Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 59 / 5 / v / Solution

Primitive causality is an initial-region completeness requirement. A state's full restriction to a suitable region $O$ determines expectations of observables in its future domain of dependence. The domain of dependence is not the whole causal future: it contains points whose past-inextendible causal curves all meet the specified initial data region. Merely being reachable by one causal curve from $O$ is not enough.

A standard local time-slice axiom makes this precise when $O$ contains a neighbourhood of a Cauchy slice of its causal development:with the regions and algebra identifications understood in the specified local net. Definitions applying primitive causality to all bounded open regions can be stronger than a time-slice property stated only for suitable causally convex Cauchy neighbourhoods; these formulations should not be conflated without extra hypotheses.

For a local hyperbolic Klein–Gordon equation, a later field is determined by the field and its normal derivative, or canonical momentum, on a Cauchy slice. Its value can be written using the causal fundamental solution, with integration supported where the past light cone meets the initial slice. If that support lies in the initial patch, the later smeared field is generated by the patch's initial field and momentum operators. Their algebra therefore generates the observables of the patch's domain of dependence. This is the local differential-equation mechanism behind the time-slice property.

The required information is the entire restricted quantum state: it assigns expectations to every observable and product in the initial algebra. Mean values of the initial field and momentum alone do not determine their fluctuations or higher correlations and thus are insufficient. Knowledge of the restricted state fixes later probability distributions and correlations; it does not fix which individual outcome a future measurement will produce.

Finally, primitive causality concerns propagation of information already encoded in initial data, while microcausality controls compatibility of spacelike measurements and quantum no-signalling controls possible interventions. The spectrum condition adds a separate positivity restriction. A causal initial-value evolution, positive energy and no-signalling are complementary requirements, not equivalent definitions of quantum locality.