= Solution
<Consistent histories> assigns probabilities to suitably compatible sequences of quantum alternatives without making an external measuring apparatus fundamental. At successive times choose complete orthogonal sets of <projection operators>. In the <Heisenberg picture>, a history has class operator $C_\alpha=P_{\alpha_n}(t_n)\cdots P_{\alpha_1}(t_1)$. For an initial <density operator> $\rho$, its <decoherence functional> is
$$
D(\alpha,\beta)=\operatorname{tr}(C_\alpha\rho C_\beta^\dagger).
$$
A common sufficient consistency condition is $D(\alpha,\beta)=0$ for $\alpha\ne\beta$. Then $p(\alpha)=D(\alpha,\alpha)$ is nonnegative and coarse-grained alternatives have additive probabilities: the <interference> cross-terms vanish. The weaker requirement on real cross-terms can suffice for individual <probability> sum rules. Predictions must stay within one compatible framework; mutually incompatible history families cannot simply be combined into one classical account.
The <many-worlds interpretation> instead treats unitary quantum evolution as universally valid and interprets the correlated outcome branches as components of one global state. A measurement interaction gives
$$
(c_0|0\rangle+c_1|1\rangle)|A_{\rm ready}\rangle|E_{\rm ready}\rangle
\longrightarrow c_0|0\rangle|A_0\rangle|E_0\rangle+c_1|1\rangle|A_1\rangle|E_1\rangle.
$$
When the environment records become nearly orthogonal, <quantum decoherence> suppresses observable <interference> between stable apparatus records. Each <relative state> contains an observer correlated with a definite record; no additional physical collapse is postulated. Branch structure is an emergent approximation, rather than a unique literal splitting at every arbitrary choice of basis.
Both approaches use the ordinary unitary formalism and decoherence, but their interpretations of histories and branches differ. Decoherence alone does not prove that one unique branch is selected, nor does the unitary equation by itself settle every assumption needed to justify <Born rule> probabilities. Neither interpretation predicts ordinary <interference> experiments incorrectly merely by interpreting the same state differently.
Back to article page
