Re: Transition from classical continuum mechanics to point mechanics?
- From: Rock Brentwood <markwh04@xxxxxxxxx>
- Date: Mon, 21 Jul 2008 17:41:22 +0000 (UTC)
On Jul 18, 2:32pm, Markus Frank <m...@xxxxxxxxxxxx> wrote:
This would likely mean starting from the differential conservation laws of
continuum mechanics for mass, momentum and energy and somehow modeling
the respective densities and currents by a sum of Dirac delta functions
for each particle. I have some ideas how to do this on my own but would
like to have an "authoritative" reference to back it up.
Chapters 4 and 5 of Barut's "Electrodynamics and Classical Theory of
Fields and Particles"
http://product.half.ebay.com/_W0QQprZ293346QQtgZinfo
The potentials are developed straight from the propagator solution of
the field. When combined with the concentration of the sources on
worldlines this gives you a certain well-known potential (one variant
for the advanced solution and the other for the retarded solution).
It turns out to be MUCH simpler to do this entirely in the language of
differential forms and work with the potential 1-form A and field 2-
form F in toto in one fell swoop. Barut, however, doesn't. Nobody I
know of does. That's your exercise.
The electrodynamics for a set of point-like sources is worked out in
detail. This is not trivial. As Poisson and Green first learned back
in the early 1800's, there is that nagging problem of defining the
"internal" field for the source, because the source literally gets in
its own way with its field.
Barut recounts two basic approaches. One uses the law of action and
reaction, computing the energy and momentum flow associated with the
radiation part of the source's field and takes the source's self-kick
to be the recoil the opposite way.
One gets the radiation reaction and a certain well-known law of motion
that is of the THIRD order.
One of the most interesting features of the solution is that almost
all choices of initial data for (position, velocity, acceleratiion)
lead to physically meaningless run-away solutions (those whose gamma =
(1-(v/c)^2)^{1/2} goes up exponentially with time). To get a regular
solution, one has to constrain the initial value of the acceleration.
The constraint is an integro-differential equation that relates the
acceleration to FUTURE values of the force and source's motion. This
is an instance of what is known as RETROCAUSALITY. The response curve
is exponential and equal to 2/3 alpha X (h-bar/(mc^2)) where alpha is
the fine structure constant. For electrons this comes out to about
10^{-22} seconds.
Another dives right in and tabulates the self-field and pulls a Green
(who first advanced the idea) of subtracting out the infinity at the
source. This is done -- at the classical level -- by renormalizing the
mass of the source.
On the issue of retrocausality, here are two more observations not
noticed by Barut that will peak your interest. There is a well-known
no-go theory in relativistic dynamics which forbids the defining of a
position operator satisfying certain well-defined properties on pain
of causality violation. The scale of the causality violation is on the
same order as that described above.
Are these two features related? I don't know.
The classical monopole solution in general relativity with the same
(gauge) charge, same angular momentum and same mass as any of the
fundamental particles is a NAKED singularity. A ring singularity, like
a looking glass portal, whose scale is on the same order as above.
It, too, possess the feature of causality violation (which is why
naked singularities are normally prejudiced against by theorists) and
non-determinism.
Is the non-determinism of the right kind suitable for erecting the
structure of a quantum logic? A paper published in arXiv a while back
(by Mark Hadley, but I'm missing the reference) says so. That is: can
the quantum world, itself, literally be pulled out of classical
physics -- in an Einstein's Revenge Scenario -- by using these naked
singularities to produce all the quantum weirdness.
In fact, there's been a lot of work in recent times in (a) developing
a picture of quantum theory that interprets all the weirdness in terms
of retrocausality (possibly distantly related to Feynman), (b)
devising experimental tests.
A jumping-off point of the whole field (and all the mines in the
field) can be found under
http://en.wikipedia.org/wiki/Retrocausality
A reference is also link from there to Hawking's Chronology Protection
Conjecture ("no naked singularities").
Penrose also made mention of the enterprise in his 2006 Road to
Reality and provided further links.
The most famous of the experiments that people have tried to push in
this direction is the "delayed choice experiment".
The Feynman article "Interaction of the Absorber..." fall squarely in
line with Barut's last section, which discusses the (related) absorber
model for action-at-a-distance electrodynamics (the Absorber is
retrocausal). Thus, we come full circle back to your question --
deriving an action-at-a-distance formulation of mechanics from field
theory.
.
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- From: Markus Frank
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