Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Is Choice of Spinor Representation a Gauge Symmetry?

  1. Jul 13, 2014 #1

    stevendaryl

    User Avatar
    Staff Emeritus
    Science Advisor

    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?
     
  2. jcsd
  3. Jul 13, 2014 #2

    WannabeNewton

    User Avatar
    Science Advisor

    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.
     
  4. Jul 13, 2014 #3

    stevendaryl

    User Avatar
    Staff Emeritus
    Science Advisor

    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?
     
  5. Jul 13, 2014 #4

    stevendaryl

    User Avatar
    Staff Emeritus
    Science Advisor

    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].
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Is Choice of Spinor Representation a Gauge Symmetry?
  1. Gauge symmetry (Replies: 15)

  2. Spinor representation (Replies: 2)

  3. Spinor representations (Replies: 1)

Loading...