Editing Consistent histories (section)

==Formalism==
===Histories===

A ''homogeneous history'' <math>H_i</math> (here <math>i</math> labels different histories) is a sequence of [[Proposition]]s <math>P_{i,j}</math> specified at different moments of time <math>t_{i,j}</math> (here <math>j</math> labels the times). We write this as:

<math> H_i = (P_{i,1}, P_{i,2},\ldots,P_{i,n_i}) </math>

and read it as "the proposition <math>P_{i,1}</math>  is true at time <math>t_{i,1}</math> ''and then'' the proposition <math>P_{i,2}</math> is true at time <math>t_{i,2}</math> ''and then'' <math>\ldots</math>". The times  <math>t_{i,1} < t_{i,2} < \ldots < t_{i,n_i}</math> are strictly ordered and called the ''temporal support'' of the history.

''Inhomogeneous histories'' are multiple-time propositions which cannot be represented by a homogeneous history. An example is the logical [[logical disjunction|OR]] of two homogeneous histories: <math>H_i \lor H_j</math>.

These propositions can correspond to any set of questions that include all possibilities.
Examples might be the three propositions 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 approach is to show that classical questions such as, "where are my keys?" are consistent. In this case one might use a large number of propositions each one specifying the location of the keys in some small region of space.

Each single-time proposition <math>P_{i,j}</math> can be represented by a [[projection operator]] <math>\hat{P}_{i,j}</math> acting on the system's [[Hilbert space]] (we use "hats" to denote operators). It is then useful to represent homogeneous histories by the [[time-ordered product]] of their single-time projection operators. This is the [[history projection operator]] (HPO) formalism developed by [[Christopher Isham]] and 
naturally encodes the logical structure of the history propositions.

===Consistency===

An important construction in the consistent histories approach is the '''class operator''' for a homogeneous history:

:<math>\hat{C}_{H_i} := T \prod_{j=1}^{n_i} \hat{P}_{i,j}(t_{i,j}) = \hat{P}_{i,n_i} \cdots \hat{P}_{i,2} \hat{P}_{i,1}</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.
This definition can be extended to inhomogeneous histories as well.

Central to the consistent histories is the notion of consistency. A set of histories <math>\{ H_i\}</math> is '''consistent''' (or '''strongly consistent''') if

:<math>\operatorname{Tr}(\hat{C}_{H_i} \rho \hat{C}^\dagger_{H_j}) = 0</math>

for all <math>i \neq j</math>. Here <math>\rho</math> represents the initial [[density matrix]], and the operators are expressed in the [[Heisenberg picture]].

The set of histories is '''weakly consistent''' if 
:<math>\operatorname{Tr}(\hat{C}_{H_i} \rho \hat{C}^\dagger_{H_j}) \approx 0</math>

for all <math>i \neq j</math>.

=== Probabilities ===
If a set of histories is consistent then probabilities can be assigned to them in a consistent way. We postulate that the [[probability]] of history <math>H_i</math> is simply

:<math>\operatorname{Pr}(H_i) = \operatorname{Tr}(\hat{C}_{H_i} \rho \hat{C}^\dagger_{H_i})</math>

which obeys the [[axioms of probability]] if the histories <math>H_i</math> come from the same (strongly) consistent set.

As an example, this means the probability of "<math>H_i</math> OR <math>H_j</math>" equals the probability of "<math>H_i</math>" plus the probability of "<math>H_j</math>" minus the probability of "<math>H_i</math> AND <math>H_j</math>", and so forth.