Algebraic quantum field theory

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.

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.

Primitive causality is a dynamical completeness condition: the state restricted to a suitable initial spacetime region determines its expectations throughout that region's domain of dependence. Algebraically one requires the later domain-of-dependence observable algebra to be contained in the initial algebra, under the identifications supplied by the dynamics. A local time-slice axiom gives this property for a neighbourhood of a Cauchy slice. Knowledge of the full restricted state, including correlations, is required; knowing only mean initial fields is insufficient.

In a translation-covariant quantum field theory on Minkowski spacetime, the joint spectrum of the energy-momentum generators is required to lie in the closed future light cone. With signature $(+,-,-,-)$ this means $p^0\ge 0$ and $p^2\ge 0$. The spectrum condition implements a positive-energy stability requirement in every inertial frame. It constrains the Fourier support and analyticity of vacuum correlation functions but does not by itself assert spacelike commutation or an initial-value property.