Gauge invariance and color charge fluctuations

The nature of confinement is connected with color charge. Unfortunately, the color charge densities in QCD, the Noether charge densities associated with the global color invariance, are not invariant under local color rotations. This implies that the expectation values of the net color charge in any region of any physical state in QCD, states that satisfy the color Gauss law, are automatically zero for all components of color. In this paper it is shown that the expectation value of the square of the net color charge in a region, a measure of the color charge fluctuations, is necessarily nonzero when evaluated in physical states and the result, while depending on the scheme and scale by which the theory is regulated, is gauge invariant. This holds despite the formal lack of gauge invariance of the operator. Moreover, there is a particular combination of the color charge fluctuations for the vacuum and for a system describable by a non-trivial density matrix that is independent that has a well-defined continuum limit