Lorentz covariance of the λ(ϕ⁴)₂ quantum field theory

Abstract

We prove that the λ(ϕ⁴)₂ quantum field theory model is Lorentz covariant, and that the corresponding theory of bounded observables satisfies all the Haag-Kastler axioms. For each Poincaré transformation {a, Λ} and each bounded regionB of Minkowski space we construct a unitary operatorU which correctly transforms the field bilinear forms:Uϕ(x, t)U*=ϕ({a, Λ} (x, t)), for (x, t) ∈B. We also consider the von Neumann algebra$\mathfrak{A}(B)$ of local observables, consisting of bounded functions of the field operators ϕ(f)=ε ϕ(x, t)f(x, t)dx dt, suppf ⊂B. We define a *-isomorphism$\sigma _{\{ a,\Lambda \} } :\mathfrak{A}(B) \to \mathfrak{A}(\{ a,\Lambda \} B)$ by setting σ_{{a, Λ}}(A)=U A U*. The mapping$\{ a,\Lambda \} \to \sigma _{\{ a,\Lambda \} }$ is a representation of the Poincaré group by *-automorphisms of the normed algebra$\cup _B \mathfrak{A}(B)$ of local observables.