The structure of phase space and quantum mechanics

Sergey Nikolaevich Mayburov

Quantum Space-Time and Phase Space with fuzzy geometric structure are studied as possible formalism for quantization of massive particles and fields. In this approach the state of nonrelativistic particle m described by the fuzzy point of fuzzy (weakly) ordered Manifold X. Due to m partial ordering in X m coordinate x principal uncertainty is generic in this ansatz. It's shown that in 1+1 dimension such Fuzzy Mechanics (FM) is equivalent to Path Integral formalism of Quantum Mechanics (QM).

Franklin E. Schroeck

Nicolaas P. Landsman

J. M. Isidro

Duality transformations within the quantum mechanics of a finite number of degrees of freedom can be regarded as the dependence of the notion of a quantum, i.e., an elementary excitation of the vacuum, on the observer on classical phase space. Under an observer we understand, as in general relativity, a local coordinate chart. While classical mechanics can be formulated using a symplectic structure on classical phase space, quantum mechanics requires a complex-differentiable structure on that same space. Complex-differentiable structures on a given real manifold are often not unique. This article is devoted to analysing the dependence of the notion of a quantum on the complex-differentiable structure chosen on classical phase space. For that purpose we consider Kähler phase spaces, endowed with a dynamics whose Hamiltonian equals the local Kähler potential.

Renzo Cirelli, Alessandro Manià, Livio Pizzocchero B

We develop a geometrical approach to Schrödinger quantum mechanics, alternative to the usual one, which is based on linear and algebraic structures such as Hilbert spaces, operator algebras, etc. The starting point of this approach is the Kähler structure possessed by the set of the pure states of a quantum system. The Kähler manifold of the pure states is regarded as a “quantum phase space”, conceptually analogous to the phase space of a classical hamiltonian system, and all the constituents of the conventional formulation, in particular the algebraic structure of the observables, are reproduced using a suitable “Kähler formalism”. We also show that the probabilistic character of the measurement process in quantum mechanics and the uncertainty principle are contained in the geometrical structure of the quantum phase space. Finally, we obtain a characterization for quantum phase spaces which can be interpreted as a statement of uniqueness for Schrödinger quantum mechanics.

Manuel Gadella