<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','to olbar=no,location=no,scrollbars=yes,resizable=yes, status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usene t ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\nSergio wrote:\n> I am wondering if there have been successful attempts to develop\nthe\n> subject on pure abstract reasoning, as can be done with\nthermodynamics.\n\nOne approach:\nThe Wigner classification of the representations of the Poincare\' group\ncan be applied both to classical and quantum theory, not just quantum\ntheory.\n\nAnother approach:\nThe classical theory, in fact, factors into a "classical" classical\npart (which is the macroscopic field equations which are a subset of\nthose listed in Maxwell), plus a "non-classical" classical part: the\nconstitutive relations (which are different that those posed by\nMaxwell, in as far as Maxwell posed any).\n\nThe former part is quite general, given its huge invariance group and\ncan be considered separately from the rest. Then you do, indeed, have\nsomething quite analogous to what you see in thermodynamics. In fact\nmore than an analogy: the differential form\ntheta = -E.dD - H.dB\nsuddenly comes into prominence. Using the relations D = E + P, B = H +\nM (treating epsilon_0 = mu_0 = 1), this yields\ntheta = d(-1/2 (E^2 + H^2)) - E.dP - H.dM,\nwhich up to a total differential (and signs and constant factors) is\njust the quantity of
heat exchange dQ.\n\nThe integrability of this form gives you a
stress tensor; otherwise you\nhave a force law given in part by a stress tensor, plus an irreversible\npart expressed in terms of theta. So, one might adopt as an axiom,\nwhich provides a generalizing envelope for any set of constitutive\nrelations, that theta be an exact differential. This then implies all\nsorts of variant formulations, e.g., a quasi-Lagrangian:\nL = L(D,B); dL/dD = -E; dL/dB = -H\nand quasi-potentials derived from it by Legendre transforms, e.g., a\nquasi-Hamiltonian:\nh = h(E,H): dh/dE = D; dh/dH = B; h = D.E + B.H - L.\nThese play the analogous role of thermodynamic potentials.\n\nSo, this gives you (ironically) a set of Maxwell relations, like you\nsee in thermodynamics, for instance:\n(dD/dH)_E = (dB/dE)_H\n(del_E x D)_H = 0; (del_H x B)_E = 0.\n(Out of this, one finds the constitutive matrices epsilon^{ij} =\ndD^j/dE_i; and mu^{ij} = dB^j/dH_i are both symmetric, for instance).\n\nThe second part of the factoring then focuses on the constitutive\nrelations, themselves. Absent these, the considerations above apply\ngenerally to all sorts of 4-D spacetimes (Galilean, Minkowski, even\nEuclidean), since the field equations:\ndiv D = rho; div B = sigma\ncurl D - dH/dt = J; curl B + dE/dt = -K\nsigma = 0, K = 0\nand force/power laws\nF = rho E + J x B + sigma H - K x D\nP = J.E + K.H\nare S(GL(4) x GL(2)) invariant (allowing for non-zero sigma, K) with a\nsubgroup thereof (strictly larger than GL(4), I believe) being the\nlittle group for sigma = 0, K = 0.\n\nSo, all the information about spacetime structure is actually locked\ninto the constitutive relations themselves. For instance, the set\nD = epsilon_0 (E - v x B)\nB = mu_0 (D + v x H)\nis Galilean invariant, (provided v transforms in the obvious way under\nGalilean transformations), and essentially postulates the existence of\nan \'ether frame\'. The set\nD = epsilon_0 E; B = mu_0 H\nin contrast gives you a Minkowski spacetime (up to conformal\ninvariance).\n\nThe whole purpose of the enterprise, one may suppose you\'re seeking, is\nto get a foundation that\'s consistent and that either removes or\nexplains away the classical singularity in the force law and stress\ntensor.\n\nThat\'s precisely the advantage of this kind of framework. Because, now\nthat the duality structure (D* = E, B* = H) or (equivalently) the\nconstitutive relations (D = epsilon_0 E, B = mu_0 H) have been\nseparated out, you have more room for movement to address the general\nissues, isolate the problem(s) and resolve them.\n\nThere\'s a lot that\'s non-trivial that needs to be said with this\nfactoring. A key theorem would be the condition that results from\nrequiring that the expressions for force and power remain regular. In\nparticular, you can ask what combinations of point-like singularities\nare possible that do not result in singularity multiplied by\nsingularity. For instance, tracing out the assumption that rho is\nsingular at a point, you find from (div D = rho) that D will be too.\nThen from the law (F = rho E + ...) you find that E must NOT be\nsingular at that point. Therefore, the relation (D = epsilon_0 E)\ncannot hold in the neighborhood of that point -- which forces you to\nrevert back to the more general equations posed above.\n\nSo, the range of allowable constitutive relations and distributions of\ncharge are restricted by the consistency requirement.\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>Sergio wrote:
> I am wondering if there have been successful attempts to develop
the
> subject on pure abstract reasoning, as can be done with
thermodynamics.
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
.
.dB
suddenly comes into prominence. Using the relations
M (treating
this yields
.
.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
. So, one might adopt as an axiom,
which provides a generalizing envelope for any set of constitutive
relations, that
be an exact differential. This then implies all
sorts of variant formulations, e.g., a quasi-Lagrangian:
and quasi-potentials derived from it by Legendre transforms, e.g., a
quasi-Hamiltonian:
.
.
.
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:
.
(Out of this, one finds the constitutive matrices
and
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:
curl
curl
and force/power laws
.
.H
are S(GL(4) x GL(2)) invariant (allowing for non-zero
with a
subgroup thereof (strictly larger than GL(4), I believe) being the
little group for
.
So, all the information about spacetime structure is actually locked
into the constitutive relations themselves. For instance, the set
is Galilean invariant, (provided v transforms in the obvious way under
Galilean transformations), and essentially postulates the existence of
an 'ether frame'. The set
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
or (equivalently) the
constitutive relations
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
is
singular at a point, you find from
that D will be too.
Then from the law
..) you find that E must NOT be
singular at that point. Therefore, the relation
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.
