Spatial Geometry of the Electric Field Representation of Non-Abelian Gauge Theories

A unitary transformation \Ps [E]=\exp (i\O [E]/g) F[E] is used to simplify the Gauss law constraint of non-abelian gauge theories in the electric field representation. This leads to an unexpected geometrization because \o^a_i\equiv -\d\O [E]/\d E^{ai} transforms as a (composite) connection. The geometric information in \o^a_i is transferred to a gauge invariant spatial connection \G^i_{jk} and torsion by a suitable choice of basis vectors for the adjoint representation which are constructed from the electric field $E^{ai}$. A metric is also constructed from $E^{ai}$. For gauge group $SU(2)$, the spatial geometry is the standard Riemannian geometry of a 3-manifold, and for $SU(3)$ it is a metric preserving geometry with both conventional and unconventional torsion. The transformed Hamiltonian is local. For a broad class of physical states, it can be expressed entirely in terms of spatial geometric, gauge invariant variables.Comment: 16pp., REVTeX, CERN-TH.7238/94 (Some revision on Secs.3 and 5; one reference added

Reformulation of QCD in the language of general relativity

It is shown that there exists such collection of variables that the standard QCD Lagrangian can be represented as the sum of usual Palatini Lagrangian for Einstein general relativity and the Lagrangian of matter and some other fields where the tetrad fields and the metric are constructed from initial $SU(3)$ Yang - Mills fields.Comment: LaTEX, 25 page

Reformulating Yang-Mills theory in terms of local gauge invariant variables

An explicit canonical transformation is constructed to relate the physical subspace of Yang-Mills theory to the phase space of the ADM variables of general relativity. This maps 3+1 dimensional Yang-Mills theory to local evolution of metrics on 3 manifolds.Comment: Lattice 2000 (Gravity and Matrix Models) 3 pages, espcrc2.st

Yang--Mills Configurations from 3D Riemann--Cartan Geometry

Recently, the {\it spacelike} part of the $SU(2)$ Yang--Mills equations has been identified with geometrical objects of a three--dimensional space of constant Riemann--Cartan curvature. We give a concise derivation of this Ashtekar type (``inverse Kaluza--Klein") {\it mapping} by employing a $(3+1)$--decomposition of {\it Clifford algebra}--valued torsion and curvature two--forms. In the subcase of a mapping to purely axial 3D torsion, the corresponding Lagrangian consists of the translational and Lorentz {\it Chern--Simons term} plus cosmological term and is therefore of purely topological origin.Comment: 14 pages, preprint Cologne-thp-1994-h1

On evaluation of two-loop self-energy diagram with three propogator

Small momentum expansion of the "sunset" diagram with three different masses is obtained. Coefficients at powers of $p^2$ are evaluated explicitly in terms of dilogarithms and elementary functions. Also some power expansions of "sunset" diagram in terms of different sets of variables are given.Comment: 9 pages, LaTEX, MSU-PHYS-HEP-Lu3/9

Supersymmetric Yang-Mills Theory and Riemannian Geometry

We introduce new local gauge invariant variables for N=1 supersymmetric Yang-Mills theory, explicitly parameterizing the physical Hilbert space of the theory. We show that these gauge invariant variables have a geometrical interpretation, and can be constructed such that the emergent geometry is that of N=1 supergravity: a Riemannian geometry with vector-spinor generated torsion. Full geometrization of supersymmetric Yang-Mills theory is carried out, and geometry independent divergences associated to the inversion of a differential operator with zero modes -- that were encountered in the non-supersymmetric case -- do not arise in this situation.Comment: 21 pages, LaTex, Added discussions and references, Final version for Nuclear Physics