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Gaussian frequency separation of connected ground correlations

Statement

Assume Countable Choice. Let $A$ be a unital complex $C^{*}$ -algebra with a strongly continuous automorphism group $α_{t}$ . Let $π : A \to B (H)$ be a contractive unital $*$ -representation on a separable Hilbert space, with linear-first inner product, and let $Ω$ be a unit vector. Suppose a strongly continuous unitary group $U_{t}$ implements $π (α_{t} (A)) = U_{t} π (A) U_{t}^{*}$ , fixes $Ω$ , and has positive self-adjoint generator $i K$ satisfying

$\ker K = C Ω, σ (K) \subseteq {0} \cup [g, \infty), g > 0.$

Write $ω (A) = ⟨ π (A) Ω, Ω ⟩$ . Let $A, B \in A$ commute, put $f (t) = ω ([A, α_{t} (B)])$ , and suppose $∣ f (t) ∣ \leq L ∣ t ∣$ for $∣ t ∣ \leq η$ , for some finite $L$ and $η > 0$ . For every $q > 0$ , the scalar integral

$I_{q} (A, B) : = \int_{R} \frac{e^{- q t^{2}}}{2 π i t} f (t) d t$

is absolutely convergent, with any value assigned to its integrand at $t = 0$ , and satisfies

$∣ ω (A B) - ω (A) ω (B) - I_{q} (A, B) ∣ \leq ∥ A ∥ ∥ B ∥ e^{- g^{2} / (4 q)} .$

The gap is that of this infinite- or finite-dimensional representation; no finite-volume approximating gaps are assumed.

Statement references

Scalar spectral measures of a separable positive generator under countable choice

Facts & Assumptions

Given: The covariant positive-energy representation, its simple ground vector and frequency gap, the commuting observables and the stated $O (t)$ bound.

[A1]

Countable Choice is assumed. The Axiom of Countable Choice ( ${A C}_{ω}$ )

[F1]

The preceding scalar spectral-measure lemma gives the Fourier pairings, support above $g$ for vectors orthogonal to the ground vector, and mixed-measure total variation bounded by the product of the vector norms. Scalar spectral measures of a separable positive generator under countable choice

[F2]

The Gaussian Fourier transform, after the change from the $2 π$ -normalized convention to angular frequency, is $\int_{R} e^{- q t^{2}} e^{i t E} d t = \sqrt{π / q} e^{- E^{2} / (4 q)}$ . Fourier transform of a Gaussian

[F3]

Scalar Fubini and dominated convergence apply to absolutely integrable product integrands and to dominated sequences, respectively. Fubini's theorem for L^1 functions on a sigma-finite product Dominated convergence

[F4]

Scalar changes of variables below are $C^{1}$ affine diffeomorphisms (the nonnegative formula extends to absolutely integrable complex functions by four signed parts). The real and complex exponential derivatives and the continuous-curve fundamental theorem evaluate the scalar exponential primitive on finite intervals, followed by its decaying endpoint limit. A C^1 diffeomorphism satisfies the change-of-variables formula for nonnegative Lebesgue measurable functions The exponential function is smooth and $(\exp)^{'} = \exp$ The complex exponential is entire and its complex derivative is itself Fundamental theorem of calculus for Banach-valued continuous curves

Proof

Given: [A1], [F1]–[F4] and the Statement hypotheses; put $Q = I - ∣ Ω ⟩ ⟨ Ω ∣$ and abbreviate $π (A), π (B)$ by $a, b$ .

1.1A1F1givenalgebra

The positive and negative frequency measures $μ_{+} = μ_{Q b Ω, Q a^{*} Ω}, μ_{-} = μ_{Q a Ω, Q b^{*} Ω}$ are carried by $[g, \infty)$ and each has total variation at most $∥ A ∥ ∥ B ∥$ , since $π$ and $Q$ are contractions. Using $U_{t} Ω = Ω$ and the linear-first pairing gives $ω (A B) - ω (A) ω (B) = μ_{+} ([g, \infty)), f (t) = \int e^{i t E} d μ_{+} (E) - \int e^{- i t E} d μ_{-} (E) .$ Indeed $ω (A α_{t} (B)) = ⟨ U_{t} b Ω, a^{*} Ω ⟩$ and $ω (α_{t} (B) A) = ⟨ U_{- t} a Ω, b^{*} Ω ⟩$ . Their ground-vector terms are both $ω (A) ω (B)$ and cancel. Commutativity also gives $f (0) = 0$ .

