Natural Science Forum / Physics / Relativity / July 2005

http://www.vacuum-physics.com/Maxwell/maxwell_oplf.pdf

Paul Stowe

gsax:

 Sure. Write down the dirac lagrangian:

 L = \Psibar(p/ - m)\Psi

   p/ == \gamma^u p_u, where the \gamma^u are dirac matrices.

 Require that to be locally gauge invariant, i.e., invariant under
the change of phase, \Psi -> \Psi' = \Psi\exp(-iS), where S == S(x^u)
is a function of the spacetime variables, x^u. Then, since p_u = id_u,
(I've set \hbar = 1, since it isn't relevant here), where d_u == d/dx^u.
Then, the new lagrangian,

 L' = \Psibar'(i\gamma^u d_u - m)\Psi'

will contain an additional term due to the derivative:

 L' = \Psibar(i\gamma^u d_u - m)\Psi + \Psibar\gamma^u\psi d_u S

We can, however, define a gauge covariant derivative, D_u = d_u + ieA_u,
such that it cancels the additional term (noether's theorem). This
gives us the gauge covariant dirac lagrangian:

 L = \Psibar'(i\gamma^u D_u - m)\Psi'

 To obtain maxwell's equations, one now just takes the commutator
of the covariant derivatives:

 [D_u, D_v] = [d_u + ieA_u, d_v + ieA_v]

            = (d_u + ieA_u)(d_v + ieA_v) - (d_v + ieA_v)( d_u + ieA_u)

 Most of the terms cancel leaving just the faraday tensor:

  F_uv  = ie (d_u A_v - d_v A_u)

Maxwell's equations in manifestly covariant form are then the
equations d_u F^uv = j^v, where j^v is the four-vector current
j^v = (\rho, J). That can be separated into the inhomogeneous
maxwell equations.

 Probably, but the theory wasn't physically viable.

 

His idea was orinially based on an aether consisting of springs and wieghts
to get his ideas in place.

The abstract form used things like the dispacement current (dD/dt) taken
from electrostatics and the current loop from current electricity.

It is of course rubbish, because there is no magnetic field, but it works
(just).  The photon model is the better model as it fits better with quantum
mechanics.  The amplitude of EMR being the number of photons and the energy
is given by the planc law E=hf.

In this way you sidestep arguments about radiation and particles.  An
accelerating electron changes its energy in quanta and emits a photon each
time with an enrgy given by E=hf.  The energy change of the Electron (even
in a radio transmitter) is given by quantum mechanics.  Many electrons mean
many photons and one electron in the atenna of the transmitter connects to
one electron in the atenna of the reciever by the photon with is the
imaginary connector.  In reality it is a quantum mechanical connection like
a fyneman diagram.

Phumph has spoke!

Chris.

Before formulating field equations it is helpful to study where
Maxwell could have ... ahhh "done better" :o)
<<
76. A. K. T. Assis and K. H. Wiederkehr, "Weber quoting Maxwell,"
56. A. K. T. Assis, "The meaning of the constant c in Weber's electrodynamics,"
36. A. K. T. Assis, "Gravitation as a fourth order electromagnetic effect,"
55. A. K. T. Assis and H. Torres Silva, "Comparison between Weber's
    electrodynamics and classical electrodynamics,"
72. A. K. T. Assis, "On the first electromagnetic measurement of the
    velocity of light by Wilhelm Weber and Rudolf Kohlrausch," >>
http://www.ifi.unicamp.br/~assis/wpapers.htm

http://tw.arxiv.org/abs/physics/0506181 Darwin
http://arxiv.org/abs/physics/0204034 Coulomb gauge

Advanced potential:
http://www.google.com/search?hl=en&lr=&q=%22advanced+potential%22+green&btnG=Search

Sue...

Interconnected cogs were the mechanical analogy he used.  Look it up.
Nobody is keeping it a secret.

Maxwell's equations are covariant right out of the box.  When the
thing was completed the aether had vanished, like balancing a
checkbook.  Debits and credits mutually cancelled.

