# Gauge theory of ISO(3) group

1. Apr 10, 2015

### 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!

2. Apr 10, 2015

### Orodruin

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

3. Apr 11, 2015

### 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?

4. Apr 11, 2015

### ChrisVer

I think the translation makes it pathological... Take for example the mass term:
$m^2 \phi^2 \rightarrow m^2 (\phi^2 + 2 a \phi + a^2)$
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
$|\partial \phi|^2$ seems fine, as long as $a^I \ne a^I(x)$.
If it's local, then it's pretty similar to a local U(1).

Last edited: Apr 11, 2015
5. Apr 11, 2015

### 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?