Quantum mechanical probabilities not restricted to a moment of time

Norifumi L. Yamada, Shuji Takagi

Norifumi L. Yamada

Norifumi L. Yamada

A theoretical aspect of quantum mechanical probabilities is studied. In particular, it is investigated whether quantum mechanical probabilities can be defined for alternatives which are not restricted to a moment of time. Firstly the meaning and the status of this investigation in quantum mechanics are explained. Secondly a general framework is constructed within which we make the investigation. Thirdly a necessary mathematical tool is reviewed and extended. Lastly the general framework is applied to concrete examples with the help of the mathematical tool. Examples of alternatives are found which are not restricted to a moment of time and for which quantum mechanical probabilities can be defined with clear measurement theoretical meanings if an initial condition of the particle belongs to a specific class. (This is the doctoral thesis of the author.)

Peter Mittelstaedt

Henry Margenau, Leon W. Cohen

Stephen D. Bartlett, Terry Rudolph, Robert W. Spekkens

How would the world appear to us if its ontology was that of classical mechanics but every agent faced a restriction on how much they could come to know about the classical state? We show that in most respects it would appear to us as quantum. The statistical theory of classical mechanics, which specifies how probability distributions over phase space evolve under Hamiltonian evolution and under measurements, is typically called Liouville mechanics, so the theory we explore here is Liouville mechanics with an epistemic restriction. The particular epistemic restriction we posit as our foundational postulate specifies two constraints. The first constraint is a classical analog of Heisenberg's uncertainty principle; the second-order moments of position and momentum defined by the phase-space distribution that characterizes an agent's knowledge are required to satisfy the same constraints as are satisfied by the moments of position and momentum observables for a quantum state. The second constraint is that the distribution should have maximal entropy for the given moments. Starting from this postulate, we derive the allowed preparations, measurements, and transformations and demonstrate that they are isomorphic to those allowed in Gaussian quantum mechanics and generate the same experimental statistics. We argue that this reconstruction of Gaussian quantum mechanics constitutes additional evidence in favor of a research program wherein quantum states are interpreted as states of incomplete knowledge and that the phenomena that do not arise in Gaussian quantum mechanics provide the best clues for how one might reconstruct the full quantum theory.