**Field Theory Seminar**

**James
T. Wheeler: **

** Classical
mechanics in biconformal space**

** Scalar
and spinor wave equations in biconformal space**

December
2-4, 2009

**Classical mechanics**

**Physical interpretation of the Weyl vector**

The Weyl
vector is the gauge vector associated with dilatations. Gauge transformations
change it by a gradient. We are interested in possible physical interpretations
of this gauge vector.

H. Weyl
proposed a physical model in which the Weyl vector is
identified with the vector potential. However, this means that atoms traversing
different paths, such that the two paths enclose non-vanishing electromagnetic
flux, will show a measurable difference in physical size. This would lead to a
broadening of atomic spectral lines far beyond what is actually seen.

Another possible physical
interpretation is as the Lagrange density for a particle. Here, too, we expect
a change by a gradient to have no physical effect. We explore this model.

**Relating the Weyl
vector to the physical variables**

In the biconformal gauging of the
conformal group, the base manifold has symplectic structure, so we may
interpret the solution in terms of canonically conjugate coordinates, (x,y). Gravitational theory on this space gives a unique
solution for the form of the Weyl vector, w = - y_{a} dx^{a}.

Two issues arise.

First, the units are wrong. The
coordinate y_{a} has geometric units of
inverse length, while the physical momentum has units of momentum. This is
easily fixed by supposing that the reduced Planck constant connects momentum
and inverse length, in the same way that, in spacetime, the speed of light
connects physical units of time with geometric units of length.

Second, the sign is reversed. The
spacetime metric, together with the definition of momentum as proportional to
the tangent vector to a spacetime curve, tells us that the metric for momentum
is the Minkowski metric. However, the Killing metric for the conformal group
induces an inner product on the y_{a}
subspace that has the opposite sign. The relationship between y_{a} and p_{a} must therefore also include
an imaginary unit.

**Hamiltonian mechanics**

The Newtonian assumption of
universal time now leads to Hamiltonian mechanics.

Since we cannot vary the assumed ÒuniversalÓ
time, the canonical Poisson brackets put constraints on p_{0},
requiring it to be dependent upon the remaining phase space variables. Varying
the integral of the Weyl vector with respect to the
independent variables leads immediately to HamiltonÕs equations.

HamiltonÕs principal function is
found by evaluating the action on the solution curves. On this restricted set
of curves, the action is a *function*, not a *functional*. This means that on this restricted class of curves, the
integral around closed loops vanishes. As a result, no physical size change
occurs between any two particles following solutions to HamiltonÕs equations.

**Spinors**** and the scalar wave equation**

From
the solution for flat biconformal space, together with the induced Killing
metric, we can write the line element and scalar wave equation on biconformal
space. We try a 1-parameter separation ansatz for its
solution

Now,
how do we choose alpha? It is tempting to choose it to cancel the two
non-quadratic terms. However, remembering that the y-subspace has metric h_{ab}, we can choose a to give the usual wave
equation on this subspace. This choice leaves a constant term, so we have a
Klein-Gordon type equation.

When
we write the simplest Dirac equation, and ask for it to square to ÒtheÓ wave
equation, we find there is necessarily a scalar curvature term. Normally, such
a term is associated with the conformal wave equation, but the factor here is
not correct for the conformal case, and no mention is made of conformal
symmetry.