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. Fig. ... It has been suggested that Quantum mechanics, philosophy and controversy be merged into this article or section. ... The Copenhagen interpretation is an interpretation of quantum mechanics formulated by Niels Bohr and Werner Heisenberg while collaborating in Copenhagen around 1927. ... In theoretical physics, quantum cosmology is a young field attempting to study the effect of quantum mechanics on the earliest moments of the universe after the Big Bang. ... For a non-technical introduction to the topic, please see Introduction to quantum mechanics. ...

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: Fig. ... Template:Unite See also projection (linear algebra). ...

The symbol T indicates that the factors in the product are ordered chronologically according to their values of t_i,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 are my keys?" are consistent. In this case one might use a very large set of projections each one specifying the location of the keys 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 H_i is simply A density matrix is a self-adjoint (or Hermitian) positive-semidefinite matrix, (possibly infinite dimensional), of trace one, that describes the statistical state of a quantum system. ... The Heisenberg Picture of quantum mechanics is also known as Matrix mechanics. ...

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 quantum mechanics, quantum decoherence is the mechanism by which quantum systems interact with their environments to exhibit probabilistically additive behavior - a feature of classical physics - and give the appearance of wavefunction collapse. ...

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. A wave function is a mathematical tool that quantum mechanics uses to describe any physical system. ... The mathematical concept of a Hilbert space (named after the German mathematician David Hilbert) generalizes the notion of Euclidean space in a way that extends methods of vector algebra from the plane and three-dimensional space to spaces of functions. ... The Hamiltonian, denoted H, has two distinct but closely related meanings. ...

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 mathematical concept of a Hilbert space (named after the German mathematician David Hilbert) generalizes the notion of Euclidean space in a way that extends methods of vector algebra from the plane and three-dimensional space to spaces of functions. ...

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. Murray Gell-Mann (born September 15, 1929 in Manhattan, New York City, USA) is an American physicist who received the 1969 Nobel Prize in physics for his work on the theory of elementary particles. ... James Hartle is an American physicist. ... Roland Omnès is a French physicist and a well-known proponent of the modern Copenhagen interpretation of quantum mechanics based on consistent histories. ... Robert Griffiths is an American Physicist at Carnegie Mellon University. ... Wojciech Hubert Zurek is a well-known physicist, as a Laboratory Fellow at Los Alamos National Laboratory. ...

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. Quantum Philosophy is a book by the physicist Roland Omnès, in which he aims to show the non-specialist reader how modern developments in quantum mechanics allow the recovery of our common sense view of the world. ... In quantum mechanics, the EPR paradox is a thought experiment which challenged long-held ideas about the relation between the observed values of physical quantities and the values that can be accounted for by a physical theory. ...

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.
• R. B. Griffiths, Consistent Quantum Theory, Cambridge University Press, 2003.