Lattice Gauge Theories as Problems of Constructive Quantum Field Theory

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The Aim of this program is to give a mathematically rigid theory for models appeared in quantum field theory. In particular the following two themes are concrete targets ; Ultraviolet stability for Abelian Higgs-Kibble model in a three dimensional finite lattice, which is a model in quantum electrodynamics, and existence of its continuous limite of the lattice space and the required physical axioms satisfied by the continuous limited space.For two years research we can not get the completely results for the above problems. But we can got the following interesting results. First we can give a rigorous definition of the renormalization transform used in theoretical physics. That is formally defined by making use of Dirac's δ -function and one of it's properties, that is, Faddeev-Popov procedure is justified by the formal invariance of Haar measure for δ-function. But we defined a renormalization transform as the measure and prove the all required properties. Secondly, we can prove the ultra-violet stability for three dimensional Abelian Higgs-Kibble model. Now we are preparing this result. In order to verify this uniform estimate we need new Ward-Takahasi identity appearing in new type graphs.