Thermo Field Dynamics of a Quantum Algebra and its Application to the Anderson Model.

Quantum field theory was extended beyond its usual realm, by considering the situation where creation and annihilation operators do not satisfy the usual commutation relations but, rather, they form a "quantum algebra". This was done within the context of a thermal quantum field theory known as "thermo field dynamics". Most of the standard techniques of quantum field theory needed to be generalized, starting with finding an operator which annihilates the vacuum, making a generalized Wick's theorem, and developing Feynman rules. This generalization was explored by considering four models which are, in order of increasing level of difficulty, (1) a single, localized, N-fold degenerate fermionic state whose occupancy is restricted to a maximum of one fermion, (2) the first model with an interaction that merely shifts the energy of the state, (3) the single -site N-fold degenerate infinite-U Anderson model, and (4) the lattice Anderson model. The generalized Wick's theorem was found to break down, beyond a certain point in the reduction, necessitating the use of a time-splitting technique to complete the reduction. Feynman rules show a two-sector structure, having different rules in each sector; one of these sectors has very complex rules. No consistent self -energy expansion could be found for the Anderson model in this "bad" sector. Spontaneous vertices were found, which arise even in the absence of any interaction. These led to the creation of a "starting point function", which is a diagram connected to the starting point of a propagator. Non -cancelling vacuum diagrams were found. In the "bad" sector, these "vacuum diagrams" become multiply connected to the main diagrams. A method was found in which to disconnect these vacuum diagrams, yielding a renormalized perturbation expansion. Diagrammatic analysis of the single-site and lattice Anderson models led to a topological classification of self-energies, and self-consistent Dyson equations, in the "good" sector. By ignoring the "bad" sector, limited success was found in reproducing known results. Specifically, the single-site result showed a phase transition to a Kondo resonance state at the correct Kondo temperature, and the lattice result showed a renormalized band structure with a mass enhancement comparable to that of heavy-fermion materials.