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In lattice gauge theories the only gauge invariant observables are constructed from Wilson loops and local field strength observables are reconstructed as zero size limits of Wilson loops. Furthermore the formalism of lattice gauge theories has no obstruction to working with finite/discrete gauge groups. Assuming that you treat the Wilson loops as your only observables even in the continuum limit, it doesn't seem in principle impossible to have a continuum limit of a discrete group gauge theory, because (unlike the classical case) the Wilson loop observable is a probability distribution over a discrete set of possibilities, which can vary continuously with the shape of the loop. So it seems that there are only a few possibilities:

  • The parameters of a finite group gauge theory cannot be tuned in the infinite volume or zero lattice spacting limit (or both) to get a non-trivial theory (I think I read that the infinite volume limit makes sense in at least some situations, which isn't very surprising since $\mathbb{Z}_2$ gauge theories are very similar to Ising models anyways).
  • The continuum limit exists, but it might exhibit certain pathologies:
  • Arbitrary Wilson loops need to be constructed as a sequence of approximations by Wilson loops along the lattice, so this limit may fail to exist even if the limit of, say, rectangular Wilson loops exist (I've read that Wilson loops with corners pose additional regularization problems in even free continuum abelian gauge theory; although now that I think about it I feel uneasy about this process even in the ordinary Lie group gauge theory case, because of issues related to trying to consider a smooth curve as a limit of increasingly zig-zaggy curves. If you try to calculate path lengths with this process it can be problematic if you're not careful, for example).
  • Even in that case one could attempt to define a 'field strength' as a limit of smaller and smaller well behaved (e.g. rectangular) Wilson loops, but this limit may also fail to exist (but this doesn't seem that bad. If you try to talk about the time derivative of Brownian motion you have to smear it, and the fluctuations cause expectation values to diverge in the no-smearing limit, but Brownian motion is still a well defined continuum process).
  • There may be no way to recover spatial symmetries in the lattice limit, beyond the existing translation and discrete rotational symmetries of the lattice.
  • The limit exists in the Euclidean case but it fails to have the properties necessary to analytically continue it to the Lorentzian case.

Are these the only possible issues? And does the continuum limit exist? I'm curious about both topological cases (wich seem far more likely to be well-behaved) and theories with dynamical degrees of freedom.

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    $\begingroup$ Continuum formulation of $Z_N$ gauge theory is known: it is just a $U(1)$ gauge theory with charge-$N$ (bosonic) matter field, in the Higgs phase. $\endgroup$ – Meng Cheng Sep 1 '15 at 0:02
  • $\begingroup$ I assume you don't know anything about the general non-abelian case then? $\endgroup$ – James Hanson Sep 1 '15 at 0:54
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    $\begingroup$ I don't know much, but I would guess that you can have a continuum description of non-Abelian discrete gauge theory by embedding the gauge group into a sufficiently large Lie group (i.e. dihedral group into $\mathrm{SU}(2)$), and use some complicated Higgs field to break the symmetry down. $\endgroup$ – Meng Cheng Sep 1 '15 at 0:57

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