Time-slice axiom

The time-slice axiom says that an algebra generated in an arbitrarily thin suitable neighbourhood of a Cauchy surface generates the algebra of its causal development. For $O$ such a neighbourhood in its development $D(O)$, this is $\mathcal{A}(O)=\mathcal{A}(D(O))$. Precise global and local versions can differ in the regions to which they apply. For a local hyperbolic field equation, later fields can be represented using the initial field and its conjugate momentum in the corresponding domain of dependence.