Classical and Quantum Observables

Enrico G. Beltrametti, Slawomir Bugajski

Dirk Aerts

We analyze the difference between classical mechanics and quantum mechanics. We come to the conclusion that this difference can be found in the nature of the observables that are considered for the physical system under consideration. Classical mechanics can only describe a certain kind of what we called ‘‘classical observable.’’ Quantum mechanics can only describe another kind of observable; it cannot describe, however, classical observables. To perform this analysis, we use a theory where every kind of observable can be treated and which is in a natural way a generalization of both classical and quantum mechanics. If in a study of a physical system in this theory we restrict ourselves to the classical observables, we rediscover classical mechanics as a kind of first study of the physical system, where all the nonclassical properties are hidden. If we find that this first study is too rough we can also study the nonclassical part of the physical system by a theory which is eventually quantum mechanics.

Abhay Ashtekar

Christof Wetterich

The classical observables of position and momentum are not well adapted to particles in a microphysical situation where typical probability distributions are characterized by a substantial dispersion. We propose the use of more robust quantum observables for probabilistic classical particles. The quantum observables are statistical observables which do not take fixed values for a given classical position and momentum. Solutions of the Liouville equation are discussed in the quantum formalism for classical statistics. Statistical observables are represented by non-commuting operators. No classical correlation function is defined for these observables and Bell's inequalities do not apply. We demonstrate for a general potential how a quantum system emerges from classical statistics. For the particular cases of a harmonic potential and a Coulomb potential we investigate subsystems which describe all features of a quantum particle. This covers the discrete energy spectrum of the hydrogen atom and quantum harmonic oscillator. We discuss the interference for the double-slit experiment. Conserved statistical observables may also be relevant for the probabilistic dynamics of dust or planets.

Roshan Raj

It presents the evolution of the quantum observables from their classical counterpart and distinctly analyses the meaning of observable in context to both the classical and the quantum realms, with an emphasis on W. Heisenberg's realization of observable and unobservable [1925]. It is stated that quantum observables are, in the first place, indeed distinctive to classical observables since each appears unobservable to the other; however, to associate physical essence as possessed by the latter, the expectation value of the former is determined. Further, the possibility of mapping quantum information to the classical ones through the action of the operator may be realized after re-interpreting analogies between the quantum commutator and Poisson brackets.

Kevin Costello, Owen Gwilliam