1.2F2F3F4givenconstructalgebra

For $ε > 0$ , the exponential primitive and its limit at infinity in [F4] give $1 / (ε + i t) = \int_{0}^{\infty} e^{- ε w} e^{- i t w} d w$ . The Gaussian factor makes the resulting double integral absolutely integrable, with majorant $e^{- q t^{2}} e^{- ε w}$ of finite integral. Hence [F2] and [F3] give $h_{q, ε} (E) : = \int_{R} \frac{e^{- q t^{2}} e^{i t E}}{2 π (ε + i t)} d t = \frac{1}{2 \sqrt{π q}} \int_{0}^{\infty} e^{- ε w} e^{- (E - w)^{2} / (4 q)} d w .$ It lies between zero and one and converges as $ε ↓ 0$ to $h_{q} (E)$ given by the same right side without $e^{- ε w}$ . For $E \geq g$ , substituting $u = (E - w) / (2 \sqrt{q})$ in its complementary Gaussian tail gives $0 \leq 1 - h_{q} (E) = h_{q} (- E) \leq \frac{1}{2} e^{- E^{2} / (4 q)} \leq \frac{1}{2} e^{- g^{2} / (4 q)} .$ For completeness, the tail estimate is $\int_{z}^{\infty} e^{- u^{2}} d u \leq e^{- z^{2}} \int_{0}^{\infty} e^{- v^{2}} d v$ for $z \geq 0$ , obtained by $u = z + v$ and $(z + v)^{2} \geq z^{2} + v^{2}$ ; the Gaussian integral in [F2] normalizes the final half.

1.3F3givenalgebra

The unregularized time integrand is integrable near zero because $∣ f (t) ∣ / ∣ t ∣ \leq L$ there. Away from zero, $∣ f (t) ∣ \leq 2 ∥ A ∥ ∥ B ∥$ , so the Gaussian divided by $∣ t ∣$ is an integrable majorant on each tail. For $t \neq 0$ one has $∣ ε + i t ∣ \geq ∣ t ∣$ . Dominated convergence in time thus proves $I_{q, ε} \to I_{q} (A, B)$ , and also establishes the asserted absolute convergence without a principal-value convention.

2.1F1F3step 1.1step 1.2algebra

Insert the two measures of step 1.1 into the regularized integral $I_{q, ε} = \int e^{- q t^{2}} f (t) / (2 π (ε + i t)) d t$ . Scalar Fubini is legitimate at fixed $ε > 0$ : their finite variations and the integrable Gaussian divided by $ε$ give an absolute majorant. Equivalently apply it to the four positive measures in each polarization formula of [F1]. Therefore $I_{q, ε} = \int h_{q, ε} (E) d μ_{+} (E) - \int h_{q, ε} (- E) d μ_{-} (E) .$ The bound $0 \leq h_{q, ε} \leq 1$ and finite variations permit dominated convergence on these measures. Its limit is $\int h_{q} (E) d μ_{+} - \int h_{q} (- E) d μ_{-}$ .

3.1step 1.1step 1.2step 2.1step 1.3algebra∎

By steps 1.1, 2.1 and 1.3, the difference from the connected correlation is the sum of an integral of $1 - h_{q} (E)$ against $μ_{+}$ and an integral of $h_{q} (- E)$ against $μ_{-}$ . Both measures are carried by $E \geq g$ . Step 1.2 bounds each multiplier by $\frac{1}{2} e^{- g^{2} / (4 q)}$ , and step 1.1 bounds each variation by $∥ A ∥ ∥ B ∥$ . The two halves give exactly the stated estimate, with no factor from replacing $A, B$ by centered operators.

Remarks

The source's Gaussian separation argument motivates this local real-time estimate. Here the spectral measures are constructed by the preceding Countable-Choice lemma, and the regularized integral and both limiting operations are proved explicitly. No external clustering theorem is used as a premise.

Depends on

Dependency tree · 2 levels

Sources

Bruno Nachtergaele and Robert Sims, Lieb–Robinson Bounds and the Exponential Clustering Theorem, arXiv:math-ph/0506030v3; section 3.2, PDF pp. 7–10, Gaussian separation and Lemma 1; the scalar regularization proof is reconstructed locally. Full 12-page article retrieved and read.