The two greatest constructions of theoretical physics in the twentieth century were quantum
mechanics, culminating in Quantum Field Theory and the standard
model of particles and interactions, and General
Relativity, Einstein's geometric theory of gravitation. These two theories,
refined and verified to extraordinary accuracy, are beautiful mathematical
descriptions of the dynamics of the physical universe. The fact that they
have been found fundamentally incompatible stands as the greatest failure of
twentieth century theoretical science, and provides the greatest challenge at
the dawn of the twenty-first century.
My own small research effort
aims at a unification of these two great theories. All previous attempts have
started from the same premise, that the established methods of quantization,
canonical or history based, may be applied to convert the classical equations
of General Relativity (or string theory) into a quantum field theory of
gravitation compatible with the standard model. It is possible that such an
attempt may yet succeed, and this line of research is being pursued by the
larger community; however, I have chosen to approach the problem from the
opposite side. I begin with the premise that the character of the universe is
fundamentally geometric, as suggested by General Relativity, and that all
particles and interactions, as well as the quantum structure, may emerge from
an elegant geometric description. I attempt not to quantize geometry, but to
find the geometric basis for particles, interactions, and the quantum by
beginning with a simple geometric model and dissecting and sorting the
dynamics of its components into agreement with the standard model. It will be
a neat trick if I can pull it off.
The first clue that this is going
to work comes from Kaluza-Klein theory, or the theory of hyperspace. If we
assume the universe to be a manifold with many of the dimensions (all other
than the four of space and time) curled up compactly (so that if you travel a
tiny distance in one of these directions you come back to where you started)
and that the manifold behaves according to the laws of General Relativity (the
dynamics of the manifold must extremize the curvature) then the fields in
space-time will be the space-time metric (gravity) and some scalar fields AND
some gauge fields identical in character and dynamics to the familiar gauge
fields of the standard model, the electromagnetic, weak, and strong fields
(providing the right compact manifold is chosen).
(I've written up a
very brief Introduction to Kaluza-Klein Theory if you'd like to see the
mathematical details.)
But Kaluza-Klein theory only naturally
generates the bosonic half of the standard model; the spinor fields -- the
fermions such as the electrons and quarks that constitute all matter -- are
notably absent. At least they were, before I found where they might be
hiding. It turns out that the spinor fields may be related to the extra, gauge degrees of freedom in the orthonormal frame (aka vielbein, or
tetrad) description of GR. By using the BRST formulation of gauge fixing along with a Clifford Algebra version of GR, the spinor fields emerge naturally as the complimentary gauge ghost fields. This, I think, is really cool.
(See "Clifford Geometrodynamics" for how
this works.)
So, things are going well. By doing Kaluza-Klein theory in the Clifford Algebra vielbein formulation we can get gauge fields, scalars, and spinors, with their corresponding Lagrangians --
obtaining these standard model elements from pure geometrodynamics. All that is needed now is to find the right shape for the compact Kaluza-Klein space to give the known fermion multiplets -- my current occupation.
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