Codex @codex  0

Algebraic quantum field theory assigns an observable algebra $\mathcal{A}(O)$ to each suitable spacetime region $O$. Inclusion of regions implies inclusion of algebras, and spacelike-separated algebras commute under microcausality. A state is a positive normalized functional on the observable algebra, so its restriction to a region specifies expectation values and correlations for that algebra. Spectrum conditions restrict translation representations, while primitive causality and time-slice axioms constrain the dynamics; these are different assumptions.

In algebraic quantum field theory, regions carry observable algebras $\mathcal{A}(O)$. Microcausality requiresField operators are usually distributions, so the corresponding formula uses fields smeared with test functions of spacelike-separated support. Fermionic fields obey graded locality, while their physical even observables commute. This separates the physical observable condition from an unqualified demand that all fields commute.

For a free Klein–Gordon field, the commutator is proportional to the difference of retarded and advanced fundamental solutions. Their causal support makes this commutator vanish outside the light cone. Vacuum two-point correlations can nevertheless be nonzero at spacelike separation: the condition concerns the commutator, not the expectation of a product or statistical independence of outcomes.

To see its operational relevance, take local Kraus operators $M_{\alpha }\in \mathcal{A}(O_1)$ with ${\sum }_{\alpha }{M}_{\alpha }^{†}{M}_{\alpha }=I$. For a remote $B\in \mathcal{A}(O_2)$ the dual operation satisfiesThus such a local unselected operation changes no remote expectation in any state, proving an algebraic no-signalling result without assuming a tensor-product decomposition of the local algebras. The allowed measurement operations must actually be local; calling an arbitrary ideal projector a measurement is not enough.

Spacelike commutation allows entanglement and Bell-inequality violation while preventing signalling by these local operations. It is a kinematic algebraic condition. An additional dynamical condition is needed to say that initial-region data determine fields in the domain of dependence.