Manuscript En05-C@1 · En05-C.tex · section · aa213ea6

Constraint-preserving regional algebras

Stage D: a finite constrained Z2 model › Constraint-preserving regional algebras · lines 861–906

En05-C@1#eq:gauss, En05-C@1#eq:gaugenet

Let $Γ = (V, E)$ be a finite connected graph without self-loops; parallel links are allowed. Associate a qubit to each link:

H_{kin} = e \in E ⨂ C^{2} .

Let $X_{e}, Z_{e}$ be its Pauli operators. The Gauss operators are

G_{v} = e ∋ v \prod X_{e} .

They commute, and $\prod_{v} G_{v} = 1$ .

Define

H_{phys} = {ψ : G_{v} ψ = ψ for every v}, P = orthogonal projection onto H_{phys} .

The tensor product above is kinematic. We do not assume that the constrained physical Hilbert space factors across a link-region partition.

This finite pure-gauge choice is a specified discrete model. The broader Hamiltonian lattice-gauge setting is reviewed in the original formulation of Kogut and Susskind KogutSusskind; no Hamiltonian from that work is a premise here.

For $S \subseteq E$ , let

F (S) = {a \in B (H_{kin}) : a acts identically outside S, [a, G_{v}] = 0 \forall v} .

Its physical image is

A (S) = q_{phys} (F (S)), q_{phys} (a) = a ∣_{H_{phys}}, M = B (H_{phys}) .

Displayed formulas in this passage

unlabeled · line 865
$H_{kin} = e \in E ⨂ C^{2} .$
En05-C@1#eq:gauss · line 870
$G_{v} = e ∋ v \prod X_{e} .$
unlabeled · line 877
$H_{phys} = {ψ : G_{v} ψ = ψ for every v}, P = orthogonal projection onto H_{phys} .$
unlabeled · line 892
$F (S) = {a \in B (H_{kin}) : a acts identically outside S, [a, G_{v}] = 0 \forall v} .$
En05-C@1#eq:gaugenet · line 898
$A (S) = q_{phys} (F (S)), q_{phys} (a) = a ∣_{H_{phys}}, M = B (H_{phys}) .$

Constraint-preserving regional algebras

Displayed formulas in this passage

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