Spin 1/2 from Gluons

TL;DR

This paper constructs the quantum operator for winding number in QCD's theta vacuum, revealing how large gauge transformations induce spin-1/2 representations and lead to spontaneous Lorentz and color symmetry breaking.

Contribution

It introduces a construction of the winding number operator in QCD and demonstrates how large gauge transformations affect Lorentz and gauge symmetries, linking infrared effects to symmetry breaking.

Findings

Large gauge transformations induce spin-1/2 representations.

Lorentz invariance is spontaneously broken in charged sectors.

Infrared clouds and escort fields relate to Goldstone modes.

Abstract

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.