Robert B. Griffiths

Consistent histories and the interpretation of quantum mechanics

The usual formula for transition probabilities in nonrelativistic quantum mechanics is generalized to yield conditional probabilities for selected sequences of events at several different times, called “consistent histories,” through a criterion which ensures that, within limits which are explicitly defined within the formalism, classical rules for probabilities are satisfied. The interpretive scheme which results is applicable to closed (isolated) quantum systems, is explicitly independent of the sense of time (i.e., past and future can be interchanged), has no need for wave function “collapse,” makes no reference to processes of measurement (though it can be used to analyze such processes), and can be applied to sequences of microscopic or macroscopic events, or both, as long as the mathematical condition of consistency is satisfied. When applied to appropriate macroscopic events it appears to yield the same answers as other interpretative schemes for standard quantum mechanics, though from a different point of view which avoids the conceptual difficulties which are sometimes thought to require reference to conscious observers or classical apparatus.

Lattice-gas model of multiple layer adsorption☆

Abstract The phase diagram for a lattice-gas model of physical adsorption on a homogeneous substrate has been calculated in a mean-field approximation. The first-order phase transitions corresponding to the addition of successive layers terminate in individual critical points as the temperature increases. The critical temperatures of successive layers increase, monotonically with layer number, from the critical temperature of the two-dimensional lattice gas to that of the bulk three-dimensional lattice gas. This last feature is probably an artifact of the mean-field approximation.

Rigorous Results for Ising Ferromagnets of Arbitrary Spin

The following results for spin‐½ Ising ferromagnets are extended to the case of arbitrary spin: (1) the theorem of Lee and Yang, that the zeros of the partition function lie on the unit circle in the complex fugacity plane; (2) inequalities of the form ≥ , where A and B are products of spin operators; (3) the existence of spontaneous magnetization on suitable lattices. Results (2) and (3) are also extended to the infinite‐spin limit in which the spin variable is continuous on the interval −1 ≤ x ≤ 1.

Mathematical properties of position-space renormalization-group transformations

Properties of “position-space” or “cell-type” renormalization-group transformations from an Ising model object system onto an Ising model image system, of the type introduced by Niemeijer, van Leeuwen, and Kadanoff, are studied in the thermodynamic limit of an infinite lattice. In the case of a KadanofF transformation with finitep, we prove that if the magnetic field in the object system is sufficiently large (i.e., the lattice-gas activity is sufficiently small), the transformation leads to a well-defined set of image interactions with finite norm, in the thermodynamic limit, and these interactions are analytic functions of the object interactions. Under the same conditions the image interactions decay exponentially rapidly with the geometrical size of the clusters with which they are associated if the object interactions are suitably short-ranged. We also present compelling evidence (not, however, a completely rigorous proof) that under other conditions both the finite- and infinite-p (“majority rule”) transformations exhibit peculiarities, suggesting either that the image interactions are undefined (i.e., the transformation does not possess a thermodynamic limit) or that they fail to be smooth functions of the object interactions. These peculiarities are associated (in terms of their mathematical origin) with phase transitions in the object system governed not by the object interactions themselves, but by a modified set of interactions.

Consistent histories and quantum reasoning

A system of quantum reasoning for a closed system is developed by treating nonrelativistic quantum mechanics as a stochastic theory. The sample space corresponds to a decomposition, as a sum of orthogonal projectors, of the identity operator on a Hilbert space of histories. Provided a consistency condition is satisfied, the corresponding Boolean algebra of histories, called a framework, can be assigned probabilities in the usual way, and within a single framework quantum reasoning is identical to ordinary probabilistic reasoning. A refinement rule, which allows a probability distribution to be extended from one framework to a larger (refined) framework, incorporates the dynamical laws of quantum theory. Two or more frameworks which are incompatible because they possess no common refinement cannot be simultaneously employed to describe a single physical system. Logical reasoning is a special case of probabilistic reasoning in which (conditional) probabilities are 1 (true) or 0 (false). As probabilities are only meaningful relative to some framework, the same is true of the truth or falsity of a quantum description. The formalism is illustrated using simple examples, and the physical considerations which determine the choice of a framework are discussed. \textcopyright{} 1996 The American Physical Society.

Correlations in Ising ferromagnets. III. A mean-field bound for binary correlations

An inequality relating binary correlation functions for an Ising model with purely ferromagnetic interactions is derived by elementary arguments and used to show that such a ferromagnet cannot exhibit a spontaneous magnetization at temperatures above the mean-field approximation to the Curie or “critical” point. (As a consequence, the corresponding “lattice gas” cannot undergo a first order phase transition in density (condensation) above this temperature.) The mean-field susceptibility in zero magnetic field at high temperatures is shown to be an upper bound for the exact result.

EPR, Bell, and quantum locality

Maudlin has claimed that no local theory can reproduce the predictions of standard quantum mechanics that violate Bell’s inequality for Bohm’s version (two spin-half particles in a singlet state) of the Einstein-Podolsky-Rosen problem. It is argued that, on the contrary, standard quantum mechanics itself is a counterexample to Maudlin’s claim, because it is local in the appropriate sense (measurements at one place do not influence what occurs elsewhere there) when formulated using consistent principles in place of the inconsistent appeals to “measurement” found in current textbooks. This argument sheds light on the claim of Blaylock that counterfactual definiteness is an essential ingredient in derivations of Bell’s inequality.

Correlations in separated quantum systems: A consistent history analysis of the EPR problem

The familiar problem of two separated, noninteracting spin‐ 1/2 particles in a state of zero total spin is analyzed using the consistent history interpretation of quantum mechanics and shown to behave in many respects like a classical system of two noninteracting objects whose individual properties are unknown but strongly correlated with each other. There is no action at a distance between the particles and a measurement on one has no effect whatsoever on the other. However, the result of a measurement of a spin component of one of the particles can be used to infer (correctly) its value prior to the measurement, and also the corresponding spin component of the other particle at all times prior to when that particle interacts with something else. In these respects the quantum system behaves like its classical counterpart. On the other hand, the paradoxical (nonclassical) aspects of the quantum situation seem to be precisely those already present in the quantum theory of a single particle.

Quantum-error-correcting codes using qudit graph states

Graph states are generalized from qubits to collections of

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