Quantum Mechanical Quantization Research Articles

Lectures on Geometric Quantization

These lectures notes are meant as an introduction to geometric quantization. In Section 1, I begin with presentation of the historical background of quantum mechanics. I continue with discoveries in the theory of representations of Lie groups, which lead to emergence of geometric quantization as a part of pure mathematics. This presentation is very subjective, flavored by my own understanding of the role of geometric quantization in quantum mechanics and representation theory. Section 2 is devoted to a review of geometry of Hamiltonian systems. Geometric quantization is discussed in the next two sections: prequantization in Section 3 and polarization in Section 4. In particular, I discuss geometric quantization with respect to polarizations given by Kahler structure, cotangent bundle projection and completely integrable system. More advanced topics, like metaplectic structure, pairing of polarizations, and commutation of quantization and reduction, are not included.

Large-Nexpansion as a semiclassical approximation to the third-quantized theory

The semiclassical theory for the large-N field models is developed from an unusual point of view. Analogously to the procedure of the second quantization in quantum mechanics, the functional Schr\odinger large-N equation is presented in a third-quantized form. The third-quantized creation and annihilation operators depend on the field \ensuremath{\varphi}(x). If the coefficient of the ${\ensuremath{\varphi}}^{4}$ term is of order $1/N$ (this is the usual condition of applicability of the $1/N$ expansion), one can rescale the third-quantized operators in such a way that their commutator will be small, while the Heisenberg equations will not contain large or small parameters. This means that the classical equation of motion is an equation on the functional \ensuremath{\Phi}[\ensuremath{\varphi}(\ensuremath{\cdot})]. This equation, being a nonlinear analogue of the functional Schr\odinger equation for the one-field theory, is investigated. The exact solutions are constructed and the renormalization problem is analyzed. We also perform a quantization procedure about found classical solutions. The corresponding semiclassical theory is a theory of a variable number of fields. The developed third-quantized semiclassical approach is applied to the problem of finding the large-N spectrum. The results are compared with formulas obtained by known methods. We show that not only the known but also new energy levels can be found.

Noncommutative differential geometry, quantization, and smooth symmetries of the C*-algebras associated to topological dynamics

Noncommutative differential geometric structures are considered for a class of simple C*-algebras. This structure is defined in terms of smooth Lie group actions on the C*-algebra in question together with a certain quantization mapping motivated directly by the known cohomological obstructions for the quantum mechanical quantization correspondence. We show that such a quantization mapping may be constructed for the C*-algebras associated to antisymmetric bi-characters and for the Cuntz/Cuntz-Krieger C*-algebras associated to topological dynamics. A certain curvature obstruction is defined in terms of the quantization mapping. It is shown that existence of smooth Lie group actions is determined by the curvature obstruction.