The theory of decoherent histories is checked for the requirement of statistical independence of subsystems. Strikingly, this is satisfied only when the decoherence functional is diagonal in both its real and imaginary parts. Although the weakened condition of consistency (or weak decoherence), allowing a nondiagonal imaginary part, is sufficient for the assignment of probabilities, it may easily violate the statistical independence of subsystems. Therefore, weakened consistency conditions and various related generalizations of the concept of decoherent histories appear to be ruled out. The same conclusion is obtained independently, by claiming a plausible dynamical robustness of decoherent histories.
The theory of decoherent histories is checked for the requirement of statistical independence of subsystems. Strikingly, this is satisfied only when the decoherence functional is diagonal in both its real and imaginary parts. Although the weakened condition of consistency (or weak decoherence), allowing a nondiagonal imaginary part, is sufficient for the assignment of probabilities, it may easily violate the statistical independence of subsystems. Therefore, weakened consistency conditions and various related generalizations of the concept of decoherent histories appear to be ruled out. The same conclusion is obtained independently, by claiming a plausible dynamical robustness of decoherent histories.
In the decoherent histories approach to quantum theory, attention focuses on the conditions under which probabilities may be assigned to sets of quantum histories. A variety of conditions have been proposed, but the most important one is decoherence, which means that the interference between every pair of histories in the set is zero. Weaker conditions have been considered, such as consistency, or linear positivity, but these are ruled out by the requirement of consistent composition of subsystems, proposed by Diosi. Here we propose a new condition which we call partial decoherence, and is the requirement that every history has zero interference with its negation. This is weaker than decoherence and stronger than linear positivity (but its relation to consistency is less simply defined -- it is neither stronger nor weaker). Most importantly, it satisfies the Diosi condition. A strengthened Diosi condition is proposed, which partial decoherence narrowly fails, due to an unusual property of inhomogeneous histories. In an appendix an example is given of a set of histories which are consistent but not decoherent.
We study a formulation of quantum mechanics in which the central notion is that of a quantum-mechanical history---a sequence of events at a succession of times. The primary aim is to identify sets of ``decoherent'' (or ``consistent'') histories for the system. These are quantum-mechanical histories suffering negligible interference with each other, and, therefore, to which probabilities may be assigned. These histories may be found for a given system using the so-called decoherence functional. When the decoherence functional is exactly diagonal, probabilities may be assigned to the histories, and all probability sum rules are satisfied exactly. We propose a condition for approximate decoherence, and argue that it implies that most probability sum rules will be satisfied to approximately the same degree. We also derive an inequality bounding the size of the off-diagonal terms of the decoherence functional. We calculate the decoherence functional for some simple one-dimensional systems, with a variety of initial states. For these systems, we explore the extent to which decoherence is produced using two different types of coarse graining. The first type of coarse graining involves imprecise specification of the particle's position. The second involves coupling the particle to a thermal bath of harmonic oscillators and ignoring the details of the bath (the Caldeira-Leggett model). We argue that both types of coarse graining are necessary in general. We explicitly exhibit the degree of decoherence as a function of the temperature of the bath, and of the width to within which the particle's position is specified. We study the diagonal elements of the decoherence functional, representing the probabilities for the possible histories of the system. To the extent that the histories decohere, we show that the probability distributions are peaked about the classical histories of the system, with the distribution of their initial positions and momenta given by a smeared version of the Wigner function. We discuss this result in connection with earlier uses of the Wigner function in this context. We find that there is a certain amount of tension between the demands of decoherence and peaking about classical paths.
The decoherent (consistent) histories formalism has been proposed as a means of eliminating measurements as a fundamental concept in quantum mechanics. In this formalism, probabilities can be assigned to any description which satisfies a particular consistency condition. The formalism, however, admits incompatible descriptions which cannot be combined, unlike classical physics. This seems to leave an ambiguity in the choice of the description. I argue that this ambiguity is removed by considering the observer as a physical system.
I review the decoherent (or consistent) histories approach to quantum mechanics, due to Griffiths, to Gell-Mann and Hartle, and to Omnes. This is an approach to standard quantum theory specifically designed to apply to genuinely closed systems, up to and including the entire universe. It does not depend on an assumed separation of classical and quantum domains, on notions of measurement, or on collapse of the wave function. Its primary aim is to find sets of histories for closed systems exhibiting negligble interference, and therefore, to which probabilities may be assigned. Such sets of histories are called consistent or decoherent, and may be manipulated according to the rules of ordinary (Boolean) logic. The approach provides a framework from which one may predict the emergence of an approximately classical domain for macroscopic systems, together with the conventional Copenhagen quantum mechanics for microscropic subsystems. In the special case in which the total closed system naturally separates into a distinguished subsystem coupled to an environment, the decoherent histories approach is closed related to the quantum state diffusion approach of Gisin and Percival.
The geometry of decoherence in generalized "consistent histories" quantum theory is explored, revealing properties of the theory that are independent of any particular application of it. It is shown how the decoherence functional of a closed quantum system may be regarded as an Hermitian form on the space of linear operators on the Hilbert space of the system. This identification makes manifest a number of structural properties of decoherence functionals. For example, a bound on the maximum number of histories in a consistent set is determined. When the decoherence functional is positive -- as in conventional quantum mechanics -- it defines a semi-inner product on the space of history operators. This shows that consistent sets of histories are just orthogonal sets in this inner product. It further implies the existence in general of Cauchy-Schwarz and triangle inequalities for positive decoherence functionals. The geometrical significance of the ILS theorem classifying all possible decoherence functionals is illuminated, and a version of the ILS theorem for decoherence functionals on class operators is given. The class of history operators consistent according to a given decoherence functional is found, and, conversely, it is shown how to construct the decoherence functionals according to which a given set of histories is consistent. More generally, the "geometric" point of view here developed supplies a powerful unified language with which to solve problems in generalized quantum theory.
We investigate the quantum theory of closed systems based on the linear positivity decoherence condition of Goldstein and Page. The objective of any quantum theory of a closed system, most generally the universe, is the prediction of probabilities for the individual members of sets of alternative coarse-grained histories of the system. Quantum interference between members of a set of alternative histories is an obstacle to assigning probabilities that are consistent with the rules of probability theory. A quantum theory of closed systems therefore requires two elements: (1) a condition specifying which sets of histories may be assigned probabilities and (2) a rule for those probabilities. The linear positivity condition of Goldstein and Page is the weakest of the general conditions proposed so far. Its general properties relating to exact probability sum rules, time neutrality, and conservation laws are explored. Its inconsistency with the usual notion of independent subsystems in quantum mechanics is reviewed. Its relation to the stronger condition of medium decoherence necessary for classicality is discussed. The linear positivity of histories in a number of simple model systems is investigated with the aim of exhibiting linearly positive sets of histories that are not decoherent. The utility of extending the notion of probability to include values outside the range of 0--1 is described. Alternatives with such virtual probabilities cannot be measured or recorded, but can be used in the intermediate steps of calculations of real probabilities. Extended probabilities give a simple and general way of formulating quantum theory. The various decoherence conditions are compared in terms of their utility for characterizing classicality and the role they might play in further generalizations of quantum mechanics.