Is Choice of Spinor Representation a Gauge Symmetry?

  • #1
stevendaryl
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Main Question or Discussion Point

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

[itex]\gamma^\mu \gamma^\nu + \gamma^\nu \gamma^\mu = 2 \eta^{\mu \nu}[/itex]

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

[itex]\gamma'^\mu = U \gamma^\mu U^{-1}[/itex]
[itex]\psi' = U \psi[/itex]
[itex]\bar{\psi'} = \bar{\psi} U^{-1}[/itex]

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

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 [itex]U(x^\mu)[/itex] differently at every point, if we make the corresponding change to the gauge field?
 

Answers and Replies

  • #2
WannabeNewton
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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.
 
  • #3
stevendaryl
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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?
 
  • #4
stevendaryl
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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 [itex]U[/itex] at each point seems like a generalization of the choice of the phase [itex]e^{i \phi}[/itex] at each point. That's the special case where [itex]U = e^{i \phi} I[/itex]. 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 [itex]U[/itex].
 

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