# [time 15] eformulating Weyl's Unified Field Theory

Stephen P. King (stephenk1@home.com)
Sat, 13 Mar 1999 21:50:16 -0500

Here we go!

Michael J. Martin, University of Missouri

Reformulating Weyl's Unified Field Theory of 1919

In 1918-19, Hermann Weyl introduced the first classical unified
field theory, a theory that united electromagnetism and general
relativity into one geometrical framework. This theory, the forerunner
of modern gauge theory, involved transforming the metric 2-tensor with a
"scale factor" that, in turn, transformed the line element, causing
lengths of parallel transported vectors, as well as their orientations,
to change. This length dependence was later dismissed, and reformulated
by London and Fock as wave function phase dependence. We attempt another
reformulation, whereby the volume element varies on parallel transport
(volume instead of length) unless it is transformed into a tensor
density. This transformation introduces a gauge potential, the
contracted Christoffel symbol, which maintains invariance under
covariant differentiation.
The electromagnetic potential, in pure gauge form, is shown to be
equivalent to a constant multiple of the contracted Christoffel symbol.
The gravitational field is introduced into wave mechanics via the
contracted Christoffel symbol, which appears as an integrable phase
factor. The contracted Christoffel symbol and the electromagnetic
4-potential are related via a simple gauge transformation under this
program.

Density Transformations in a Gauge Unification of Gravity and
Electromagnetism

The Lagrangian which describes an electron wavefunction in QED
remains invariant under local phase transformations through a gauge
transformation of the electromagnetic 4-potential, which is the gauge
connection appearing in the QED covariant derivative. We postulate that
similarly, the Lagrangian which describes the electron wavefunction in
both an electromagnetic and gravitational field remains invariant under
local phase transformations of the wave function density, via the
introduction of the contracted Christoffel symbol as gauge connection in
the covariant derivative. The resulting transformations, of the QED
Lagrangian density, are explored. The quanta of the gravitational field
would appear to be both massless and of infinite range, as the
transformations are nearly identical to U(1). We explore the case of
wave function --> wave function density, and wave function
density-->wave function density, where a density is defi.....

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