consistency conditions for probability of quantum histories

Giuseppe Nisticò

Thomas F. Jordan, Eric D. Chisolm

J. J. Halliwell

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.

Marcelo Losada, Roberto Laura, Olímpia Lombardi

In this article, we reconstruct the Frauchiger and Renner argument, taking into account that the assertions of the argument are made at different times. To do this, we use a formalism of quantum histories, namely the Theory of Consistent Histories. We show that the supposedly contradictory conclusion of the argument requires computing probabilities in a family of histories that does not satisfy the consistency condition, i.e., an invalid family of histories for the theory.

Thomas Marlow

Norifumi Yamada

An effort is continued to define quantum probabilities for continuous histories. The sum-over-paths approach used in Prog. Theor. Phys. 87, 77 (1992) is critically reviewed, improved, and applied again. Consistency between wave nature and particle nature is the criterion used to judge the definability of the probabilities and is formulated as the path classifiability condition (C1) and the no-interference condition (C2). A set of classes of histories satisfying these two conditions is considered as a space-time analog of observables. In particular, Feynman's paths for a nonrelativistic particle are considered as histories and the definability of probabilities for classes of paths is investigated, where classes are defined by classifying paths according to their behavior with respect to a rectangular space-time region \ensuremath{\Omega}\ensuremath{\equiv}\ensuremath{\Delta}X\ifmmode\times\else\texttimes\fi{}\ensuremath{\Delta}T. Confining ourselves to cases where C1 is satisfied, we examine C2 for some sets of classes of paths. Although C2 does not hold in general, some examples are found where C2 holds. In all the examples, the initial wave function is restricted. In some examples, the location of \ensuremath{\Omega} and/or the potential in the region also affect the success of C2. Due to these restrictions, the resultant probabilities for classes of paths take reasonable values. A puzzling example in which probabilities cannot be defined for histories contrary to intuition is resolved by considering an appropriate coarse graining of classes of paths. Considering a rectangular potential barrier, we show that the reflection and the transmission probabilities are special cases of probabilities for histories, and also that probability densities of transmission and reflection times cannot be defined. This study may be taken to be a study of the consistent-histories approach not with discrete histories defined by products of projection operators but with continuous histories defined by Feynman's paths in configuration space. \textcopyright{} 1996 American Institute of Physics.