Natural Science Forum / Physics / Research / February 2008

From 2001 December 14, sci.physics.research
http://groups.google.com/group/sci.physics.research/browse_thread/thread/6a23142
6b3a313c0/4b63fb3c38091eb8?hl=en&lnk=gst&q=Fancy+Schmancy+Forms#4b63fb3c38091eb8

From b...@math-cl-n03.math.ucr.edu (John Baez):

There's a double irony here. Maxwell primarily wrote in the language of
differential forms. This isn't clearly seen at first sight, because the
vocabulary of terminology wasn't available at the time. But once you
filter through the verbiage (e.g., "expressions referred to surface
integrals" = 2-forms; etc.) you begin to see that everything was
primarily thought of (and, indeed, written) with differential forms ...
albeit under integral signs.

The lynchpin of this realisation is that ...
  Maxwell Used The Grassmann Algebra!

That is: there is a Grassmann sighting in the treatise -- a place
where the rule dxdy = -dydx was explicitly stated.

But, I would venture that the root of the resistance lies completely
in a different issue -- the contorsion required to fit the Maxwell
equations
  div D = rho; curl H - dD/dt = J
into differential forms in the particular way you fit them in.

The question isn't whether this

gives you what is commonly cited as the Maxwell equations, but whether
it is correct. The second relation is NOT the Maxwell equations -- as
Maxwell had originally set them out. The Maxwell equations would be

dG = J.

What you're doing here is slipping in the Lorentz relations G = *F.
That is the root of the resistance, and the source of all contentions.

First of all, the correct relation is G = epsilon_0 c *F. What modern
theoretical convention has done in the way of "setting units equal to
1" is remove from the picture physics that is highly relevant -- here,
the physics of the permittivity of the vacuum.

Secoind, Maxwell made a major issue of specifically stating that no
such constant relation could hold; the vacuum had to be a dielectric,
too. In subsequent years, a few decades after Lorentz shut off the
vacuum by writing G = epsilon_0 *F (translated in modern notation),
people came to the realization that something inconsistent comes out
of this.

* This relation gives you a Lagrangian that is quadratic in the field
gradients with constant coefficients.

* The constancy of the coefficients yields Green's functions with
singular characteristic surfaces (i.e., one gets singularities arising
from the fact that the extra degree of freedom associated with the
coefficients has been turned off)

* The singularity of the Green's functions leads to singular self-
energy and self-force laws.

* The quantum fields inherit the singularity of the Green's functions
(via the propagators) and become operator-valued distributions,
instead of ordinary operators as they ought to be.

* The distributions collide in non-linear expressions -- the stress
tensor and interacting field equations (i.e. the quantized version of
the self-energy and self-force divergences).

These (except for the last couple items) were the very things that
Maxwell had sought to avoid, arguing as such in the first chapters of
the treatise.

In EM, G and F can NOT be related by duality. At best, this can only
be taken as an approximation that holds asymptotically away from
sources.

More generally, what's happening here (as well as in other field
theories) is that something more general than the Hodge star operator
is at work. Something that might not even be local, but may have a
simple integral form.

In non-Abelian gauge theories, the non-triviality of the G <-> F
relation shows up particularly in SU(3) gauge theory, where one also
acquires a parity-violating contribution:

G = epsilon c *F + theta F.

Interestingly, this inverts to

F = mu c *G - lambda G

where the mu, lambda matrices are rather intricate expressions when
written in terms of epsilon and theta. The theta term is well-known in
QCD. That's the vacuum phase term.

But it's present even in electromagnetism. When taking the *correct*
classical limit (i.e., the classical theory arrived at when
integrating out the contributions from the extra fields (i.e.^2, when
treating the vacuum surrounding sources as a dielectric)), one obtains
a similar expression with non-trivial epsilon and theta coefficients.

[Another Wizard suddenly materializes:
one might expect the same considerations apply to gravity. Writing the
frame field as theta^a (a = 0, 1, 2, 3); and the curvature 2-form as
Omega^{ab} (a,b = 1, 2, 3, 4), a Lagrangian (with a parity-violating
term) can be written:
  L = a epsilon_{abcd} theta^a ^ theta^b ^ Omega^{cd}
  + b theta^a ^ theta^b ^ Omega_{ab}
  + c epsilon_{abcd} theta^a ^ theta^b ^ theta^c ^ theta^d.

Here, the role of "dual" fields (or the Maxwell "displacement" field
that loop quantum gravity theorists misname the E field!) are played
by
  G^{ab} = a S^{ab} + b/2 *S^{ab}
where
  S^{ab} = theta^a ^ theta^b
is the 2-frame basis.
This is sometimes quoted as "the area is the electric field". The
correct statement is: the area is the DISPLACEMENT field, not the
electric field! The electric field (E) and magnetic field (B)'s role
is played by the curvature 2-form and torsion.

The a and b coefficients are analogous to epsilon and theta.

The same argument that applied to the problems with the Lorentz
relations holds here too. The dual fields cannot be related simply by
duality. Something more general than duality, which may even be non-
local and probably even has a simple integral expression (think of
Regge calculus) may apply here, instead.