The configuration space quasi-probability formulation of quantum mechanics

Philipp Roser, Matthew T. Scoggins

Kazuki Fukuda

George Allen Baker Jr.

We postulate a formulation of quantum mechanics which is based solely on a quasi-probability function on the classical phase space. We then show that this formulation is equivalent to the standard formulation, and that the quasi-probability function is exactly analogous to the density matrix of Dirac and von Neumann. We investigate the theory of measurement in this formulation and derive the following remarkable results. As is well known, the correspondence between classical functions of both the position and conjugate momentum and quantum mechanical operators is ambiguous because of noncommutativity. We show that the solution of this correspondence problem is completely equivalent to the solution of the eigenvalue problem. This result enables us to give a constructive method to compute eigenvalues and eigenfunctions.

Mustafa Amin

Classical mechanics, in the Koopman-von Neumann formulation, is described in Hilbert space. It is shown here that classical canonical transformations are generated by Hermitian operators that are in general noncommutative. This naturally brings about uncertainty relations inherent in classical mechanics, for example between position and the generator of space translations, between momentum and the generator of momentum translations, and between dynamical time and the Liouvillian, to name a few. Further, it is shown that the Wigner representation produces a quasi-probability distribution that can take on negative values. Thus, two of the hallmark features of quantum mechanics are reproduced, and become apparent, in a Hilbert-space formulation of classical mechanics.

Bryce de Witt

Patrick M. Duerr, Alexander Ehmann

The paper takes up Bell's (1987) "Everett (?) theory" and develops it further. The resulting theory is about the system of all particles in the universe, each located in ordinary, 3-dimensional space. This many-particle system as a whole performs random jumps through 3N-dimensional configuration space - hence "Tychistic Bohmian Mechanics" (TBM). The distribution of its spontaneous localisations in configuration space is given by the Born Rule probability measure for the universal wavefunction. Contra Bell, the theory is argued to satisfy the minimal desiderata for a Bohmian theory within the Primitive Ontology framework (for which we offer a metaphysically more perspicuous formulation than is customary). TBM's formalism is that of ordinary Bohmian Mechanics (BM), without the postulate of continuous particle trajectories and their deterministic dynamics. This "rump formalism" receives, however, a different interpretation. We defend TBM as an empirically adequate and coherent quantum theory. Objections voiced by Bell and Maudlin are rebutted. The "for all practical purposes"-classical, Everettian worlds (i.e. quasi-classical histories) exist sequentially in TBM (rather than simultaneously, as in the Everett interpretation). In a temporally coarse-grained sense, they quasi-persist. By contrast, the individual particles themselves cease to persist.