Existence of Charged States of the Yang-Mills Field

Abstract

For a tentative choice of configuration space Ω, it is proved that the Yang‐Mills field, self‐interacting but not coupled to other fields, has states with a nonvanishing isospin component if gauge‐invariant quantization is used. This is shown by proving existence of a solution for the elliptic boundary‐value problem ▿_β▿^βζⁱ(x) = 0 on all of 3‐dimensional Euclidean space, subject to the asymptotic condition ζⁱ = cⁱ + O(r⁻¹), ∂_β∂^βζⁱ = O(r⁻⁴) as r → ∞, where cⁱ are constants; ▿_β is the covariant derivative belonging to the spatial Yang‐Mills potentials b_βⁱ(x). The existence proof is a modification of Schauder's proof to an unbounded domain. Ω consists of all numerical real multiplet functions b_βⁱ(x) which are of order O(r⁻²) as r → ∞, have ∂_βb^βi = O(r⁻⁴), and satisfy certain smoothness conditions. Also, for this configuration space, the problem of existence of equivalent transverse potentials is reduced to a simpler uniqueness problem. In the classical theory, the existence of solutions ζⁱ implies that the constraint equation can be satisfied for any choice of the ``covariant‐transverse'' part of B<sup>0βi</sup> within a very large class, by a unique ``covariant‐longitudinal'' part of B^0βi, if the potentials b_αⁱ(x) have the full SU(2) as holonomy group.