**George Jorjadze**

## Hamiltonian Reduction and Quantization on Symplectic Manifolds

**abstract:**

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.