|Mar26-06, 04:00 AM||#1|
On The Quantum Dynamics of Moving Bodies
"On the Quantum Dynamics of Moving Bodies"
may be found in the list under http://federation.g3z.com/Physics
This writeup is still at a heuristic level, not mathematically
airtight. The major problem, at present, is trying to find where the
corresponding theorems *reside* (C*-algebra framework, representation
theory or otherwise) -- somewhat analogous to the difficulty faced
prior to the discovery of (a?/the?) proper formulation of the Stone-von
Nonetheless, it's quite interesting, revealing, and has a lengthy
discussion on the general issue of how one might generalize the
infrastructure of quantum theory so as to enable its safe cohabitation
with classical theories, like GR.
It was something I originally intended for the 2005 anniversary edition
of Annalen (hence the language in the introduction), but it will have
to await further resolution of the above issues...
There's a nice derivation of the Helmholz conditions in the appendix,
which supplements the derivation posed for the field theoretic case
("The Helmholz Conditions and Field Equations", also found in the
archive above). Part of the intent is to show the close parallel with
the derivation posed in the rest of the paper, leading to the idea that
the structure being uncovered in the writeup is simply the
non-commutative generalization of Helmholz, itself.
It's possible that someone with more mathematically keen eyes may see
how to actually produce from this a non-commutative analogue of the
derivation in the appendix, within some kind of non-commutative version
of the usual jet bundle formalism used for lagrangians, so that the
spirit of the rest of the writeup can be captured.
The combination of the equal-time commutator relatoins [q(t),q(t)] = 0
and 2nd order euqations of motion q''(t) = a(q(t),q'(t)) for a system
with configuration coordinates q(t) = (q^1(t),q^2(t),...) imposes
compatibility constraints on the dynamics strikingly similar to the
Helmholz conditions which -- in classical dynamics -- determine the
existence of a Lagrangian and Hamiltonian.
Following Hojman and Shepley (1991), we will investigate the
consequences of this correspondence, this time without the need to
resort to the classical limit. Our focus will be restricted to the
special case where the [q,q'] commutators are c-numbers. For closed
systems with a finite number of degrees of freedom, this implies the
reduction to the combination of a classical subsystem and quantum
subsystem, the latter being canonically quantized with respect to a
Hamiltonian quadratic in the conjugate momenta and evolving in
superselection sectors provided by the former. This includes, as
limiting cases, purely classical or purely quantum systems.
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