For nonabelian Yang-Mills in the Coulomb phase, can soft gluons render the charge orientation of charged particles indefinite? Let's say the gauge group is a nonabelian simple Lie group G. Suppose there is a charged particle in the nontrivial irreducible representation r. It might appear the particle can take on a definite state in r. However, we need to infrared regulate by taking the partial trace over soft gluon quanta with total energy less than $\epsilon$. Work in the $R_\xi$ gauge. A gauge rotation is made which only depends upon the distance from the origin. Vary it sufficiently slowly. The longitudinally polarized "soft" gluons from this gauge transformation contributes a total energy of less than $\epsilon$ if varied slowly enough. This corresponds to an internal charge rotation of the element of r.

Does taking the partial trace over soft gluons render the orientation of the charge in a mixed state? Soft gluons carry nonabelian charges too.

Suppose we have finite number of charged particles. Soft gluons in the limit only rotates the total charge of the system. The relative charge orientations of the individual charges at a fixed separation won't be affected in the limit of $\epsilon \to 0^+$.


The boundary conditions state that in the limit of going to spatial infinity, the value of the gauge transformation has to approach the identity. Spatial infinity breaks gauge invariance in the Coulomb phase. You can't change the charge orientation in this manner.

However, in the $R_\xi$ gauge, the orientation of the charge in the interior is entangled with the charge of the chromoelectric field contributions of the soft gluons in such a way that even though the total charge orientation remains fixed, the charge of the "undressed" particle remains in a superposition over all possible orientations.

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