Science > Physics > Quantum Gravity Via Expansion-Contraction 61.2: Spin Via Nonstandard Analysis, Lie Algebra, and PI
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Science > Physics |
| User: |
"OsherD" |
| Date: |
02 Jan 2007 01:15:33 AM |
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Quantum Gravity Via Expansion-Contraction 61.2: Spin Via Nonstandard Analysis, Lie Algebra, and PI |
From Osher Doctorow
In continuation of 61.0 and 61.1, let's take a look at spin in
classical and quantum mechanics.
It turns out that the reason why spin in Q.M. isn't the same as
classical spin as rotation about an axis passing thru the particle/body
or even about an axis passing through another body is basically due to
the difference between Lie Algebras and Lie Groups. The former are
expressible as Nonstandard Analysis versions of the latter more or
less, with our old friend the generalized exponential (generalized from
real, complex, matrix analysis to Lie Algebras/Groups) which is so
central to PI playing a key role. Elements of the Lie Algebra g
corresponding to Lie group G are called infinitesimal generators of G,
with the subgroup of G generated by a neighborhood N of identity e
being the identity component of G, where elements of the Lie algebra g
are "infinitesimally close" to the identity e. See Answers.com's "Lie
group" and also their "Spin" which incorporates their own and
Britannica's and Wikipedia's expositions, at a simple enough level for
most readers.
The infinitesimal of Nonstandard Analysis in terms of the "lower unit"
corresponds to PI's 0, while the "infinite" (1 divided by the
infinitesimal) or "upper unit" corresponds to PI's 1 more or less, so
we can just put spin into PI as a "pre-geometry" type of thing which in
the classical case becomes rotation or intrinsic angular momentum with
its orbital angular momentum analog or similar process. The numerous
torsion generalizations/modifications of GR in the research literature
relate to this too, and we can consider rotation as a type of either
pre-expansion or "identity" expansion or tendency to expand in physics
or even an effective "null expansion".
Osher Doctorow
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| User: "OsherD" |
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| Title: Re: Quantum Gravity Via Expansion-Contraction 61.2: Spin Via Nonstandard Analysis, Lie Algebra, and PI |
02 Jan 2007 01:39:38 AM |
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From Osher Doctorow
As I pointed out in my older threads, there are two types of motion in
the Universe which are very different:
1) one-direction-at-a-time motion (curvilinear, linear), as in
classical physics and GR usually
2) many-directions-at-a-time (simultaneously) motion of a body, usually
known as expansion/contraction, common in cosmology/astrophysics,
condensed matter physics, gravitational collapse, radiation from a
source, etc.
Rotation technically belongs to (2) because the parts of a rotating
body including particles on and in the surface of the body are moving
in different directions although there is a certain similarity in
clockwise vs counterclockwise type of orientation. This refers to a 3
dimensional body in Euclidean or Euclidean-like 3-space or moderately
curved deformations of the former.
PI and the Riccati Differential Equation dy/dt = A(t) + B(t)y + C(t)y^2
mostly refer to (2), although (1) can be regarded as a limiting case of
(2) as the number of directions (or a continuous modification of that
number) approaches 1. Adherents of (1) who try to deal with (2)
obviously will usually have a difficult time and typically have many
simultaneous equations to solve in that context. Quantum Theory
usually tries to "abolish" motion altogether insofar as trajectories or
anything resembling them are concerned, but it can't get rid of volume
or even boundary conditions which are more or less enough for
expansion-contraction, so (2) would arguably be its best "home".
Osher Doctorow
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