In a recent appearance on the Joe Rogan Experience, Lawrence Krauss explained Gauge symmetry with a chess board analogy and specifically talked about gauge symmetry being applied to the definition of which species of particles have positive and negative charge, and that symmetry leading to the electro-magnetic interaction. In the chess board analogy, at each position on the board the color of the square on the board tells you how to assign positive and negative to protons/electrons.

Here's my understanding of how gauge theories work. Space-time is defined by a differentiable manifold. At each point/event there is at least one vector space (may or may not be a tangent space). The vector spaces are all isomorphic to each other.

This means that the set of linear transformations from the vector space to itself (automorphisms), which is a Lie group, can be mapped onto the process of moving vectors from space to space in the manifold via parallel transport (for tangent spaces) or Wilson lines (a generalization of a Dyson series). Note that this mapping leans on mapping the parameter differentiability of Lie group within a space to the manifold position differentiability of the resulting maps between spaces.

The above processes admit a local symmetry that corresponds to changing the bases of the vector space at each point in the space-time manifold, called the gauge transformation. The gauge transformation is a map from the manifold to the elements of a Lie group of the vector spaces, often one that preserves vector lengths. So, for example, if the vector spaces are two dimensional then the Lie group might be $SO(2) = U(1)$ (depending on how you choose to write it). This local transformation induces a transformation on the gauge field that depends, partly, on the derivative of the gauge transformation. Thus, it's my understanding that only continuous symmetries can be gauged. It's also my understanding that the symmetry between positive and negative charge is discrete.

In the particular case of electromagnetism, the gauge symmetry that leads to electromagnetism is a $U(1)$ symmetry on a $\mathbb{C}^1$ vector space. This leads to a gauge transformation of the form: $$\psi(x) \rightarrow \mathrm{e}^{iq\alpha(x)} \psi(x),$$ that is a local change in the complex phase angle of some complex field.

The point being, it's the phase angle of the wave function that is the object of the gauge transformation, not it's charge. The charge, $q$, enters the picture with the role of the momentum conjugate to the phase angle.

My question is this: is there some duality in the electromagnetic gauge transformation that allows us to view it as a transformation in how we assign charges to particles, instead of a transformation of the phase angle of the field?

  • $\begingroup$ I'm not sure what you mean by "transformation in the assignment of charge particles", could you elaborate? $\endgroup$ – ACuriousMind Apr 3 '17 at 10:16
  • $\begingroup$ Moreover, it is not true that only continuous symmetries can be gauged. $\endgroup$ – Antoine Apr 3 '17 at 10:18
  • $\begingroup$ @ACuriousMind I'll flesh out my understanding and what Krauss described. $\endgroup$ – Sean E. Lake Apr 4 '17 at 2:21

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