Numerical and analytical investigations of the lattice gauge models at finite temperatures by renormalization group approach

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1. The lattice gauge theory (LGT) at finite temperatures is investigated in the Migdal renormalization group (RG) approach. Starting from various bare theories defined by various bare coupling constants, we arrive at a unique renormalized theory after sufficient number of Migdal RG transformations. This behavior is seen with a set of infinite number of coupling constants corresponding to the infinite number of irreducible representations. Each action is represented by a point in the infinite dimenensional coupling constant space. RG transformations drive the point to a unique renormalized trajectory independent of the bare action. Migdal RG transformation at T=0 is expressed by isotropic scale transformations. To investigate T<>0 systems, asymmetric scale transformation where the scale in timelike direction is fixed is necessary. We investigated SU(2) and SU(3) LGT's at T<>0. RG flow at T<>0 and internal energy are calculated. We found clear phase transition in the RG flow. The internal energy shows clear deconfinement transition. It rises from zero in low T to a finite value in high T. It shows approximate Stefan-Boltzmann law in high T region. Migdal RG approach including fermion is postponed for future studies.2. Non-perturbative method found in studies of quantum chromo dynamics is applied to quantum gravity. The gravity with higher derivatives is expressed by 0(4) variables. By rewriting them by variables with SU(2)xSU(2) gauge symmetry, it is shown the symmetry is spontaneously broken. The Einstein term arises as a result.3. The Lagrangian including coordinates which are bounded is quantized in the path integral method. According to Faddeev-Senjanovic method, namely by introducing the constraint in <delta> -function form, the quantum Hamiltonian of the system on D-dimensional sphere is obtained.