Consistent histories
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In quantum mechanics, the consistent histories approach is intended to give a modern interpretation of quantum mechanics, generalising the conventional Copenhagen interpretation and providing a natural interpretation of quantum cosmology. The theory is based on a consistency criterion that then allows the history of a system to be described so that the probabilities for each history obey the rules of classical probability while being consistent with the Schrödinger equation.
According to this interpretation of quantum mechanics, the purpose of a quantum-mechanical theory is to predict probabilities of various alternative histories. A history is defined as a sequence (product) of projection operators at different moments of time:
The symbol T indicates that the factors in the product are ordered chronologically according to their values of ti,j: the "past" operators with smaller values of t appear on the right side, and the "future" operators with greater values of t appear on the left side.
These projection operators can correspond to any set of questions that include all possibilities. Examples might be the three projections meaning "the electron went through the left slit", "the electron went through the right slit" and "the electron didn't go through either slit". One of the aims of the theory is to show that classical questions such as, "where is my car?" are consistent. In this case one might use a very large set of projections each one specifying the location of the car in some small region of space.
A history is a sequence of such questions, or—mathematically—the product of the corresponding projection operators. The role of quantum mechanics is to predict the probabilities of individual histories, given the known initial conditions.
Finally, the histories are required to be consistent, i.e.
for i,j different. Here ρ represents the initial density matrix, and the operators are expressed in the Heisenberg picture. The consistency requirement allows us to postulate that the probability of the history Hi is simply
which guarantees that the probability of "A or B" equals the probability of "A" plus the probability of "B" minus the probability of "A and B", and so forth. The interpretation based on consistent histories is used in combination with the insights about quantum decoherence. Quantum decoherence implies that only special choices of histories are consistent, and it allows a quantitative calculation of the boundary between the classical domain and the quantum domain.
In some views the interpretation based on consistent histories does not change anything about the paradigm of the Copenhagen interpretation that only the probabilities calculated from quantum mechanics and the wave function have a physical meaning. In order to obtain a complete theory, the formal rules above must be supplemented with a particular Hilbert space and rules that govern dynamics, for example a Hamiltonian.
In the opinion of others this still does not make a complete theory as no predictions are possible about which set of consistent histories will actually occur. That is the rules of CH, the Hilbert space, and the Hamiltonian must be supplemented by a set selection rule.
The proponents of this modern interpretation, such as Murray Gell-Mann, James Hartle, Roland Omnès, Robert B. Griffiths, and Wojciech Zurek argue that their interpretation clarifies the fundamental disadvantages of the old Copenhagen interpretation, and can be used as a complete interpretational framework for quantum mechanics.
In Quantum Philosophy, Roland Omnès provides a less mathematical way of understanding this same formalism. The consistent histories approach can be interpreted as a way of understanding which sets of classical questions can be consistently asked of a single quantum system, and which sets of questions are fundamentally inconsistent, and thus meaningless when asked together. It thus becomes possible to demonstrate formally why it is that the questions which Einstein, Podolsky and Rosen assumed could be asked together, of a single quantum system, simply cannot be asked together. On the other hand, it also becomes possible to demonstrate that classical, logical reasoning often does apply, even to quantum experiments – but we can now be mathematically exact about the limits of classical logic.
[edit] References
- R. Omnès, Understanding Quantum Mechanics, Princeton University Press, 1999. Chapter 13 describes consistent histories.
- R. Omnès, Quantum Philosophy, Princeton University Press, 1999. See part III, especially Chapter IX.
