Quantum Theory in Algebraic and Geometric Approaches

This chapter is devoted to different approaches to quantum mechanics.It starts with a summary of the main principles of quantum theory in the standard approach when states are represented by density matrices and pure states are vectors in Hilbert space but the main focus is on the algebraic approach where the states are represented by positive functionals on associative algebra with involution and on the geometric approach where the set of states is the primary object. The relation between different approaches is investigated; in particular, the GNS construction that permits us to consider states of algebraic approach as vectors in Hilbert space is explained. It is shown that quantum mechanics can be considered as a deformation of classical mechanics; this leads to a notion of Weyl algebra. Weyl algebras, their close relatives called Clifford algebras, their representations, symbols of operators, and simplest Hamiltonians are studied. ( This is called quantization and second quantization in standard textbooks.) We derive decoherence from adiabatic interaction with a random environment; this allows us to obtain standard formulas for probabilities from first principles. (The proof of decoherence is based on adiabatic perturbation theory.) Finally, we discuss the main notions of statistical physics.