The Geometry of Gauge Theories.

Overview

I am fascinated by the connections between Riemannian geometry, gauge theories, and quantum mechanics. The connections between gauge theory and geometry are mentioned in nearly every textbook on quantum field theory. How this geometry can be visualized is not often mentioned.

Gauge theories can be expressed using the principal bundle constructions. Principal bundles are defined with respect to a symmetry group, and one can have a distinct representations of the symmetry group act on different objects in the theory. Because of the multiple representations, not every principle bundle can be mapped back into a single vector bundle construction.

In contrast, traditional general relativity is defined around coordinate reparameterization invariance, typically represented on a vector bundle. All vector objects are sections of the same vector bundle. For this reason, the vector bundle associated with the tangent space is relatively easy to understand, especially when thought of in terms of an embedding space. A natural basis for the tangent space is provided by space-time coordinates. Vectors are geometrical objects on the tangent space that can be expressed in terms of different basis choices, but the vector itself is a basis-independent object. Although one cannot think easily in more than three dimensions, Riemannian geometry with the aid of an embedding space provides a visual tool to understand the geometrical significance of the curvature that leads to gravitational force.

What about the geometrical significance of gauge theory. Principle bundles add a great deal of structure beyond the vector bundles at the heart of coordinate reparameterization invariance in general relativity. Cahill and I used a vector-bundle construction, shown in the figure below, which in many ways parallels Riemannian geometry with an embedding space, to visually represent gauge-invariant quantities. We use a trivial bundle as a type of embedding space. This trivial bundle allows one to compare vectors at different space-time points. The idea is analogous to embedding a 2-dimensional sphere in 3-dimensional Eucledian space to understand the role of parallel transport in the covariant derivatives of Riemannian geometry. We have mostly studied U(1) gauge theories represented as SO(2) gauge theories. What one would normally think of as the SO(2) gauge fibre is now seen as an two-dimensional real vector bundle inserted within the trivial vector bundle of larger dimension. The wave-function is gauge-invariant vector field from the embedding space that can be described in terms of any basis that spans the two-dimensional gauge fibre. A gauge transformation changes the basis by which one describes the gauge fibre. The variation of the gauge-fibre at different space-time points manifests itself as electric and magnetic fields. As explained in the papers below, this curvature gives rise to electric and magnetic forces. The geometrical nature also provides a geometrical explanation for the ubiquity of the electrons charge in nature.

Example Animations

• PosAndNegWFVectorsMovie.avi Quantum field theory shows objects with positive vs negative frequencies have opposite charges. Here one can visualize the wave functions two, non-interacting particles of opposite charges rotating in opposite directions on the gauge fibre. These two animations show the same phenomena: a positive and negative scalar particle, which do not interact with each other, moving in a small orbit due to a constant background magnetic field. The first image shows a wide-angle view of the amplitude squared of the two wave functions. The second image shows the vector-bundle representation of the geometry for a zoom-in of the region where the two particles are overlapping but traveling in the same direction. In the second image, one can see the wave functions rotating in opposite directions. A detailed explanation of this calculation can be found in our paper.