Is it possible to construct a gauge theory with local ISO(3) symmetry?

[Einj]

Hello everyone. Does anyone know if it is possible to build a gauge theory with a local ISO(3) symmetry (say a Yang-Mills theory)? By ISO(3) I mean the group composed by three-dimensional rotations and translations, i.e. if ##\phi^I## are three scalars, I'm looking for a symmetry under:
$$
\phi^I\to O^{IJ}\phi^J,
$$
with ##O^{IJ}\in SO(3)## and under:
$$
\phi^I\to\phi^I+a^I.
$$

Thanks!

 

[Orodruin]

This group is not compact due to the translations. Usually this leads to problems with ghosts.

 

[Einj]

I don't want to quantize the theory. I'm just looking for a Lagrangian for classical fields invariant under local ISO(3). Is this still pathological?

 

[ChrisVer]

I think the translation makes it pathological... Take for example the mass term:
[itex] m^2 \phi^2 \rightarrow m^2 (\phi^2 + 2 a \phi + a^2) [/itex]
Maybe you can drop the third term off your Lagrangian, but the second term doesn't seem right... and I don't think there is a way to get rid of it.
If the field is massless, then the kinetic terms work fine with the translations and a term like
[itex] |\partial \phi|^2[/itex] seems fine, as long as [itex]a^I \ne a^I(x)[/itex].
If it's local, then it's pretty similar to a local U(1).

 

[Einj]

Yes, I'm talking about a Lagrangian that only depends on derivative of the fields in the form ##\mathcal{L}(|D_\mu\phi|^2)##, where the covariant derivative must be found by imposing the right transformation rules under an ISO(3) gauge transformation. In particular, I'm writing an infinitesimal transformation as ##U=1+i\alpha^ap^a+i\beta^aJ^a##, with ##p^a## and ##J^a## being the generators of the shifts and rotations. Do you think this is possible?