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Gauge theory of ISO(3) group

  1. Apr 10, 2015 #1
    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:

  2. jcsd
  3. Apr 10, 2015 #2


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    This group is not compact due to the translations. Usually this leads to problems with ghosts.
  4. Apr 11, 2015 #3
    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?
  5. Apr 11, 2015 #4


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    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).
    Last edited: Apr 11, 2015
  6. Apr 11, 2015 #5
    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?
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