Quantum decoherence: a logical perspective

SEBASTIAN FORTIN

Río de Janeiro

Congreso; UNILOG 2013: 4th World Congress on Universal Logic; 2013

UNILOG

There are different perspectives to address the problem of the classical limit of quantum mechanics. The orthodox treatment introduces the phenomenon of decoherence as the key to solve this problem (Bub 1997). The mainstream approach of decoherence is the so-called “environment induced decoherence”, developed by Zurek and his collaborators (see, e.g., Zurek 1981, 2003, Paz & Zurek 2002). In the context of this approach, the goal is to know whether the state becomes diagonal or not (Schlosshauer 2007). If the state becomes diagonal, then it acquires a structure to a mixed state of classical mechanics; this feature leads to the usual interpretation of the decohered state from a classical viewpoint.In our group, we have developed a general theoretical framework for decoherence based on the study of the evolution of the expectation values of certain relevant observables of system (Castagnino, Fortin, Laura & Lombardi 2008, Castagnino, Fortin & Lombardi 2010). According to this framework, decoherence is a phenomenon relative to the relevant observables selected in each particular case (Lombardi, Fortin & Castagnino, 2012). This new approach and the orthodox treatment of decoherence are equivalent from a mathematical point of view. Nevertheless, there are good reasons to think that the treatment of decoherence by means of the behavior of the observables of the system instead that of its states may have conceptual advantages.The purpose of this work is to argue that the main advantage of the study of decoherence in terms of the Heisenberg representation is that this approach allows us to analyze the logical aspects of the classical limit. On the one hand, we know that the lattices of classical properties are distributive or Boolean (Boole 1854): when operators are associated with those properties, they commute with each other. On the other hand, it is well-known that the lattices of quantum properties are non-distributive, a formal feature manifested in the existence of non-commuting observables (Cohen 1989, Bub 1997). In spite of this difference, there are certain quantum systems which, under certain particular conditions, evolve in a special way: although initially the commutator between two operators is not zero, due to the evolution it tends to become zero (Kiefer & Polarski 2009). Therefore, in these systems should be possible to show that, initially, they can be represented by a non-Boolean lattice, but after a definite time a Boolean lattice emerges: this process, that could be described from the perspective of the expectation values of the system’s observables, deserves to be considered as a sort of decoherence that leads to the classical limit. In other words, from this perspective the classical limit can be addressed by studying the dynamical evolution of Boolean lattices toward Boolean lattices.