Re: extra dimensions
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Posted by ronron on December 11, 19100 at 14:05:41:
In Reply to: Re: extra dimensions posted by FireTag on December 07, 19100 at 13:50:53:
i hope not to confuse the dimension of a state vector
with the dimension of spacetime. i refer the reader
to von Neumann's work "Matematische Grundlagen der
Quantenmekanik" (Mathematical Foundations of Quantum
Mechanics), wherein he puts QM on a firm mathematical
footing where state vectors are shown to be self-adjoint
operators in an infinite dimensional L-2 metric
space (Hilbert Space). this has nothing to do with
spacetime, as QM was not developed for
spacetime. relativity is a classical theory
throughout, QM is a modern one, developing as an empirical
science from Plank's blackbody work, then with the
photoelectric effect, etc. one of the problems unification theories had was not
having "enough room" to cover everything the Standard
Model and QFT accounted for. string theory appears to
have enough room, but both QM and GR need to be
contained therein. so i am back to my original question
of how to "embed" infinite dimensional L-2 metric
space, which is the state space for QM, into a space
of 11 or 26 dimensions. here i am referring to
topological dimension. you had mentioned the infinities, etc. you refer to
the need to invoke various functionals such as the
dirac delta function, etc, so that certain integrals
can be evaluated. the von Neumann work shows that
this is not necessary and puts QM on rigorous
mathematical ground without them. it's a wonderfully
elegant treatment and i strongly refer the interested
reader to it. it contains the famous "hidden parameter
proof" where he shows that no hidden parameters could
be added to QM that would render it a deterministic
theory. so he proves rigorously that either QM is
wrong, or it the statistical interpretation is the
most one can ever achieve with it. so it answers
Einstein's question about playing dice. god either
plays dice or QM is wrong. JVN did not even require
that such hidden parameters, even if they existed,
be self-adjoint operators in Hilbert space. it's
a beautiful and elegant theory. it (hidden parameter proof) is
a little hard to follow because it's
written in notation that existed before the modern
dirac formalism. for instance, von Neumann writes a
projection operator as P[zeta] (where the brackets
are subscripts). the same thing in the Dirac
notation would be : |zeta>bra and ket vectors.you mention the equivalence of operator and
DE formulations. sure, no argument there.
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