Heisenberg–Schrödinger Equivalence as a Collapse of Boundary-Channel Duality

This paper proves a finite-dimensional collapse-and-obstruction theorem for the relation between Heisenberg observable evolution and Schrödinger state evolution. The starting datum is a completion-locked boundary-channel datum: a boundary Hilbert space, a locked boundary observable algebra, a Heisenberg comparison on boundary observables, and an admissible state-side channel. The basic duality law is the trace pairing TrX Φ(ρ) = TrΦ*(X) ρ, restricted, when appropriate, to a locked boundary algebra A∂. The theorem proves that this pairing undergoes Heisenberg–Schrödinger collapse exactly in the full-algebra reversible case: A∂ = B(H∂) and the Heisenberg dual is a ∗-automorphism, equivalently the state-side channel is unitary conjugation. Outside this reversible full-algebra case, the same expectation pairing is a boundary-channel law rather than a unitary full-algebra equivalence. Three rigorous obstructions certify noncollapse relative to the locked boundary datum: a proper boundary algebra is state- and channel-compressive; a nonmultiplicative Heisenberg dual cannot be implemented by unitary conjugation on the full algebra; and a many-to-one bulk/history realization can induce identical boundary laws from inequivalent channels or histories. Thus finite-dimensional closed-system unitary quantum mechanics is identified as the special case in which the boundary algebra is the whole kinematic algebra and the observable-state interface has no quotient, no proper boundary restriction, and no channel-valued loss of reversibility. The theorem has immediate consequences for measurement theory. Nonselective projective measurement is a boundary-channel operation: it fixes the commutative measured algebra while removing off-boundary coherences. Selective measurement is conditioning on a boundary record, not state-uniform retuning of the unconditioned boundary law. Bipartite no-signaling is likewise a fixed-boundary theorem: a remote completely positive trace-preserving channel fixes the local boundary algebra in the Heisenberg picture. Random-unitary symmetry-selection channels and finite rigidity-gap quotient selections supply two realization mechanisms. Their conserved observables are determined by the active commutant, while their low-temperature or equilibrium limits are controlled by explicit channel-norm bounds. License note: Distributed under CC BY-NC-ND 4.0.