Griffiths ’ “quantum trajectories ” formalism is extended to describe weak decoherence. The decoherence conditions are shown to severely limit the complexity of histories composed of fine-grained events. In response to the increasingly popular opinion that the Copenhagen interpretation of quantum mechanics raises more questions than it answers [7] and a desire to treat the entire universe quantum mechanically, Gell-Mann and Hartle [1, 2, 3] have worked to create an alternative interpretation of quantum theory, expanding upon earlier work by Griffiths [4] and Omnès [6]. Their scheme emphasizes not individual events but Griffiths ’ notion of a history, a sequence of events at a succession of times, and they assert that the
Griffiths ’ “quantum trajectories ” formalism is extended to describe weak decoherence. The decoherence conditions are shown to severely limit the complexity of histories composed of fine-grained events. In response to the increasingly popular opinion that the Copenhagen interpretation of quantum mechanics raises more questions than it answers [7] and a desire to treat the entire universe quantum mechanically, Gell-Mann and Hartle [1, 2, 3] have worked to create an alternative interpretation of quantum theory, expanding upon earlier work by Griffiths [4] and Omnès [6]. Their scheme emphasizes not individual events but Griffiths ’ notion of a history, a sequence of events at a succession of times, and they assert that the
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.
We introduce and review the consistent (decoherent) histories approach to quantum mechanics due to Griffiths, to Gell-Mann and Hartle, and to Omnes. We will not attempt an in depth look at this approach as it would be impossible to treat it in such a short review. The emphasis is on understanding the broader meaning of the consistency and decoherence. The consistent history approach to quantum mechanics provides a precise conceptual framework for describing how a closed quantum system develops in time. The approach also provides a framework from which we may observe the emergence of an approximately classical domain for macroscopic systems, together with the conventional Copenhagen quantum mechanics for microscopic subsystems. We will study the formalism of decoherence and look at several approaches to decoherence. In examining a system of quantum Brownian motion and using the methods of Feynman and Vernon, we derive an equation of motion for the density matrix, the master equation of the system. This leads us to a brief overview of decoherence. PACS Nos.: 03.65.Ta, 03.65.w, 03.65.Yz, 04.60.m, 05.30.d
I give an informal overview of the decoherent histories approach to quantum mechanics, due to Griffiths, to Omnès, and to Gell-Mann and Hartle is given. Results on the connections between decoherence, records, correlation and entropy are described. The emphasis of the presentation is on understanding the broader meaning of the conditions of consistency and decoherence, and in particular, the extent to which they permit one to assign definite properties to the system. The quantum Brownian motion model is briefly discussed. (To appear in proceedings of the workshop, "Stochastic Evolution of Quantum States in Open Systems and Measurement Processes", Budapest, March, 1993, edited by L.Diosi).
I briefly review the "decohering histories" or "consistent histories" formulation of quantum theory, due to Griffiths, Omnès, and Gell-Mann and Hartle (and the subject of my graduate work with George Sudarshan). I also sift through the many meanings that have been attached to decohering histories, with an emphasis on the most basic one: Decoherence of appropriate histories is needed to establish that quantum mechanics has the correct classical limit. Then I will describe efforts to find physical mechanisms that do this. Since most work has focused on density matrix versions of decoherence, I'll consider the relation between the two formulations, which historically has not been straightforward. Finally, I'll suggest a line of research that would use recent results by Sudarshan to illuminate this aspect of the classical limit of quantum theory.
The probabilistic element of quantum theory can be combined with the unitary time evolution of Schr\"odinger's equation in a natural and consistent way using the idea of a quantum trajectory, the quantum analog of the trajectory traced out in phase space as a function of time by a point representing the state of a closed classical system. A family of quantum trajectories can be defined using bases for the quantum Hilbert space at different times chosen so that an appropriate noninterference condition, related to the Gell-Mann and Hartle notion of medium decoherence, is satisfied. The result is a generalization of the consistent histories approach to quantum mechanics.
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.