Editing Consistent histories (section)

===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>.