Lectures on quantization of gauge systems

A gauge system is a classical field theory where among the fields there are connections in a principal G-bundle over the space-time manifold and the classical action is either invariant or transforms appropriately with respect to the action of the gauge group. The lectures are focused on the path integral quantization of such systems. Here two main examples of gauge systems are Yang-Mills and Chern-Simons.Comment: 63 pages, 22 figures. Based on lectures given at the Summer School "New paths towards quantum gravity", Holbaek, Denmark, 200

The Dirac operator and gamma matrices for quantum Minkowski spaces

Gamma matrices for quantum Minkowski spaces are found. The invariance of the corresponding Dirac operator is proven. We introduce momenta for spin 1/2 particles and get (in certain cases) formal solutions of the Dirac equation.Comment: 25 pages, LaTeX fil

On invariants of graphs related to quantum sl(2)\mathfrak{sl}(2) at roots of unity

We show how to define invariants of graphs related to quantum $\mathfrak{sl}(2)$ when the graph has more then one connected component and components are colored by blocks of representations with zero quantum dimensions

Quantum Chains with GLq(2)GL_q(2) Symmetry

Usually quantum chains with quantum group symmetry are associated with representations of quantized universal algebras $U_q(g)$ . Here we propose a method for constructing quantum chains with $C_q(G)$ global symmetry , where $C_q(G)$ is the algebra of functions on the quantum group. In particular we will construct a quantum chain with $GL_q(2)$ symmetry which interpolates between two classical Ising chains.It is shown that the Hamiltonian of this chain satisfies in the generalised braid group algebra.Comment: 7 pages,latex,this is the completely revised version of my paper which is submitted for publicatio

Affine Toda field theory as a 3-dimensional integrable system

The affine Toda field theory is studied as a 2+1-dimensional system. The third dimension appears as the discrete space dimension, corresponding to the simple roots in the $A_N$ affine root system, enumerated according to the cyclic order on the $A_N$ affine Dynkin diagram. We show that there exists a natural discretization of the affine Toda theory, where the equations of motion are invariant with respect to permutations of all discrete coordinates. The discrete evolution operator is constructed explicitly. The thermodynamic Bethe ansatz of the affine Toda system is studied in the limit $L,N\to\infty$. Some conjectures about the structure of the spectrum of the corresponding discrete models are stated.Comment: 17 pages, LaTe