A new Quantum Mechanics on phase space
A complex function is associated to each bounded linear operator
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arXiv subjects
Antonio Cassa
Publications and source records attributed to Antonio Cassa.
A realistic and deterministic description of a quantum system
From the analysis of the measurement process we make the hypothesis that we have to add to the quantum state psi a label z and a special function alpha in order to describe completely the preparation of a (pure) quantum system . Given such a terne every observable A receives a univocal value . Therefore all physical quantities of the system are well defined independently from the measuring process. Making the unknown parameter z to vary in its space we get the probabilities prescribed by quantum mechanics.We show also that every evolution of quantum states can be extended to an evolution of the ternes.
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All quantum expectation values as classical statistical mean values
Given a physical quantum system described by a Hilbert H, for any bounded quantum observable (a bounded self-adjoint operator) T it is possible to define several ''hidden observable'' functions f:H->R associated to T and for any quantum mixed state (a density matrix) D it is possible to define several ''hidden mixed states'' (probability measures) m on H associated to D in such a way that the following equality is verified: Trace[ b(T). D] =integral[b(f(psi)).dm(psi) whatever is the continuous function b:R->R. This formula gives a general way to express any expectation value computable in a quantum theory as a classical statistical mean value.
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Quantum physics with a hidden variable
Every quantum physical system can be considered the ''shadow'' of a special kind of classical system. The system proposed here is classical mainly because each observable function has a well precise value on each state of the system: an hypothetical observer able to prepare the system exactly in an assigned state and able to build a measuring apparatus perfectly corresponding to a required observable gets always the same real value. The same system considered instead by an unexpert observer, affected by the ignorance of a hidden variable, is described by a statistical theory giving exactly and without exception the states, the observables, the dynamics and the probabilities prescribed for the usual quantum system.
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Quantum systems as classical systems
A characteristical property of a classical physical theory is that the observables are real functions taking an exact outcome on every (pure) state; in a quantum theory, at the contrary, a given observable on a given state can take several values with only a predictable probability. However, even in the classical case, when an observer is intrinsically unable to distinguish between some distinct states he can convince himself that the measure of its ''observables'' can have several values in a random way with a statistical character. What kind of statistical theory is obtainable in this way? It is possible, for example, to obtain exactly the statistical previsions of quantum mechanics? Or, in other words, can a physical system showing a classical behaviour appear to be a quantum system to a confusing observer? We show that from a mathematical viewpoint it is not difficult to produce a theory with hidden variables having this property. We don't even try to justify in physical terms the artificial construction we propose; what we do is to give a general and rigorous argument showing how the interplay between the classical and quantum mechanics we offer is interpretable as the difference between an imaginary very expert observer and another nonexpert observer. This proves also that besides the well known theorems concerning the impossibility of hidden variables (cfr. Von Neumann [Neu] and Jauch-Piron [J-P]) there is also room for a result in favor of the possibility.
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