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Global unitary dynamics in the Gauss sector

Statement

Assume the model data of Exact compact U(1) rotor plus finite CAR data and the Axiom of Choice. Let $H$ be the self-adjoint Hamiltonian on the exact domain supplied by Rotor interacting Hamiltonian with exact domain, and let $P$ be the physical projection of Gauss operators, gauge group and nonempty physical projection. For any scalar $ℏ > 0$ , the family $U (t) = e^{- i t H / ℏ}, t \in R,$ is a strongly continuous unitary group on $H_{k i n}$ , preserves $D (H)$ , and satisfies $\lim_{h \to 0} \frac{U (h) ψ - ψ}{h} = - \frac{i}{ℏ} H ψ (ψ \in D (H)) .$ It commutes with every gauge transformation $U (α)$ and with $P$ . Hence $H_{p h y s} = ran P$ is a reducing subspace and $U (t) ∣_{H_{p h y s}}$ is a strongly continuous unitary group with skew-adjoint infinitesimal generator $- i H ∣_{D (H) \cap H_{p h y s}} / ℏ$ ; its self-adjoint Stone operator is $- H ∣_{D (H) \cap H_{p h y s}} / ℏ$ . Equivalently, the restricted physical Hamiltonian $H ∣_{D (H) \cap H_{p h y s}}$ is self-adjoint. This is an exact fixed-graph result; no continuum dynamics are asserted.

Facts & Assumptions

Given: The finite-graph model, its gauge representation and physical projection, the self-adjoint operator $H$ , and the Axiom of Choice.

[A1]

The Axiom of Choice is the principle that every family of nonempty sets has a choice function. The Axiom of Choice

[F1]

Gauss operators, gauge group and nonempty physical projection: the operators $U (α)$ form a strongly continuous unitary representation of $T^{V}$ and $P$ is the orthogonal projection onto their joint fixed space $H_{p h y s}$ .

[F2]

Rotor interacting Hamiltonian with exact domain: $H$ is self-adjoint on $D (H_{0})$ and $U (α) H U (α)^{†} = H$ there; $U (α)$ preserves that domain.

[F3]

Stone's theorem: unitary groups and self-adjoint generators: assuming the Axiom of Choice, every self-adjoint operator $T$ determines the strongly continuous group $e^{i t T}$ , whose generator has domain exactly $D (T)$ and equals $i T$ there; a strongly continuous unitary group is uniquely determined by its generator.

Proof

Given: [A1], [F1]–[F3], and $ℏ > 0$ .

1.1A1F2F3algebra

Since $H$ is self-adjoint, $T = - H / ℏ$ is self-adjoint. Stone's theorem [F3] gives $U (t) = e^{i t T} = e^{- i t H / ℏ}$ as a strongly continuous unitary group with the stated derivative at $t = 0$ and exact generator domain $D (H)$ .

2.1F3step 1.1algebra

For $x \in D (H)$ and fixed $t$ , the difference quotient at zero for $U (h) U (t) x = U (t) U (h) x$ equals $U (t) (U (h) x - x) / h$ and converges to $- i U (t) H x / ℏ$ . The exact domain characterization in [F3] therefore gives $U (t) x \in D (H)$ and $H U (t) x = U (t) H x$ .

2.2F2F3step 1.1

For each $α$ , the conjugated family $V_{α} (t) = U (α) U (t) U (α)^{†}$ is a strongly continuous unitary group. By [F2], its generator is $- i H / ℏ$ on exactly $D (H)$ , the same generator as $U (t)$ . Uniqueness in Stone's theorem gives $V_{α} (t) = U (t)$ for every $t$ .

3.1F1step 2.2algebra

Averaging the commutation $U (α) U (t) = U (t) U (α)$ over normalized Haar measure gives $P U (t) = U (t) P$ . Thus $U (t)$ and its inverse preserve $ran P = H_{p h y s}$ , and the restricted group is unitary and strongly continuous on that closed Hilbert subspace.

4.1A1F3step 3.1algebra∎

For $x \in H_{p h y s}$ , the restricted difference quotient is the same norm limit as in $H_{k i n}$ . It exists exactly when $x \in D (H)$ by [F3]; therefore the restricted group generator has domain $D (H) \cap H_{p h y s}$ and is $- i H / ℏ$ there. Stone's theorem on $H_{p h y s}$ identifies its self-adjoint Stone operator as $- H ∣_{D (H) \cap H_{p h y s}} / ℏ$ . Since multiplication by the nonzero real scalar $- ℏ$ preserves self-adjointness, the restricted physical Hamiltonian is self-adjoint.

Global unitary dynamics in the Gauss sector

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources