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Is there any gauge in which Yang-Mills theory (4d, non-SUSY) can be written as a local action containing only the propagating modes?

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If there is one, it will not be explicitly Lorentz invariant, so please specify what you mean by "local". Would you allow a field which couples only with space-derivatives and no time derivatives? How about just one space derivative, does that count as local? Would you allow a straight-up Lagrange-multiplier field which has no derivatives? –  Ron Maimon Nov 18 '11 at 6:17
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1) The usual definition of a local action forbids the Lagrangian density ${\cal L}(x)$ to contain inverse differential operators (=integration operators) like $\partial_\mu^{-1}$, which is a non-local operation.

If OP requires the action to be on manifestly local form in the above sense, then the answer is No, to the best of my knowledge.

2) On the other hand, if one does not require a manifestly local form, then it is possible to write the gauge-fixed Yang-Mills action in $d\geq 2$ spacetime dimensions using propagating transversal gauge potentials only.

Consider the light-cone formulation (which implies that evolution/propagation is wrt. the light-cone coordinate $x^+$ as opposed to $x^0\equiv t$) in the light-cone gauge $A^+=0$.

Then the gauge potential $A^{-}$ becomes a non-propagating auxiliary field, which can be integrated out. This yields a Lagrangian density that only contains $d-2$ propagating transversal fields, but with inverse powers of $\partial_-$, and hence non-local.

See e.g. Warren Siegel, Fields, p. 210, arXiv:hep-th/9912205, for further details.

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