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The analysis carried out in this thesis is based on the Hamiltonian approach to the Lattice Gauge theory. On defining a Hamiltonian for the gauge degrees of freedom, one deals with a quantum mechanics with a well-defined Hamiltonian Operator; And its advantages are discussed in this thesis. While considering the partition function approach to field theory, a lattice is used to regularise the theory. The case of SU(2) and 2+1 dimensional theory is specialized and the physics of Hamiltonian is given by the physics of coupled symmetric tops. It is sufficient to understand the quantum mechanics of a rigid operator, for understanding the gauge degree of freedom, on a single link. The SU(2) lattice Gauge theory is shown to be exactly equivalent to a U(1) gauge theory on a Kagome lattice. Further new dynamical variables which create or annihilate a unit of an additive color invariant electric flux are introduced. It is also shown that the concepts used in SU(2) can be extended to SU(3) case. It is shown that SU(3) lattice gauge theory on a square lattice can be rewritten as a certain abelian gauge theory which has U(1) x U(1) local gauge invariance. It is the precise realization of the t'Hooft's conjecture that for confinement U(1)^ (N-1) gauge theory is relevant in the SU(N) case. This reformulation is based on the concept that the Physical subspace of the Hilbert Space of lattice gauge theories can be explicitly labeled using certain gauge invariant local operators. A generalization to SU(N) groups of the triangle rule for addition of angular momenta and its extension to SU(3) are discussed. |
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