Wightman axioms

Definition

Let $\R^4$ be Minkowski space with the form $x \cdot y = x^0 y^0 - \sum_{k=1}^3 x^k y^k$, and let $\mathcal{P}$ be the identity component of the Poincaré group, acting by $x \mapsto \Lambda x + a$. Let $\mathcal{H}$ be a separable complex Hilbert space, and for each Schwartz function $f \in \mathcal{S}(\R^4)$ (see Schwartz space) let $\phi(f)$ be a linear operator on $\mathcal{H}$. The Wightman axioms for a real scalar field $\phi$ are the following.

1. Relativistic quantum mechanics: there is a strongly continuous unitary representation $U$ of $\mathcal{P}$ on $\mathcal{H}$. The joint spectrum of the generators $P^0, \dots, P^3$ of the translations $U(a, I) = e^{i P \cdot a}$ lies in the cone $\set{p : p^0 \ge \abs{\mathbf{p}}}$. There is a unit vector $\Omega$, unique up to a phase, with $U(a, \Lambda)\Omega = \Omega$ for all $(a, \Lambda) \in \mathcal{P}$.
2. Domain and continuity: there is a dense subspace $D \subseteq \mathcal{H}$ with $\Omega \in D$ and $U(a, \Lambda) D \subseteq D$. For every $f$, $D$ lies in the domain of the unbounded operator $\phi(f)$, and $\phi(f) D \subseteq D$. For all $\chi, \psi \in D$ the map $f \mapsto \ip{\chi}{\phi(f)\psi}$ is a tempered distribution. Moreover $\ip{\chi}{\phi(f)\psi} = \ip{\phi(\bar f)\chi}{\psi}$ for $\chi, \psi \in D$.
3. Covariance: $U(a, \Lambda)\, \phi(f)\, U(a, \Lambda)^{-1} = \phi(f_{(a, \Lambda)})$ on $D$, where $f_{(a, \Lambda)}(x) = f(\Lambda^{-1}(x - a))$.
4. Locality: if the supports of $f$ and $g$ are spacelike separated, that is, $(x - y) \cdot (x - y) < 0$ for all $x \in \operatorname{supp} f$ and $y \in \operatorname{supp} g$, then $\phi(f)\phi(g)\psi = \phi(g)\phi(f)\psi$ for all $\psi \in D$.
5. Cyclicity of the vacuum: the finite linear combinations of vectors $\phi(f_1) \cdots \phi(f_n)\Omega$, $n \ge 0$, are dense in $\mathcal{H}$.