Algebraic free Weyl net on Green backgrounds

For a Green-data background of Green-hyperbolic free scalar interface, let $(V(M),{\sigma }_M)$ be the real symplectic vector space of Causal propagator induces the free scalar symplectic quotient. Define its algebraic Weyl $\ast$-algebra $\mathfrak{W}(M)$ to be the complex vector space of finite formal sums of basis symbols $W_M(v)$, $v\in V(M)$, with bilinear product and conjugate-linear involution specified on generators by

$W_M(v)W_M(w):=e^{-i{\sigma }_M(v,w)\mathrm{/}2}{W}_M(v+w),W_M(v{)}^{\ast }:=W_M(-v)\mathrm{.}$

The unit is $W_M(0)$. This is the algebraic Weyl algebra: this definition assigns no norm, Hilbert-space representation, or $C^{\ast }$-completion. For an open set $O\subseteq M$, define the local algebra ${\mathfrak{W}}_M(O)$ as the unital $\ast$-subalgebra generated by the symbols $W_M([f])$ with real $f\in {\mathcal{D}}_g(M)$ and $\mathrm{supp}f\subseteq O$. No norm closure is taken.

An open subset $O\subseteq M$ is causally convex if every locally absolutely continuous future- or past-directed causal curve segment $\gamma :[a,b]\to M$ with both endpoints in $O$ satisfies $\gamma ([a,b])\subseteq O$. For an embedding between the oriented backgrounds specified in Green-hyperbolic free scalar interface, orientation-preserving means that its differential carries positively oriented tangent bases to positively oriented tangent bases, and time-orientation-preserving means that it maps each nonzero future-directed causal tangent vector to a future-directed causal tangent vector, using the time orientations specified by $N_U,N_M$. An embedding $\iota :U\to M$ is isometric if $g_M(d{\iota }_{p}v,d{\iota }_{p}w)=g_U(v,w)$ for every $p\in U$ and $v,w\in T_{p}U$.

An admissible Green-background morphism $\iota :(U,g_U,N_U,P_U,E_{+}^U,E_{-}^U)\to (M,g_M,N_M,P_M,E_{+}^M,E_{-}^M)$ is an orientation-preserving, time-orientation-preserving isometric smooth embedding with open causally convex image, for which $P_M{\iota }_{\ast }f={\iota }_{\ast }{P}_{U}f$ on compact smooth real tests. Here ${\iota }_{\ast }f$ is extension by zero from the open image, which is smooth and compactly supported because $\mathrm{supp}f$ is compact in $U$. The morphism is Cauchy when its image contains a Cauchy hypersurface of $(M,g_M,N_M)$, meaning a spacelike embedded hypersurface met exactly once by every inextendible causal curve, as in Spacelike hypersurfaces, acausal sets, Cauchy sets, domains of dependence and Cauchy horizons. The associated quotient map, when well-defined, sends $[f]\in V(U)$ to $[{\iota }_{\ast }f]\in V(M)$, and the Weyl map is required to send $W_U([f])$ to $W_M([{\iota }_{\ast }f])$.

Green-hyperbolic free scalar interface

Causal propagator induces the free scalar symplectic quotient

Causal curves, chronological and causal reachability, and inextendibility

Chronology, causality, strong causality, global hyperbolicity and time functions

Spacelike hypersurfaces, acausal sets, Cauchy sets, domains of dependence and Cauchy horizons

Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model

Only the real symplectic form and the algebraic Weyl relations are part of this definition. In particular, no assertion about Reeh–Schlieder, local factor type, existence of a state, or a $C^{\ast }$-completion is made.