In consistent-history quantum theory, a description of the time development of a quantum system requires choosing a framework or consistent family, and then calculating probabilities for the different histories that it contains. It is argued that the framework is chosen by the physicist constructing a description of a quantum system on the basis of questions he wishes to address, in a manner analogous to choosing a coarse-graining of the phase space in classical statistical mechanics. The choice of framework is not determined by some law of nature, though it is limited by quantum incompatibility, a concept that is discussed using a two-dimensional Hilbert space (spin half particle). Thus certain questions of physical interest can only be addressed using frameworks in which they make (quantum-mechanical) sense. The physicist's choice does not influence reality, nor does the presence of choices render the theory subjective. On the contrary, predictions of the theory can, in principle, be verified by experimental measurements. These considerations are used to address various criticisms and possible misunderstandings of the consistent-history approach, including its predictive power, whether it requires a new logic, whether it can be interpreted realistically, the nature of ``quasiclassicality,'' and the possibility of ``contrary'' inferences.
In consistent-history quantum theory, a description of the time development of a quantum system requires choosing a framework or consistent family, and then calculating probabilities for the different histories that it contains. It is argued that the framework is chosen by the physicist constructing a description of a quantum system on the basis of questions he wishes to address, in a manner analogous to choosing a coarse-graining of the phase space in classical statistical mechanics. The choice of framework is not determined by some law of nature, though it is limited by quantum incompatibility, a concept that is discussed using a two-dimensional Hilbert space (spin half particle). Thus certain questions of physical interest can only be addressed using frameworks in which they make (quantum-mechanical) sense. The physicist's choice does not influence reality, nor does the presence of choices render the theory subjective. On the contrary, predictions of the theory can, in principle, be verified by experimental measurements. These considerations are used to address various criticisms and possible misunderstandings of the consistent-history approach, including its predictive power, whether it requires a new logic, whether it can be interpreted realistically, the nature of quasiclassicality,'' and the possibility of contrary'' inferences.
Precise rules are developed in order to formalize the reasoning processes involved in standard non-relativistic quantum mechanics, with the help of analogies from classical physics. A classical or quantum description of a mechanical system involves a {\it framework}, often chosen implicitly, and a {\it statement} or assertion about the system which is either true or false within the framework with which it is associated. Quantum descriptions are no less ``objective'' than their classical counterparts, but differ from the latter in the following respects: (i) The framework employs a Hilbert space rather than a classical phase space. (ii) The rules for constructing meaningful statements require that the associated projectors commute with each other and, in the case of time-dependent quantum histories, that consistency conditions be satisfied. (iii) There are incompatible frameworks which cannot be combined, either in constructing descriptions or in making logical inferences about them, even though any one of these frameworks may be used separately for describing a particular physical system. A new type of ``generalized history'' is introduced which extends previous proposals by Omnès, and Gell-Mann and Hartle, and a corresponding consistency condition which does not involve density matrices or single out a direction of time. Applications which illustrate the formalism include: measurements of spin, two-slit diffraction, and the emergence of the classical world from a fully quantum description.
There are good motivations for considering some type of quantum histories formalism. Several possible formalisms are known, defined by different definitions of event and by different selection criteria for sets of histories. These formalisms have a natural interpretation, according to which nature somehow chooses one set of histories from among those allowed, and then randomly chooses to realise one history from that set; other interpretations are possible, but their scientific implications are essentially the same. The selection criteria proposed to date are reasonably natural, and certainly raise new questions. For example, the validity of ordering inferences which we normally take for granted – such as that a particle in one region is necessarily in a larger region containing it – depends on whether or not our history respects the criterion of ordered consistency, or merely consistency. However, the known selection criteria, including consistency and medium decoherence, are very weak. It is not possible to derive the predictions of classical mechanics or Copenhagen quantum mechanics from the theories they define, even given observational data in an extended time interval. Attempts to refine the consistent histories approach so as to solve this problem by finding a definition of quasiclassicality have so far not succeeded. On the other hand, it is shown that dynamical collapse models, of the type originally proposed by Ghirardi-Rimini-Weber, can be re-interpreted as set selection criteria within a quantum histories framework, in which context they appear as candidate solutions to the set selection problem. This suggests a new route to relativistic generalisation of these models, since covariant definitions of a quantum event are known.
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.
In a recent paper Kent has pointed out that in consistent histories quantum theory it is possible, given initial and final states, to construct two different consistent families of histories, in each of which there is a proposition that can be inferred with probability one, and such that the projectors representing these two propositions are mutually orthogonal. In this note we stress that, according to the rules of consistent history reasoning two such propositions are not contrary in the usual logical sense namely, that one can infer that if one is true then the other is false, and both could be false. No single consistent family contains both propositions, together with the initial and final states, and hence the propositions cannot be logically compared. Consistent histories quantum theory is logically consistent, consistent with experiment as far as is known, consistent with the usual quantum predictions for measurements, and applicable to the most general physical systems. It may not be the only theory with these properties, but in our opinion, it is the most promising among present possibilities.
Abstract: The Consistent Histories (CH) formalism aims at a quantum mechani-cal framework which could be applied even to the universe as a whole. CH stresses the importance of histories for quantum mechanics, as opposed to measurements, and maintains that a satisfactory formulation of quantum mechanics allows one to assign probabilities to alternative histories of a quantum system. It further proposes that each realm, that is, each set of histories to which probabilities can be assigned, provides a valid quantum-mechanical account, but that different realms can be mutually incom-patible. Finally, some of its proponents offer an “evolutionary ” explanation of our existence in the universe and of our preference for quasiclassical descriptions of nature. The present work questions the validity of claims offered by CH proponents asserting that it solves many interpretational problems in quantum mechanics. In particular, we point out that the interpretation of the framework leaves vague two crucial points, namely, whether realms are fixed or chosen and the link between measurements and histories. Our claim is that by doing so, CH overlooks the main interpretational prob-lems of quantum mechanics. Furthermore, we challenge the evolutionary explanation offered and we critically examine the proposed notion of a realm-dependent reality. 1
The Consistent Histories (CH) formalism attempts to construct a quantum framework which can be used without the need to introduce observers external to the studied system. The prime motivation in mind is the application of the formalism to the universe as a whole. In order to achieve this, CH maintains that a formulation of quantum mechanics should allow for the assignment of probabilities to alternative histories of a system. Therefore, it provides an observer-independent criterion to decide which sets of histories can be given probabilities and states rules to determine them. The framework establishes that each realm, that is, each set of histories to which probabilities can be assigned, provides a valid quantum-mechanical account of a system. Furthermore, the version of CH first presented in [1, 2] proposes an "evolutionary" explanation of our existence in the universe and of our preference for quasiclassical descriptions of nature. The present work critically evaluates claims to the effect that the formalism offered in [1, 2] solves many interpretational problems in quantum mechanics. In particular, it is pointed out that the interpretation of the proposed framework leaves vague two crucial points, namely, whether realms are fixed or chosen and the link between measurements and histories. The claim of this work is that by doing so, CH overlooks the main interpretational problems of quantum mechanics. Furthermore, we challenge the evolutionary explanation offered and we critically examine the proposed notion of a realm-dependent reality.