Spin 12 from gluons

Abstract

The theta vacuum in QCD is obtained from the standard vacuum, after twisting by the exponential of the Chern–Simons term. However, a question remains–what is the quantum operator [Formula: see text] for winding number 1? We construct this operator [Formula: see text] in this note. The Poincaré rotation generators commute with it only if they are augmented by the spin [Formula: see text] representation of the Lorentz group, coming from large gauge transformations. This result is analogous to the well-known “spin-isopin” mixing result due to Jackiw and Rebbi [Phys. Rev. Lett. 36, 1116 (1976)], and Hasenfratz and ’t Hooft [Phys. Rev. Lett. 36, 1119 (1976)]. There is a similar result in fuzzy physics literature of Balachandran, Kurkcuoglu and Vaidya [Lectures on Fuzzy and Fuzzy Susy Physics (World Scientific Publishing Company, 2007)]. This shows that states can drastically affect representations of observables. This fact is further shown by charged states dressed by infrared clouds. Following Mund, Rehren and Schroer [arXiv:hep-th/2109.10342], we find that Lorentz invariance is spontaneously broken in these sectors. This result has been extended earlier to QCD (Balachandran, Nair, Pinzul, Reyes-Lega and Vaidya [arXiv:2112.08631 [hep-th]], reference given in the Final Remarks) where even the global QCD group is shown to be broken. It is argued that the escort fields of Mund, Rehren and Schroer [arXiv:hep-th/2109.10342] are the Higgs fields for Lorentz and color breaking. They are string-localized fields where the strings live in a union of de Sitter spaces. Their oscillations and those of the infrared cloud can generate the associated Goldstone modes.