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The Hidden Spatial Geometry of Non-Abelian Gauge Theories | | D.Z. Freedman
; P.E. Haagensen
; K. Johnson
; J.I. Latorre
; | | Date: |
8 Sep 1993 | | Subject: | hep-th hep-ph | | 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 $phi^{ij} = B^{ia} B^{ja}$. Arguments are given that the tensor $G_{ij} = (phi^{-1})_{ij},det B$ is a more appropriate variable. When the Hamiltonian is expressed in terms of $phi$ or $G$, the quantity $Gamma^i_{jk}$ appears. The gauge field Bianchi and Ricci identities yield a set of partial differential equations for $Gamma$ in terms of $G$. One can show that $Gamma$ is a metric-compatible connection for $G$ with torsion, and that the curvature tensor of $Gamma$ 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. | | Source: | arXiv, hep-th/9309045 | | Services: | Forum | Review | PDF | Favorites | |
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