Existence of Charged States of the Yang-Mills Field
- 1 November 1970
- journal article
- research article
- Published by AIP Publishing in Journal of Mathematical Physics
- Vol. 11 (11) , 3258-3274
- https://doi.org/10.1063/1.1665125
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 bβi(x). The existence proof is a modification of Schauder's proof to an unbounded domain. Ω consists of all numerical real multiplet functions bβi(x) which are of order O(r−2) as r → ∞, have ∂βbβi = 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 B0βi within a very large class, by a unique ``covariant‐longitudinal'' part of B0βi, if the potentials bαi(x) have the full SU(2) as holonomy group.Keywords
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