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.

Spinors: c

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.

Spinors: a

Hamiltonian mechanics

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

Since we cannot vary the assumed “universal” time, the canonical Poisson brackets put constraints on p₀, 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 Hamilton’s equations.

Hamilton’s 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 Hamilton’s equations.

Spinors: b

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

Spinors: d

Spinors: e

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.

Spinors: f

Spinors: g

When we write the simplest Dirac equation, and ask for it to square to “the” 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.

Spinors: h

Spinors: i

Spinors: k