Consistent histories

An approach assigning probabilities to compatible families of time-ordered quantum alternatives. A decoherence functional measures interference between histories; when its relevant cross-terms vanish, coarse-grained probabilities add as in probability theory. Different incompatible families must not be combined as if they were one classical history space. The formalism does not require a fundamental external measurement apparatus.

For alternatives represented by projection operators at successive times, set $C_{\alpha }=P_{{\alpha }_n}(t_n)\cdots P_{{\alpha }_1}(t_1)$ in the Heisenberg picture. The displayed functional uses the initial density operator $\rho$. Combining two histories gives $D(\alpha +\beta ,\alpha +\beta )=D(\alpha ,\alpha )+D(\beta ,\beta )+2\mathrm{Re}D(\alpha ,\beta )$. Thus vanishing real cross-terms gives the probability sum rule; the stronger commonly used condition $D(\alpha ,\beta )=0$ for distinct histories is sufficient as well. Its diagonal is nonnegative because each $C_{\alpha }\rho C_{\alpha }^{†}$ is positive.