# Gauge theory with non-dynamical gauge field

1. Apr 11, 2015

### Einj

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. Apr 13, 2015

### vanhees71

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.

3. Apr 15, 2015

### haushofer

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 :)