Exact Intertwiner Preservation Under the Hamiltonian Constraint at the Four-Valent Vertex with j = 1/2 (Paper 48 in the Intertwiner Series)

AbstractWe prove that the intertwiner quantum number k ∈ {0, 1} at the four-valent spin-network vertex with edge spins j = 1/2 is exactlypreserved under the action of the Thiemann Hamiltonian constraint. The proof rests on a single kinematical fact — the volumeoperator ˆV is proportional to the identity on the two-dimensional gauge-invariant intertwiner space Hint —combined with a generaloperator identity. For any operator A on the full unconstrained kinematical space Hkin =N4p=1 V1/2 and any gauge-invariant states|k⟩, |k′⟩ ∈ Hint:⟨k′|[A, ˆV ]|k⟩ = ⟨k′|A ˆV |k⟩ − ⟨k′|ˆV A|k⟩ = v0⟨k′|A|k⟩ − v0⟨k′|A|k⟩ = 0,where v0 = (√3/8)1/2 ℓ3P and we have used ˆV |k⟩ = v0|k⟩ on the right and the Hermiticity of ˆV to write ⟨k′|ˆV = v0⟨k′| on the left.Since the Thiemann Hamiltonian is constructed entirely from commutators [A, ˆV ] sandwiched between gauge-invariant externalstates, it preserves k exactly. The proof holds for finite holonomies, products of holonomies, and arbitrary operators — not onlyinfinitesimal generators— and requires no effective-theory approximation.