- which forces you to
revert back to the more general equations posed above.

So, the range of allowable constitutive relations and distributions of
charge are restricted by the consistency requirement.