turkmath.org

Geometric quantization gives a way to pass from a classical mechanical description of a physical system to a quantum mechanical description of the same system. Symplectic manifolds are mathematical models for a classical system. The geometric quantization helps us to understand the quantum description of this system by a Hilbert space. Key ingredient needed in the geometric quantization procedure is called polarization. Let U be a compact Lie group and K a closed subgroup such that the quotient space U/K is a compact symmetric space. We apply geometric quantization to the cotangent bundle of U/K, for which we have two natural choices for a polarization: the vertical polarization and the holomorphic polarization. Geometric quantization then gives us two Hilbert spaces, one a space of functions on U/K and the other a space of holomorphic functions on the cotangent bundle which can be identifies with the complexification U^C/K^C. The object of this talk is to see the relationship between these two Hilbert spaces. As particular cases of compact symmetric spaces, we consider compact Lie groups and the 5-sphere S^5.