Understanding Color Confinement through Quantum Reference Frames and Relational Observables

Abstract

We present a formulation for color confinement on the basis of quantum reference frames (QRFs) and relational observables. In the QRF approach to color confinement, colored quantities are not defined as isolated local fields, but rather as relational observables with respect to a color frame or a dressing field. The Gauss law removes compactly supported local color charge from the physical bulk algebra, while boundary fluxes, Wilson lines, and holonomies may remain as semi-local data. This separates gauge invariance from frame independence. Color confinement is characterized under the fixed protocol by the absence of a finite-energy, physically accessible, asymptotically isolated nontrivial color-charged representation in the infinite volume limit. A failure to globalize a long-distance color frame is one possible mechanism contributing to this absence, which clarifies the role played by topological defects in color confinement. As concrete examples, we examine $(1+1)$-dimensional $SU(2)$ Yang--Mills theory and $(1+1)$-dimensional $U(1)$ gauge--Higgs model. We also take up the two-dimensional U(1) gauge--Higgs model on ${\mathbb{H}}^2$ (Euclidean $Ad{S}_2$) and three-dimensional $SU(2)$ gauge--Higgs model on ${\mathbb{H}}^3$ (Euclidean $Ad{S}_3$) which are obtained based on conformal equivalence by dimensional reduction of four-dimensional SU(2) Yang--Mills theory restricted to symmetric-instanton sectors. Through explicit calculations in these examples and in controlled sectors, we provide nontrivial checks for internal consistency of the present formulation. We also discuss prospects for four-dimensional Yang--Mills theory and gauge--Higgs theories.

73 pages, 17 figures, completely revised version