Consistent Histories

Consistent-histories approaches assign probabilities to sequences of quantum properties when interference between the sequences satisfies a consistency condition. Measurement is treated as an ordinary physical process within a history. The distinctive interpretive rule is that incompatible frameworks must not be combined into one chain of reasoning.

Required background. Born rule: Translate a specified state and measurement into outcome probabilities.

Histories and their consistency

For alternatives represented by Heisenberg-picture projectors at times $t_1<\cdots<t_n$, define

$$C_\alpha=P_{\alpha_n}(t_n)\cdots P_{\alpha_1}(t_1), \qquad D(\alpha,\beta)=\operatorname{Tr}(C_\alpha\rho C_\beta^\dagger).$$

The diagonal $D(\alpha,\alpha)$ is nonnegative. Interference terms obstruct the classical addition rule for coarse-grained alternatives. A common sufficient condition is medium decoherence, $D(\alpha,\beta)=0$ for $\alpha\ne\beta$. Weaker conditions and approximate versions need separate qualification.

The decoherence-functional page derives this functional and works through examples. Here the question is what the resulting probabilities mean.

The single-framework rule

A framework specifies a compatible exhaustive family of alternatives and a consistency condition. Within an accepted framework, ordinary probability reasoning applies. Statements from incompatible frameworks cannot be joined as if they formed one classical event algebra.

For a spin-$1/2$ system, a framework using $Z$ projectors and one using $X$ projectors are both legitimate at a single time. They do not jointly define events “$Z=+1$ and $X=+1$” as commuting sharp properties. Combining them is not merely using an unknown classical joint distribution.

Measurement without a special collapse law

An apparatus correlates a microscopic property with a later macroscopic record. In a suitable consistent framework, conditional probabilities can relate the record to an earlier property. This aims to justify ordinary experimental inference without a separate physical projection event.

The inference is framework-dependent. It should not be promoted to a unique complete description of every property at every time. A history family is more than a list of performed measurements: it can describe unmeasured alternatives, provided its consistency conditions and interpretation are respected.

Probability, classicality, and ontology

Born-like history probabilities are part of the formalism; consistency tells us where classical sum rules can be applied. Decoherence supports approximately classical families, especially for coarse macroscopic variables.

Proponents regard multiple frameworks as complementary descriptions rather than competing physical worlds. Critics ask why a particular quasiclassical family is appropriate, what happens without a unique selection rule, and how to understand apparently incompatible retrodictions. Those are substantive interpretive questions, not algebraic failures of the decoherence functional.

The approach does not, merely by invoking histories, establish Bell-locality or a unique classical trajectory. Its account of locality requires its own definitions and assumptions.

Exercises

Two history amplitudes contribute to the same final event. Why is their total probability generally not the sum of their separate squared norms?

Solution

For class operators $C_\alpha,C_\beta$, the grouped event has probability $\operatorname{Tr}[(C_\alpha+C_\beta)\rho(C_\alpha+C_\beta)^\dagger]$. This equals the two diagonal terms plus $2\operatorname{Re}D(\alpha,\beta)$. The consistency condition removes the interference contribution for the relevant grouping.

References

• R. B. Griffiths, “Consistent histories and the interpretation of quantum mechanics,” Journal of Statistical Physics 36, 219–272 (1984), doi:10.1007/BF01015734.
• R. B. Griffiths, “Consistent quantum measurements” (2015), arXiv:1501.04813.
• R. Omnès, “Consistent interpretations of quantum mechanics,” Reviews of Modern Physics 64, 339–382 (1992), doi:10.1103/RevModPhys.64.339.