Is Choice of Spinor Representation a Gauge Symmetry?

[stevendaryl]

In the Dirac equation, the only thing about the gamma matrices that is "fixed" is the anticommutation rule:

\gamma^\mu \gamma^\nu + \gamma^\nu \gamma^\mu = 2 \eta^{\mu \nu}

We can get an equivalent equation by taking a unitary matrix U and defining new spinors and gamma-matrices via:

\gamma'^\mu = U \gamma^\mu U^{-1}
\psi' = U \psi
\bar{\psi'} = \bar{\psi} U^{-1}

(Actually, it occurs to me now that U doesn't need to be unitary. But if it's not unitary, we need to define \bar{\psi'} = \psi'^\dagger (U U^\dagger)^{-1} \gamma'^0, rather than \bar{\psi'} = \psi'^\dagger \gamma'^0)

My question is whether this freedom to choose a representation is a gauge symmetry. Is there a corresponding gauge field so that we are free to choose U(x^\mu) differently at every point, if we make the corresponding change to the gauge field?

 

[WannabeNewton]

No. It is no more a gauge symmetry than the ability to express the electric and magnetic fields in terms of cartesian basis vectors or spherical polar basis vectors.

 

[stevendaryl]

No. It is no more a gauge symmetry than the ability to express the electric and magnetic fields in terms of cartesian basis vectors or spherical polar basis vectors.

Well, the choice of a different basis at each point in spacetime IS a gauge symmetry, isn't it? Can't GR be described in those terms?

 

[stevendaryl]

No. It is no more a gauge symmetry than the ability to express the electric and magnetic fields in terms of cartesian basis vectors or spherical polar basis vectors.

To me, the choice of the matrix U at each point seems like a generalization of the choice of the phase e^{i \phi} at each point. That's the special case where U = e^{i \phi} I. The choice of phase is the gauge symmetry associated with electromagnetic interactions. I was wondering if there was a more general gauge symmetry that involved more complicated choices of U.