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The density matrix in the de Broglie-Bohm approach

Maroney, O J E

Published Nov 21, 2003• 22 citations
OpenAlex

Abstract

If the density matrix is treated as an objective description of individual systems, it may become possible to attribute the same objective significance to statistical mechanical properties, such as entropy or temperature, as to properties such as mass or energy. It is shown that the de Broglie-Bohm interpretation of quantum theory can be consistently applied to density matrices as a description of individual systems. The resultant trajectories are examined for the case of the delayed choice interferometer, for which Bell appears to suggest that such an interpretation is not possible. Bell's argument is shown to be based upon a different understanding of the density matrix to that proposed here.

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If the density matrix is treated as an objective description of individual systems, it may become possible to attribute the same objective significance to statistical mechanical properties, such as entropy or temperature, as to properties such as mass or energy. It is shown that the de Broglie-Bohm interpretation of quantum theory can be consistently applied to density matrices as a description of individual systems. The resultant trajectories are examined for the case of the delayed choice interferometer, for which Bell appears to suggest that such an interpretation is not possible. Bell's argument is shown to be based upon a different understanding of the density matrix to that proposed here.
I will try to interest you in the de Broglie–Bohm version of non-relativistic quantum mechanics. It is, in my opinion, very instructive. It is experimentally equivalent to the usual version insofar as the latter is unambiguous. But it does not require, in its very formulation, a vague division of the world into ‘system’ and ‘apparatus,’ nor of history into ‘measurement’ and ‘nonmeasurement.’ So it applies to the world at large, and not just to idealized laboratory procedures. Indeed the de Broglie–Bohm theory is sharp where the usual one is fuzzy, and general where the usual one is special. This is not a systematic exposition, but only an illustration of the ideas with a particularly nice example, and then some remarks on the role of the density matrix – in tribute to the title of this conference. No one more eloquently than John A. Wheeler has presented the delayed-choice double slit experiment. A pulsed particle source S (see Fig. 1) is so feeble that not more than one particle is emitted per pulse. The associated wave pulse B falls on a screen with slits 1 and 2. The transmitted pulses B′ are focussed by off-centred lenses into intersecting plane wave trains which fall finally on particle counters C 1 and C 2 – unless a photographic plate P is pushed into the region where the two wave trains interpenetrate. The decision, to interpose the plate or not, is made only after the pulse has passed the slits.
Abstract The fundamentals of the theory of density matrices are reviewed. The density matrix is introduced first in terms of state vectors for pure states over which a probability distribution is given. It is then shown that the applications of the density matrix do not require knowledge of the particular probability distribution over pure states that may have been used for its original determination. It is stressed that density matrices, like state vectors, describe ensembles rather than single systems, as determinism in quantum theory (except for some conservation laws) is of a statistical character only. The entropy of a quantum‐mechanical system is defined as the average per system of the entropy of an ensemble of systems described by the quantum‐mechanical (pure or mixed) state. For pure states this entropy is zero, in agreement with the fact that state vectors constitute the maximal information that can be given about a quantum‐mechanical system. As the entropy should be independent of the particular probability distribution over pure states that may have been used to form the mixed state, it is calculated from statistical considerations about an imaginary ensemble in which the eigenvalues of the density matrix are probabilities over pure states given by the eigenfunctions of the density matrix. The entropy is calculated for a mixture of two nonorthogonal states. The time evolution of the density matrix, in general, is not a unitary transformation, as observable systems are not closed and therefore in the Hamiltonian contain unknown external parameters that are not the same for the entire mixture. Therefore transitions between pure states and mixed states take place. Those from pure to mixed states take place spontaneously and irreversibly under increase of entropy. Transitions from mixed states to pure states can be fabricated; an example is given. As Wigner's thought experiment of mixing coherently the two beams from a Stern‐Gerlach apparatus according to estimates of Bohm should be impossible by loss of coherence in the gradient magnetic field, it is indicated how Wigner's ideas may be realized by using, instead, polarized light beams from birefringent crystals. After a brief review of the quantum theory of measurements and the various arguments usually given for the reducibility of state after the measurement, Wigner's argument against blind use of these arguments is interpreted as a limit of validity of the familiar probability rule of quantum theory, which should not be applied after an unsuccessful measurement. By splitting light beams coherently by glass plates, one can by the same apparatus have measurements both of the relative intensities of two coherent beams, and of interference effects between the two beams. This shows that complementarity applies to the individual systems in the beam, as they reach the glass plate where a choice must be made between two complementary possibilities of future measurement. Finally, by describing Schrödinger's cat by a density matrix and asking about its time dependence, it is shown that, if it were possible to measure the interference effects between living and dead cats used by some people as a justification for writing down a state vector for the cat, then this measurement could be used to revive some dead cats.
There are three possible interpretations of the wave function in the de Broglie-Bohm theory: taking the wave function as corresponding to a physical entity or a property of the Bohmian particles or a law. In this paper, we argue that the first interpretation is favored by an analysis of protective measurements. The de Broglie-Bohm theory is an alternative to standard quantum mechan-ics initially proposed by de Broglie (1928) and later developed by Bohm (1952). According to the theory, a complete description of a quantum system is provided by the configuration defined by the positions of its particles together with its wave function. The wave function follows the linear Schrödinger equation and never collapses. The motion of the particles, which are usually called Bohmian particles, follows the so-called guiding equation. Although the de Broglie-Bohm theory is mathematically equivalent to quantum mechanics, there is no clear consensus with regard to its physical interpretation. In particular, the interpre-
The Statistical Interpretation of quantum theory is formulated for the purpose of providing a sound interpretation using a minimum of assumptions. Several arguments are advanced in favor of considering the quantum state description to apply only to an ensemble of similarily prepared systems, rather than supposing, as is often done, that it exhaustively represents an individual physical system. Most of the problems associated with the quantum theory of measurement are artifacts of the attempt to maintain the latter interpretation. The introduction of hidden variables to determine the outcome of individual events is fully compatible with the statistical predictions of quantum theory. However, a theorem due to Bell seems to require that any such hidden-variable theory which reproduces all of quantum mechanics exactly (i.e., not merely in some limiting case) must possess a rather pathological character with respect to correlated, but spacially separated, systems.
Examines a suggestion by Bell (1984) that a deterministic model might be furnished by the continuum limit of an objective stochastic model which he has proposed for quantum field theory on a lattice. The virtues of models of Bell's type for general quantum systems are discussed, and it is shown that for a system of a fixed number of particles such a model does yield the deterministic de Broglie-Bohm model in an appropriate limit. It is argued, however, that a deterministic limit is unlikely to emerge from a quantum field theory capable of describing particle production and decay. An interpretation of quantum field theory is proposed which has the advantages of Bell's interpretation but is closer to conventional quantum mechanics.