Abstract:
Considering one-dimensional nonminimally-coupled lattice gauge theories, a class of nonlocal one-dimensional systems is presented, which exhibits a phase transition. It is shown that the transition has a latent heat, and, therefore, is a first order phase transition.

Abstract:
A general formlulation for discrete-time quantum mechanics, based on Feynman's method in ordinary quantum mechanics, is presented. It is shown that the ambiguities present in ordinary quantum mechanics (due to noncommutativity of the operators), are no longer present here. Then the criteria for the unitarity of the evolution operator is examined. It is shown that the unitarity of the evolution operator puts restrictions on the form of the action, and also implies the existence of a solution for the classical initial-value problem.

Abstract:
We consider one dimensional lattice gauge theories constructed by the minimal coupling prescription. It is shown that these theories are exactly solvable in the thermodynamic limit. After considering the most general case, we discuss some special cases on finite lattices, and also work out some examples. There is no phase transition in these minimally coupled theories.

Abstract:
Using the simple path integral method we calculate the $n$-point functions of field strength of Yang-Mills theories on arbitrary two-dimensional Riemann surfaces. In $U(1)$ case we show that the correlators consist of two parts , a free and an $x$-independent part. In the case of non-abelian semisimple compact gauge groups we find the non-gauge invariant correlators in Schwinger-Fock gauge and show that it is also divided to a free and an almost $x$-independent part. We also find the gauge-invariant Green functions and show that they correspond to a free field theory.

Abstract:
By using the path integral method , we calculate the Green functions of field strength of Yang-Mills theories on arbitrary nonorientable surfaces in Schwinger-Fock gauge. We show that the non-gauge invariant correlators consist of a free part and an almost $x$-independent part. We also show that the gauge invariant $n$-point functions are those corresponding to the free part , as in the case of orientable surfaces.

Abstract:
Using the path integral method, we calculate the partition function and the generating functional (of the field strengths) of the generalized 2D Yang-Mills theories in the Schwinger--Fock gauge. Our calculation is done for arbitrary 2D orientable, and also nonorientable surfaces.

Abstract:
Using the collective field theory approach of large-N generalized two-dimensional Yang-Mills theory on cylinder, it is shown that the classical equation of motion of collective field is a generalized Hopf equation. Then, using the Itzykson-Zuber integral at the large-N limit, it is found that the classical Young tableau density, which satisfies the saddle-point equation and determines the large-N limit of free energy, is the inverse of the solution of this generalized Hopf equation, at a certain point.

Abstract:
The phase structure of the generalized Yang--Mills theories is studied, and it is shown that {\it almost} always, it is of the third order. As a specific example, it is shown that all of the models with the interaction $\sum_j (n_j-j+N)^{2k}$ exhibit third order phase transition. ($n_j$ is the length of the $j$-th row of the Yang tableau corresponding to U($N$).) The special cases where the transition is not of the third order are also considered and, as a specific example, it is shown that the model $\sum_j (n_j-j+N)^2+g\sum_j (n_j-j+N)^{4}$ exhibits a third order phase transition, except for $g=27\pi^2/256$, where the order of the transition is 5/2.

Abstract:
We present the technique of derivation of a theory to obtain an $(n+1)f$-degrees-of-freedom theory from an $f$-degrees-of-freedom theory and show that one can calculate all of the quantities of the derived theory from those of the original one. Specifically, we show that one can use this technique to construct, from an integrable system, other integrable systems with more degrees of freedom.