Spin 1/2 from Gluons

The theta vacuum in QCD is the standard vacuum, twisted by the exponential of the Chern-Simons term. But what is the quantum operator $U(g)$ for winding number $1$? We construct $U(g)$ in this note. The Poincare' rotation generators commute with it only if they are augmented by the spin 1/2 representation of the Lorentz group coming from large gauge transformations. This result is analogous to the 'spin-isopin' mixing result due to Jackiw and Rebbi [1], and Hasenfratz and 't Hooft[2] and a similar result in fuzzy physics [3]. Hence states can drastically affect representations of observables. This fact is further shown by charged states dressed by infrared clouds. Following Mund, Rehren and Schroer [4], we find that Lorentz invariance is spontaneously broken in these sectors. This result has been extended earlier to QCD (references [15] given in the Final Remarks) where even the global QCD group is shown to be broken. It is argued that the escort fields of [4] are the Higgs fields for Lorentz and colour breaking. They are string-localised fields where the strings live in a union of de Sitter spaces. Their oscillations and those of the infrared clouds generate the associated Goldstone modes.