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Is there any interacting quantum field theory of massless fields with helicity $\pm 1$ which can be expressed entirely locally in terms of the field strength Fμν with no reference to vector potentials at all? Clearly, quantum electrodynamics doesn't fall into this category. The Aharonov-Bohm effect is the reason.

Classically, it's easy to come up with many such theories, but unfortunately, they happen to be nonrenormalizable. But do asymptotically safe models exist?

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i think the word on the street is that the fact that the gauge field constrained to $U(1)$ covariance would somehow be connected to renormalizability, of course you are concerned with the converse affirmation; does the absence of a interesting gauge imply nonrenormalizable? – lurscher May 18 '11 at 20:43
Before even attempting to discuss QFT and renormalizability, I would recommend you to be more specific about, e.g., the classical field content ($\vec{E}$ and $\vec{B}$ as opposed to $A_{\mu}$?), and the classical equations of motion (Maxwell equations?), and perhaps mention an action for one of the many classical theories, that it's easy to come up with, cf. v2 of the question. Then we might get a better idea of what sort of Aharonov-Bohm-free theory, that you are looking for, at the quantum level. – Qmechanic May 19 '11 at 14:46

There are simple non- renormalizable examples: the easiest is QED with elementary neutrons only. The neutrons have a magnetic dipole moment, and interact with the photon locally through the F tensor only contracted with the antisymmetric product of gamma matrices. There are no renormalizable local examples in 4d because we have a list of all of these, and the vectors always interact with charged particles, not magnetically polarized ones.

(I missed that the question asks specifically for asymptotically safe examples, and the OP was probably aware of this system. But the answer might be useful to others)

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The answer is yes. There are many examples of emergent gauge theory from quantum spin models on lattice. For example, see Phys. Rev. D68, 024501 (2003), and references there. The quantum spin models have no gauge symmetry and all their degrees of freedom are physical and, in some sense, correspond to $F_{\mu\nu}$.

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But these models are not Lorentz invariant. If you make Lorentz invariant helicity 1 theory, can you get a pure F coupling in any renormalizable model? Perturbative considerations for local actions say no. What is the low-energy limit of the model you are considering? – Ron Maimon May 28 '12 at 7:00
@Ron: The low-energy limit of the model has emergent Lorentz symmetry, and photons with only helicity $\pm 1$ modes. At lattice scale, the model is just some quantum spin model with no Lorentz symmetry and no gauge symmetry. – Xiao-Gang Wen May 28 '12 at 7:43

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