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A Brief Introduction to Consistent Histories

Consistent histories theory was born as an attempt to describe
closed systems in quantum mechanics, partly in light of a desire
to construct quantum theories of cosmology. In fact, the
Copenhagen interpretation of quantum mechanics cannot be applied
to closed systems since it rests on the notion of probabilities
defined in terms of a sequence of repeated measurements by an
external observer. Thus it enforces a, cosmologically
inappropriate, division between system and observer. The
consistent-history formulation avoids this division since it
assigns probabilities without making use of the measurements and
the associated state vector reductions.
In the standard Copenhagen interpretation of quantum theory, probability assignments to sequences of measurements are computed using the von Neumann reduction postulate which, roughly speaking, determines a measurement-induced change in the density matrix that represents the state. Therefore, to give meaning to probabilities, the notion of measurement-induced, state vector reduction is essential.

The consistent history formalism was developed in order to make sense of probability assignments but without invoking the notion of measurement. This requires introducing the decoherence functional, , which is a map from the space of all histories to the complex numbers. Specifically, given two histories (sequences of projection operators) and the decoherence functional is defined as

(1) |

where is the initial density matrix, is the Hamiltonian, and represents the `class operator' which is defined in terms of the Schrodinger-picture projection operator as

(2) |

Thus represents the history proposition `` is true at time , and then is true at time , , and then is true at time ''. It is worth noting that the class operator can be written as the product of Heisenberg-picture projection operators in the form . Generally speaking this is not itself a projection operator.

The physical meaning associated to the quantity
is that it is the probability of the history
being realized. However, this interpretation can only be ascribed in a non-contradictory way if the history
belongs to a special set of histories, namely a *consistent set* which, is a set
of histories which do not interfere with each other, i.e.
for all
. Only within a consistent set does the definition of
consistent histories have any physical meaning. In fact, it is
only within a given consistent set that the probability
assignments are consistent.
Each decoherence functional defines such a consistent set(s).

For an in-depth analysis of the axioms and definition of consistent-history theory the reader is referred to [12], [22], [27] and references therein. For the present paper only the following definitions are needed.

- A
*homogeneous history*is any sequentially-ordered sequence of projection operators - The definition of the join
is straight forward when the two histories have the same time support and differ in their values only at one point
. In this case
is a homogeneous history and satisfies the
relation
.
The problem arises when the time supports are different, in particular when the two histories and are disjoint. The join of such histories would take us outside the class of homogeneous histories. Similarly the negation of a homogeneous history would not itself be a homogeneous history.

- An
*inhomogeneous history*arises when two disjoint homogeneous histories are joined using the logical connective ``or''( ) or when taking the negation ( ) of a history proposition. Specifically, given two disjoint homogeneous histories and we can meaningfully talk about the inhomogeneous histories and . Such histories are generally not a just a sequence of projection operators, but when computing the decoherence functional they are represented by the operator and

Gel Mann and Hartle, tried to solve the problem of representing inhomogeneous histories using path integrals on the configuration space, , of the system. In this formalism the histories and are seen as subsets of the paths of Q. Then a pair of histories is said to be disjoint if they are disjoint subsets of the path space Q. Seen as path integrals, the additivity property of the decoherence functional is easily satisfied namely

where is any subset of the path space Q.

Similarly, the negation of a history proposition is represented by the complement of the subset of Q. Therefore

where 1 is the unit history

^{4}.

The above properties in (3) and 4 are well defined in the context of path integrals. But what happens when defining the decoherence functional on a string of projection operators? Gel'Mann and Hartle solved this problem by postulating the following definitions for the class operators when computing decoherence functionals:

if and are disjoint histories. The right hand side of these equations are indeed operators that represent and when computing the decoherence functional, but as objects in the consistent-history formalism, it is not really clear what and are.

In fact, as defined above, a homogeneous history is a time ordered sequence of projection operators, but there is no analogue definition for or . One might try to define the inhomogeneous histories and component-wise so that, for a simple two-time history , we would have

However, this definition of the negation operation is wrong. For is the temporal proposition `` is true at time , and then is true at time '', which we shall write as . It is then intuitively clear that the negation of this proposition should be

which is not in any obvious sense the same as (6).

A similar problem arises with the ``or'' ( ) operation. For, given two homogenous histories and , the ''or'' operation defined component-wise is

(8) |

This history would be true (realized) if both and are true, which implies that either an element in each of the pairs and is true, or both elements in either of the pairs and are true. But this contradicts with the actual meaning of the proposition , which states that either history is realized or history is realized. In fact the `or' in the proposition should really be as follows

Thus for the proposition to be true both elements in either of the pairs and have to be true, but not all four elements at the same time. If instead we had the history proposition from equation (16), , this would be equivalent to

This shows that it is not possible to define inhomogeneous histories component-wise. Moreover, the appeal to path integrals when defining is realization-dependent and does not uncover what actually is.

However, the right hand sides of equations (5) have a striking similarity to the single-time propositions in quantum logic. In fact, given two single-time propositions P and Q which are disjoint, the proposition is simply represented by the projection operator ; similarly, the negation is represented by the operator .

This similarity of the single-time propositions with the right hand side of the equations (5) suggests that somehow it should be possible to identify history propositions with projection operators. Obviously these projection operators cannot be the class operators since, generally, these are not projection operators. The claim that a logic for consistent histories can be defined such that each history proposition is represented by a projection operator on some Hilbert space is also motivated by the fact that the statement that a certain history is ''realized", is itself a proposition. Therefore, the set of all such histories could possess a lattice structure similar to the lattice of single-time propositions in standard quantum logic.

These considerations led Isham to construct the, so-called, HPO formalism. In this new formalism of consistent histories it is possible to identify the entire set with the projection lattice of some `new' Hilbert space. In the following Section we will describe this formalism in more detail.

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