The Hidden Spatial Geometry of Non-Abelian Gauge Theories

Abstract

The Gauss law constraint in the Hamiltonian form of the $SU(2)$ gauge theory of gluons is satisfied by any functional of the gauge invariant tensor variable $ϕ^{ij} = B^{ia} B^{ja}$. Arguments are given that the tensor $G_{ij} = (ϕ^{-1})_{ij}\,\det B$ is a more appropriate variable. When the Hamiltonian is expressed in terms of $ϕ$ or $G$, the quantity $Γ^i_{jk}$ appears. The gauge field Bianchi and Ricci identities yield a set of partial differential equations for $Γ$ in terms of $G$. One can show that $Γ$ is a metric-compatible connection for $G$ with torsion, and that the curvature tensor of $Γ$ is that of an Einstein space. A curious 3-dimensional spatial geometry thus underlies the gauge-invariant configuration space of the theory, although the Hamiltonian is not invariant under spatial coordinate transformations. Spatial derivative terms in the energy density are singular when $\det G=\det B=0$. These singularities are the analogue of the centrifugal barrier of quantum mechanics, and physical wave-functionals are forced to vanish in a certain manner near $\det B=0$. It is argued that such barriers are an inevitable result of the projection on the gauge-invariant subspace of the Hilbert space, and that the barriers are a conspicuous way in which non-abelian gauge theories differ from scalar field theories.