PhyslibAlpha.ProbabilisticTheory.LocalNet.GlobalSymmetry

Let $R$ be a collection of regions in a net of local systems, and let $A(X)$ denote the local system (the algebra of observables) associated with a region $X \in R$. Let $\text{ofLoc}(X, a)$ be the map that embeds a local observable $a \in A(X)$ into the global quasi-local system. For any two regions $X, Y \in R$ such that $X = Y$, and for any local observable $a \in A(X)$, the embedding of $a$ into the global system via region $X$ is identical to the embedding of its identified counterpart in $A(Y)$ via region $Y$. That is, $\text{ofLoc}(Y, \text{eqIso}(e, a)) = \text{ofLoc}(X, a)$, where $\text{eqIso}(e, a)$ is the image of $a$ under the isomorphism induced by the equality $e : X = Y$.

Given a net action $\alpha$ of a group $G$ on a net of local systems $N$, and an element $g \in G$, $\beta_{\text{Lin}}(\alpha, g)$ is the linear map from the global (quasi-local) system to itself induced by the local isomorphisms $\tau_g$. Specifically, it is the unique linear map such that for any local observable $a$ in a region $i$, its image is the local observable $\tau_g(a)$ in the transformed region $g \cdot i$.

Let $\alpha$ be a net action of a group $G$ on a net of local systems $N$. For any group element $g \in G$, local region $i$, and local observable $a$ belonging to the local system at $i$, the global linear map $\beta_g$ acts on the representation of $a$ in the global system as follows: $$\beta_g(\text{ofLoc}_i(a)) = \text{ofLoc}_{g \cdot i}(\tau_g(a))$$ where $\text{ofLoc}_i$ denotes the embedding of a local observable from region $i$ into the global (quasi-local) system, and $\tau_g$ is the local isomorphism that maps observables from region $i$ to the transformed region $g \cdot i$.

Given a net action $\alpha$ of a group $G$ on a net of local systems $N$, and an element $g \in G$, the map $\beta_g$ is the unital positive linear map (channel) from the global system (quasi-local algebra) to itself induced by the local isomorphisms $\tau_g$. This channel is the unique unital positive linear map such that for any local observable $a$ in a region $i$, its image is the local observable $\tau_g(a)$ in the transformed region $g \cdot i$.

Let $\alpha$ be a net action of a group $G$ on a net of local systems $N$ over regions $R$. For any group element $g \in G$, any region $i \in R$, and any local observable $a$ in the system at $i$, the global channel $\beta_g$ acts on the embedding of $a$ into the global system as follows: $$\beta_g(\text{ofLoc}_i(a)) = \text{ofLoc}_{g \cdot i}(\tau_g(a))$$ where $\text{ofLoc}_i$ is the canonical embedding of a local observable from region $i$ into the quasi-local (global) system, and $\tau_g$ is the local isomorphism mapping observables from region $i$ to the transformed region $g \cdot i$.

Let $\alpha$ be a net action of a group $G$ on a net of local systems $N$. The global channel $\beta_1$ induced by the identity element $1 \in G$ on the quasi-local algebra of the global system is the identity unital positive linear map.

Let $G$ be a group and $\alpha$ be a net action of $G$ on a net of local systems $N$. For any elements $g, h \in G$, let $\beta_g$ denote the global channel (the unital positive linear map on the quasi-local algebra) induced by $g$. Then the global channel associated with the product $gh$ is the composition of the global channels associated with $g$ and $h$: $$\beta_{gh} = \beta_g \circ \beta_h$$

Given a net action $\alpha$ of a group $G$ on a net of local systems $N$, and an element $g \in G$, the symmetry $\beta_g$ is the **order-automorphism** of the global system (the quasi-local algebra) induced by $g$. It is defined as the channel $\beta_g$ with its inverse given by the channel $\beta_{g^{-1}}$, effectively extending the local isomorphisms $\tau_g$ to the entire infinite system.

Given a net action $\alpha$ of a group $G$ on a net of local systems $N$, let $\mathcal{A}$ denote the global system (the quasi-local algebra). The mapping `globalβ` is the group homomorphism $\beta : G \to \text{Symmetry}(\mathcal{A})$ that assigns to each group element $g \in G$ the global symmetry $\beta_g$. This homomorphism extends the local symmetries to the entire system, satisfying $\beta_{gh} = \beta_g \circ \beta_h$ and $\beta_e = \text{id}_{\mathcal{A}}$.

Given a net action $\alpha$ of a group $G$ on a net of local systems $N$, let $\mathcal{A}$ denote the global system (the quasi-local algebra). The `covariantRep` is the canonical covariant representation of the net $N$ within $\mathcal{A}$. This structure identifies the global symmetry group homomorphism $\beta: G \to \text{Symmetry}(\mathcal{A})$ (where $\beta = \text{global}\beta_\alpha$) and ensures that the inclusion of local observables into the global system is covariant. Specifically, for any group element $g \in G$, local region $X$, and local observable $a \in N(X)$, the action of the global symmetry $\beta_g$ on the image of $a$ in $\mathcal{A}$ coincides with the image of the locally translated observable $\tau_g(a) \in N(g \cdot X)$.

Let $G$ be a group acting on a net of local systems $N$ via an action $\alpha$. Let $\mathcal{A}$ be the quasi-local algebra (global system) and $\beta: G \to \text{Symmetry}(\mathcal{A})$ be the group homomorphism that extends the local symmetries to the global system. For any state $\omega$ on $\mathcal{A}$, let $\text{stateEquiv}(\omega)$ denote the corresponding family of local states (the net state). For any group element $g \in G$, the action of the global symmetry $\beta_g$ on the state $\omega$ (defined as $\beta_g \cdot \omega = \omega \circ \beta_g^{-1}$) satisfies the identity: $$\text{stateEquiv}(\beta_g \cdot \omega) = g \cdot \text{stateEquiv}(\omega)$$ where $g \cdot \text{stateEquiv}(\omega)$ represents the transformation of the net state under the group element $g$.

Let $\alpha$ be a net action of a group $G$ on a net of local systems $N$, and let $\mathcal{A}$ be the corresponding global quasi-local system. A global state $\omega$ on $\mathcal{A}$ (an element of the state space $\mathcal{S}[\mathbb{R}, \mathcal{A}]$) corresponds to a compatible family of local states via the mapping $\text{stateEquiv}(\omega)$. The family of local states is invariant under the net action $\alpha$ if and only if the global state $\omega$ is invariant under the global symmetry $\beta_g$ for all $g \in G$. That is, $$\text{NetState.IsInvariant}(\alpha, \text{stateEquiv}(\omega)) \iff \forall g \in G, \omega \circ \beta_g = \omega,$$ where $\beta_g$ is the group homomorphism $\text{global}\beta(\alpha, g)$ representing the global symmetry acting on the quasi-local system.