Manuscript Em03-G@1 · Em03-G-v1.tex · theorem · 0bb3a8a3

Theorem: Exact ordered evolution

Ordered noncommuting source dynamics › Representation and common domain · lines 559–611 · proof included or linked

Em03-G@1#thm:unitary, Em03-G@1#eq:UG

The interaction (eq:interaction) has a unitary propagator. After both slabs have ended, its endpoint is

U_{G} = e^{i (γ_{1} + γ_{2})} e^{- i X \otimes X (a_{2})} e^{- i Z \otimes X (a_{1})} .

The individual phases are

γ_{j} = - \frac{g _{j} ^{2}}{2} \int_{I_{j}} d t \int_{I_{j} \cap {u < t}} d u s (k_{j, t}, k_{j, u}) .

The two factors preserve the same number form domain. On that domain their chronological composition solves the strong Schrödinger equation into the Hilbert space.

Proof. For one slab and source observable $O \in {X, Z}$ , $O^{2} = I$ . If $K (t) = g_{j} \int_{I_{j} \cap {u \leq t}} k_{j, u} d u$ , its propagator is

e^{i γ_{j} (t)} o = \pm 1 \sum P_{o}^{O} \otimes W (- oK (t)) .

Chronological Weyl products give exactly the displayed scalar phase. Smooth covariance-norm continuity from Lemma lem:compactregularity gives the strong product limit.

For a differentiable coefficient path $r (t)$ , the Weyl relation and the Stone derivative give, on the number form domain,

\partial_{t} W (r (t)) = [i X (r ˙ (t)) + \frac{i}{2} s (r ˙ (t), r (t))] W (r (t)) .

The scalar derivative cancels $\partial_{t} γ_{j}$ , proving the Schrödinger equation on the slab. Creation/annihilation estimates and the displacement identity show domain invariance.

The slabs do not overlap. Their endpoint propagators therefore compose in chronological order on the same domain, proving (eq:UG). Differentiating the inner product of two candidate solutions gives uniqueness. No central-commutator claim is made for overlapping $X$ - and $Z$ -couplings.

Displayed formulas in this passage

Em03-G@1#eq:UG · line 563
$U_{G} = e^{i (γ_{1} + γ_{2})} e^{- i X \otimes X (a_{2})} e^{- i Z \otimes X (a_{1})} .$
unlabeled · line 572
$γ_{j} = - \frac{g _{j} ^{2}}{2} \int_{I_{j}} d t \int_{I_{j} \cap {u < t}} d u s (k_{j, t}, k_{j, u}) .$
unlabeled · line 587
$e^{i γ_{j} (t)} o = \pm 1 \sum P_{o}^{O} \otimes W (- oK (t)) .$
unlabeled · line 597
$\partial_{t} W (r (t)) = [i X (r ˙ (t)) + \frac{i}{2} s (r ˙ (t), r (t))] W (r (t)) .$

Theorem: Exact ordered evolution

Displayed formulas in this passage

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