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arXiv:0905.2948 · Journal of Physics Conference Series

Quantum-classical interactions and measurement: a consistent description using statistical ensembles on configuration space

M Reginatto, M J W Hall

Published Jun 1, 2009• 25 citations
arXivHTMLPDF

Abstract

We provide an overview of a canonical formalism that describes mixed quantum-classical systems in terms of statistical ensembles on configuration space, and discuss applications to measurement theory. It is shown that the formalism allows a general description of the measurement of a quantum system by a classical apparatus without running into inconsistencies. An example of classical and quantum particles interacting gravitationally is also given.

DOI
10.1088/1742-6596/174/1/012038

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We provide an overview of a canonical formalism that describes mixed quantum-classical systems in terms of statistical ensembles on configuration space, and discuss applications to measurement theory. It is shown that the formalism allows a general description of the measurement of a quantum system by a classical apparatus without running into inconsistencies. An example of classical and quantum particles interacting gravitationally is also given.
A consistent description of interactions between classical and quantum systems is relevant to quantum measurement theory and to calculations in quantum chemistry and quantum gravity. A solution is offered here to this long-standing problem, based on a universally applicable formalism for ensembles on configuration space. This approach overcomes difficulties arising in previous attempts and in particular allows for a back reaction on the classical ensemble, conservation of probability and energy, and the correct classical equations of motion in the limit of no interaction. Applications include automatic decoherence for quantum ensembles interacting with classical measurement apparatuses, a generalization of coherent states to hybrid harmonic oscillators, and an equation for describing the interaction of quantum matter fields with classical gravity, which implies that the radius of a Robertson-Walker universe containing a quantum massive scalar field can be sharply defined only for particular ``quantized'' values.
Abstract Finding a physically consistent approach to modeling interactions between classical and quantum systems is a highly nontrivial task. While many proposals based on various mathematical formalisms have been made, most of these efforts run into difficulties of one sort or another. One of the first detailed descriptions was given by Sudarshan and his collaborators who, motivated by the measurement problem in quantum mechanics, proposed a Hilbert space formulation of classical–quantum interactions which made use of the Koopman–von Neumann description of classical systems. Here we use the approach of ensembles on configurations space to give a detailed account of a classical apparatus measuring the position of a quantum particle that is prepared in a superposition of two localized states. We show that the probability of the pointer of the classical apparatus is left in a state that corresponds to the probability of the quantum particle. A subsequent observation of the pointer leads to an update of its probability density. From this we can obtain information about the position of the quantum particle, leading to an update of its wave function. Since this formalism incorporates uncertainties and finite measurement precision, it is well suited for metrological applications. Furthermore, it resolves fundamental issues that appear in the case of a quantum description of the apparatus.
A recent proposal for mixed dynamics of classical and quantum ensembles is shown, in contrast to other proposals, to satisfy the minimal algebraic requirements proposed by Salcedo for any consistent formulation of such dynamics. Generalized Ehrenfest relations for the expectation values of classical and quantum observables are also obtained. It is further shown that additional desirable requirements, related to separability, may be satisfied under the assumption that only the configuration of the classical component is directly accessible to measurement, e.g., via a classical pointer. Although the mixed dynamics is formulated in terms of ensembles on configuration space, thermodynamic mixtures of such ensembles may be defined which are equivalent to canonical phase-space ensembles on the classical sector. Hence, the formulation appears to be both consistent and physically complete.
Considering the recently established arbitrariness the Schroedinger equation has to be interpreted as an equation of motion for a statistical ensemble of particles. The statistical qualities of individual particles derive from the unknown intrinsic energy components, they depend on the physical environment by way of external potentials. Due to these statistical qualities and wave function normalization, non-locality is inherent to the fundamental relations of Planck, de Broglie and Schroedinger. A local formulation of these statements is introduced and briefly assessed, the modified and local Schroedinger equation is non-linear. Quantum measurements are analyzed in detail, the exact interplay between causal and statistical reasons in a measurement process can be accounted for. Examples of individual measurement effects in quantum theory are given, the treatment of diffraction experiments, neutron interferences, quantum erasers, the quantum Zeno effect, and interaction-free measurements can be described consistent with the suggested framework. The paper additionally provides a strictly local and deterministic calculation of interactions in a magnetic field. The results suggest that quantum theory is a statistical formalism which derives its validity in measurements from considering every possible measurement of a given system. It can equally be established, that the framework of quantum physics is theoretically incomplete, because a justification of ensemble qualities is not provided.
We investigate here the concept of generalized complementarity introduced by Bohr. It is shown that it is not possible to ascertain the microscopic state of a system without, for the purpose of measurement, imposing upon it physical conditions so stringent that they preclude the application of the statistical description which was appropriate before the measurement. One may conclude that the statistical averages which appear in the formation of an ensemble should in quantum statistics be given an interpretation somewhat different from that of classical statistics. These results lead naturally to a critical survey of the foundations of quantum statistical mechanics. We discuss the physical interpretation of statistical matrices which are the quantum equivalent of the classical ensemble. By a straightforward generalization of classical methods one obtains a statistical representation of a system if a set of measurements are given. Let $\ensuremath{\rho}$ designate the statistical matrix which represents the system; the method consists in making the diagonal sum of $\ensuremath{\rho} log \ensuremath{\rho}$ a minimum, while the results of the performed measurements appear as conditions of the minimum. As an application we treat the composition of two separate systems and derive from this the general proof of the HHH theorem which was first given by Klein.