Hamiltonian reduction scheme based on the analysis of restricted 1-forms in gauge-invariant variables is constructed. The method is applied for several physically interesting gauge invariant models. For the models of Yang-Mills theory a possible mechanism of the confinement is obtained. A quantization method (E-quantization) based on the extension of phase space with further application of the constrained quantization technique is constructed. A problem of scalar product for the constrained systems is investigated. A possible solution to this problem is found. Generalization of the Gupta-Bleuler conditions is done by minimization of quadratic fluctuations of quantum constraints. Connection of E-quantization to the geometric quantization and Berezin quantization is found. The quantum distribution function is introduced. For a pure state distribution the special elliptic type equation is obtained. A possible experimental measuring of the quantum distribution is discussed.
Mathematics Subject Classification: 70G35, 81S10, 81S30, 81T13.
Key words and phrases: Phase space, Hamiltonian reduction, gauge theory, symmetry, quantization, coherent state, quantum distribution.