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Consistent Histories

Consistent Histories is an interpretation of quantum mechanics that started in 1984 with a statistical physicist Robert Griffiths, who discovered a set of criteria designed to assign conditional probabilities based on classical rules of probability. The interpretation was picked up by Roland Omnes in 1988 who gave the theory a formal logical systemisation and later by James Hartle and Murray Gell-Mann, who emphasise the role of decoherence in replacing collapse of the wave function in 1990.

The interpretation starts with with subspaces of the Hilbert space and applies to closed systems by treating them as a sequence of events or "histories". An event specificies the properties of a system as a projection operator on the Hilbert space, where:

A "homogenous history" is taken as a series of such operators. The evolution of these histories from event-to-event is stochastic and evolves according to the Schrodinger equation in Griffiths' original formulation. Only histories which satisfy a set of consistency conditions are assigned probabilities and given an interpretation in the theory. These histories do not represent real features of reality but are a useful frame work for discussing time sequences of possibilities, they behave as classical histories only to the extent that they are non-interfering. The weight of a history is given by

E represents the event or its operator. If two histories have weights which are orthogonal such that interference is negligible 

Then their weights can be added together. This may be extended to include so called "families" which is a space to include all non-interfering histories. For such families we can consider a generalised coordinate description of any classical field that is considered fundamental to physics, like the gravitational field, if the projection operators up to n include all the possible field variables, Hartle and Gell-Mann call this set exhaustive. 

We can then go on to define "alternative histories" as an exhaustive set of histories containing an alternative range. The extreme case is to give a complete set of operators at all times, a completely fine-grained history. However it is essential to the Histories approach that fine grained histories cannot be assigned probabilities.

A consistent account of probability will involve a suitable set of coarse grainned histories. The Hartle Gell-Mann approach further specifies specific conditions which apply to the early universe for projection operators. Not only must they commute but their product must be equal to zero. We can later abandon this restrictive condition for later times for an alternative set of histories. It is not necessary to apply Consitent Histories to the entire universe but if one does then the wave function is treated as pre-probability but plays no role in the ontology of the interpretation.

When one integrates the projection operators 

You obtain a decoherence function 

Once decoherence is introduced the two projection operators will diverge and represent possible histories of the system. Griffiths and Omnes began considering small levels of decoherence whereas Hartle and Gell-Mann are interested in strong decoherence. In either case decoherence replaces measurement of the system and the interpretation retains locality. The decoherence function depends on both the density matrix and Hamiltonian of the early universe, Hartle and Gell-Mann follow the time evolution of the operator back to the early universe, which is intended to solve the preferred basis problem.

What history actually occurs? According the formalism, histories are treated similar to the wave function when calculating the path of a particle, a central tendency is located along classical trajectories when the projection operator is coarse grained and the wave function very slowly spreads out. Only histories which are sufficiently coarse grained and therefore close to the classical path will have a high probability of occurring.

In order for a quasi-classical realm to correspond to the set of histories however, one must avoid a completely fine grained description but mustn't have a description which is grained too coarse such that no classically unambiguous description is any longer available.


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