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Hatto von Aquitanien
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Hatto von Aquitanien
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 Posted: Sat May 26, 2007 10:34 am Post subject: Re: Scalar field: Physics vs. Math |
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markwh04@yahoo.com wrote:
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On May 25, 4:25 pm, Hatto von Aquitanien <a...@AugiaDives.hre> wrote:
It is a common error to confuse phase space with physical space.
Well, first of all, this does not relate to anything you're replying
to.
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I do not agree. The last part of the post I was replying to discussed the
integral of the Lagrangian, etc. That is an essential precursor to
Hamilton's canonical equations used the terms of which constitute the
coordinates of phase space as Gibbs used the term. Since the essential
distinction to which I was referring between phase space and physical space
is a result of the method of deriving the Euler-Lagrange equation by
requiring the variation of the path integral of a functional to vanish, my
comment was fully justified.
| Quote: |
Second of all: Hello?! Do you know you who you're replying to? Mr.
Configuration-Space-Is-Not-Physical-Space?!
I practically invented that phrase
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Dr. Schrödinger! What a pleasant surprise! I thought you were long dead.
| Quote: |
and it is, no less, one of the
central aspects of the framework that couches what you're replying
to ... as well as one of my major bones of contention, as illustrated
in the following (which one could even be excused from having the
impression that that's where you got your reply from)
http://federation.g3z.com/Physics/index.htm#QuantumDynamics
"In contrast to the conclusions, drawn by Feynman and (before him)
Dyson, who purported to pull out the formal structure of
electromagnetic theory from the consideration of the commutator
algebra, here the Lorentz law and force are placed in their proper
context. The 'Lorentz force' resides in configuration space, not in
ordinary space. The confusion with electromagnetism occurs when one
restricts attention to a one-particle system (as Feynman and Dyson
did), where the configuration space is three-dimensional."
or
http://federation.g3z.com/Physics/index.htm#Nelson
"A deeper analysis shows that the quantum potential is generated by
the quantum deformation of a stress tensor, representing the
probabilistic flow of the system in configuration space, subject to
the continuity equation and a force law that expresses the flow of the
momentum density for the system in terms of a configuration space
Lorentz force."
As to your original question, you'd do best to do a web search on the
whole framework of bundles, "triviality", etc. I can't fill in on all
the details here. For one, I've never done a study in any depth on the
intricacies related to non-trivial bundles.
The examples that come to my mind come more from pure mathematics:
those relating to Riemann surfaces in complex analysis (the logarithm
and exponential are closely linked to an archetypical example of the
general phenomenon of non-triviality, with the logarithm technically
not being a function at all because of this issue). In a more physical
context might be projective Lie group representations; which can be
thought of a sections over complex bundles that have a Lie group
manifold as a base space (i.e. phase). The Galilei group, for
instance, (for Newtonian physics) forms a manifold whose corresponding
complex bundle is non-trivial. Out of this ultimately comes the
classical Newtonian notion of mass.
The somewhat standard model of a Maxwell monopole makes essential use
of the non-triviality of the bundle corresponding to the Maxwell field
around a singular point to construct (piecemeal) the magnetic source.
The gauge is locally trivial, but globally non-trivial. This bears
some similarity (and relation) to the integrability constraint
mentioned in the "Nelson" article above.
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