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Consistent histories

From Academic Kids

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. Some believe that this derives from the work by Hugh Everett and is a modern version of the Many-worlds interpretation. Others strongly disagree. 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:

<math>H_i = T \prod_{j=1}^{n_i} P_{i,j}(t_{i,j})<math>

The symbol <math>T<math> indicates that the factors in the product are ordered chronologically according to their values of <math>t_{i,j}<math>: the "past" operators with smaller values of <math>t<math> appear on the right side, and the "future" operators with greater values of <math>t<math> 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 for 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.

<math>\operatorname{Tr}(H_i \rho H^\dagger_j) = 0<math>

for <math>i,j<math> different. Here <math>\rho<math> 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 <math>H_i<math> is simply

<math>\operatorname{Pr}(H_i) = \operatorname{Tr}(H_i \rho H_i^\dagger)<math>

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 Omnes, 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.

References

  • R. Omnès, Understanding Quantum Mechanics, Princeton University Press, 1999. Chapter 13 describes consistent histories.
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