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Prior to the advent of quantum
mechanics, physics based on ‘physical’ pictures was the
norm. The role of mathematical modeling was to make
precise a prior understanding of a physical process; the
mathematics simply refined and helped explore a vision
of how a particular aspect of nature worked.
The unyielding inscrutability of quantum mechanics has,
I think, changed the role of mathematics within the
physics community. In the modern era, theoretical
physicists largely believe, and are guided by
mathematical models while mistrusting physical ones.
Thus we ‘interpret’ quantum mechanics, accepting
Schroedinger’s equation but placing little trust in any
associated physical picture. This is unavoidable to a
certain extent. At this point in time we do not actually
have a physical picture of the quantum world from which
to work.
Against this modern trend, but in accord
with the spirit of this forum, my own work is based on
physical pictures within classical statistical
mechanics. Recent works explore ‘zitterbewegung’ as a
physical process and have strong overlaps with David
Hestenes’ work on the Dirac equation. These works
establish a kinetic theory underlying the Dirac equation
in which the Pauli algebra appears as a manifestation of
the Euclidean geometry of space. The odd signature of
spacetime then appears as a direct result of an
intrinsic geometry that pairs spacetime paths of
opposite temporal sense. Development of
these ideas in 1+1 dimensions may be found in two papers
on the archive. The first,
Phys. Rev. A, 67, 2003, discusses the stochastic
process that gives rise to the Dirac equation. The
second,
Annals of Physics, 308(2), 2003, discusses the
connection with Schroedinger’s equation.
The extension to 3+1 dimensions and the relation to
Geometric Algebra will hopefully be posted in quant-ph
before the meeting. |
