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Gauge theory with non-dynamical gauge field

  1. Apr 11, 2015 #1
    Hello everyone, I'm trying to write down a Lagrangian invariant under local ISO(3) (rotations+shifts) transformations. I'm working at classical level and there will be no quantization of any kind so the theory shouldn't have any ghost pathology.
    However, I found that, out of the 6 gauge fields needed, 3 of them are non-dynamical, i.e. they don't admit a kinetic term in the Lagrangian but they only appear in the covariant derivative of the fields, like a source. However, they also transform non-trivially under local ISO(3) (of course).
    Is there anything pathologically wrong in it or can I just accept the fact that they are non-dynamical?

    Thanks a lot!
  2. jcsd
  3. Apr 13, 2015 #2


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    In the classical theory, the physical degrees of freedom are represented by an entire class of gauge potentials. For a massive (massless) vector field, represented by a four-vector field, only 3 (2) field degrees of freedom are physical. You need a gauge constraint to pick one representation out of the infinitely many connected by a gauge transformation. In the classical theory that's all you need.

    In quantum field theory you have to make sure that you get well-defined propagators for the gauge fields which again make it necessary to fix the gauge. At the same time you must make sure that the non-physical field-degrees of freedom do not become interacting and thus violate causality and unitarity of the S matrix. For that you have to introduce Faddeev Popov ghost fields, which is most easily seen in the path-integral formalism.
  4. Apr 15, 2015 #3


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    Isn't such a construction similar to gauging ISO(1,3) in order to obtain General Relativity? In that case I would recommend the lecture notes/book by Van Proeyen on Supergravity :)
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