Library.  Big rooms filled with books.  Maxwell is late 1800s, and
that is sufficiently contemporary for complete archives and learned
commentaries.

http://www.fuckinggoogleit.com/

You already had a reply from bilge explaining the modern view from local
gauge invariance.  The following also explores it -
http://www.upscale.utoronto.ca/GeneralInterest/DBailey/SubAtomic/Lectures/LectF1
3/Lect13.htm
Also it can be derived from Coulombs law and SR -
http://www.cse.secs.oakland.edu/haskell/SpecialRelativity.htm

Thanks
Bill

Why not just read the Treatise, itself?

Maxwell explicitly set out in the Treatise that everything set out in
it was to remain neutral with regard to the issue of whether the vacuum
was actually a vacuum or a plenum.

"Aether" is not used by Maxwell.  His theory has little or nothing to
do with anything that goes under the name "aether".  The underlying
theory (which he set out to remain neutral with regard to in all
sections following) was explicitly set out in chapter 1: it stated that
the vacuum is a dielectric capable of polarization (i.e. vacuum
polarization) and in which charge screening will take place to smear
out point source infinities (as explicitly pointed out in sections with
titles like "why there are no infinite point or line sources").

The equations don't come from anything.  They're a formulation of the
empirical laws that had been gathered to that time -- plus the element
of linking up the material current into a "total current" with the
latter including a contribution from the field (explained in virtue of
the vacuum being a dielectric), and satisfying the continuity law.

As he pointed out, the issue of what (if anything) comprises the vacuum
is left unresolved.

What you're calling "aether" is a laymanizing of a complex of unrelated
ideas that have little to do with the latter-day revisionistic
accounting of Maxwell.  One of those ideas in the complex is the
natural frame of reference associated with the vacuum.

The one thing Maxwell DID derive based on THAT idea was the particular
form of the constitutive relations, which in modern notation would have
read:
                  D = epsilon_0 (E + G x B)
where G is the velocity relative to the frame defined by the vacuum.
(The corresponding relation for B and H would therefore read B = mu_0
(H - G x D), which was briefly mentioned in the Dover edition of the
treatise).

This, however, is, in fact, NOT what is currently known as Maxwell's
theory -- which has, instead, the constitutive relations
              D = epsilon_0 E; B = mu_0 H.
(Which comes from Lorentz).

In fact, THIS discrepancy was what Eintein was referring specifically
to 100 years ago when recounting the recent developments of the theory.
Einstein's reference to "aether" was actually the reference to
Maxwell's G.  The point made is that -- after finishing out the
analysis of the behavior of the field under a change of frame of
reference (an analysis which Maxwell only half-carried out in the
Treatise, and the incompleteness of which represents a serious gaping
hole in the treatise); if one assumes the Lorentz relations are valid
-- as they are empirically -- instead of the "aether frame" relations
involving G, then the whole need for G is gone and we're back to all
frames being equivalent.

The two theories
              D = epsilon_0 (E + G x B); B = mu_0 (H - G x D)
vs.
              D = epsilon_0 E; B = mu_0 H
are NOT empirically equivalent.  The former -- which is the one which
would be invariant under Galilean Relativity and true if Newtonian
Physics were true -- is seriously out of kilter with the reality.
Swimming pools, for instance, would probably start to assume weird
colorations because of the dependence of D on B.  The latter -- which
is the one invariant under Poincare' Relativity and true if Special
Relativity holds -- is the ones you actually see.  Hence, no "aether
frame".

The idea of a frame and a G, nonetheless, returns in a different guise.
If the vacuum, itself, is a thermal state (as it clear is, since outer
space has a non-zero temperature of 3 degrees kelvin filled with the
CMB thermal radiation), then there IS, in fact, a natural velocity, G,
associated with it and a natural frame ... since every thermal state
identifies a natural frame of reference.

So, the whole notion of "aether frame" turns out to be a complete red
herring, completely ORTHOGONAL to any questions about Galilean vs.
Poincare' Relativity.  It has nothing to do with nothing, since it
applies or can apply equally in both settings.