The Phase Space Formalism for Quantum Mechanics and C* Axioms

Maciej Mączyński

Sergey Nikolaevich Mayburov

The quantum spacetime and the phase space with fuzzy structure are investigated as the possible quantization formalism. In this theory, the state of the nonrelativistic particle m corresponds to the element of fuzzy ordered set (Foset), i.e. the fuzzy point. Due to Foset partial (weak) ordering, the m space coordinate x acquires principal uncertainty σ x . It is shown that Schrödinger formalism of quantum mechanics can be completely derived from consideration of m evolution in fuzzy phase space with minimal number of axioms.

Franklin E. Schroeck

Manuel Gadella

Eduard Prugovečki

An axiomatic formulation of a quantum mechanical formalism is given. The formulation is not in terms of objects associated with the Hilbert space, but in terms of a different kind of objects for which the name ``complex probability measures'' has been chosen. It is shown that the conventional Hilbert-space formalism obeys the given axioms. A few consequences of the axioms are investigated, some of which are found useful in the second part of this work.

Stanley Gudder