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Exact local dynamics for commuting diagonal interactions

Statement

Assume Countable Choice and the finite-matrix C*-identity for the local spin algebras. Let $α_{t}$ be the thermodynamic dynamics for the diagonal interaction of Diagonal quantum Ising interaction and classical conditional energies, as constructed under the exact hypotheses of Strongly continuous thermodynamic spin automorphism group. For a local observable $A \in A_{X}$ , define the finite self-adjoint matrix $K_{X} : = \sum_{S : S \cap X \neq \emptyset} Φ (S),$ where the sum ranges over the singleton and pair supports of that Ising interaction. Then the sum is finite and $α_{t} (A) = e^{i t K_{X} / ℏ} A e^{- i t K_{X} / ℏ} (t \in R) .$ In particular, every local diagonal observable is fixed by $α_{t}$ . The formula is exact for this commuting diagonal model and uses the full local matrix algebra; it does not restrict observables to diagonal matrices.

Facts & Assumptions

Given: Countable Choice, the finite-matrix C*-identity, the finite-range diagonal interaction of Diagonal quantum Ising interaction and classical conditional energies, its free-boundary finite-volume dynamics, $ℏ > 0$ , and the thermodynamic limit stated in Strongly continuous thermodynamic spin automorphism group.

[A1]

Countable Choice is the axiom that every sequence of nonempty sets has a choice function (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[A2]

Finite matrix C-star algebra and density-state duality states that each nonzero finite local matrix algebra with its operator norm satisfies the C*-identity; its current batch-36 proof is draft and is used under its stated hypotheses.

[F1]

Diagonal quantum Ising interaction and classical conditional energies gives diagonal self-adjoint terms $Φ (S)$ with finite range and bounded per-site norm sum; each singleton or pair term has finite support.

[F2]

Strongly continuous thermodynamic spin automorphism group states, under Countable Choice and [A2], that free-boundary finite-volume dynamics converge in norm on local observables, uniformly on compact time intervals, to the exhaustion-independent group $α_{t}$ .

Proof

technique · Separate the finitely many interaction terms meeting the observable's support from terms disjoint from it

Given: [F1]–[F2], $A \in A_{X}$ local, and $ℏ > 0$ .

1.1givenA1A2F1F2algebra

Only finitely many singleton and pair supports meet the finite set $X$ , because the interaction range is finite and each site has finitely many neighbors. Hence $K_{X}$ is a finite self-adjoint diagonal matrix. Choose any increasing finite exhaustion $Λ_{n}$ that eventually contains every support $S$ meeting $X$ . For all sufficiently large $n$ , decompose $H_{Λ_{n}} = K_{X} + L_{n}$ , where $L_{n}$ is the sum of the remaining terms contained in $Λ_{n}$ . Every support in $L_{n}$ is disjoint from $X$ , so $[L_{n}, A] = 0$ ; every term is diagonal, so $[L_{n}, K_{X}] = 0$ . Therefore $e^{i t H_{Λ_{n}} / ℏ} A e^{- i t H_{Λ_{n}} / ℏ} = e^{i t K_{X} / ℏ} A e^{- i t K_{X} / ℏ}$ for all real $t$ .

2.1F2step 1.1algebra∎

The right side of step 1.1 is independent of $n$ . Passing to the norm limit supplied by [F2] gives the claimed exact formula for $α_{t} (A)$ . If $A$ is diagonal, it commutes with the diagonal matrix $K_{X}$ , so the formula reduces to $α_{t} (A) = A$ .

Exact local dynamics for commuting diagonal interactions

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources