Research proof index

Derivative bounds for q-expansion coefficients. Given ‖a n‖ ≤ n^k, produces bounds ‖a n * 2πin * exp(2πin z)‖ ≤ 2π * n^(k+1) * exp(-2πn * y_min) on compact K ⊆ {z : 0 < z.im}. This is a key hypothesis for D_qexp_tsum_pnat.

Quantum relative entropy when σ has full rank -/ theorem qRelativeEnt_rank {ρ σ : MState d} [σ.M.NonSingular] : (𝐃(ρ‖σ) : EReal) = ⟪ρ.M, ρ.M.log - σ.M.log⟫ := by apply qRelativeEnt_ker simp [HermitianMat.nonSingular_ker_bot] section lowerSemicontinuous_1 variable {d : Type*} [Fintype d] [DecidableEq d] open scoped InnerProductSpace RealInnerProductSpace...

When a quadratic form is equivalent to a weighted sum of squares, negative weights correspond to vectors where the form takes negative values. This is a concrete realization of a 1-dimensional negative definite subspace, contributing to O'Neill's index ν (Definition 18, p. 47).

For a positive definite quadratic form, the negative dimension (index) is zero. O'Neill states (p. 47) that "ν = 0 if and only if b is positive semidefinite." Since positive definite implies positive semidefinite (Definitions 17 (1) and (2), p. 46), a positive definite form must have index ν = 0.

The canonical commutation relations: [xᵢ, pⱼ] = iℏ δᵢⱼ𝟙. -/ lemma position_commutation_momentum : ⁅𝐱 i, 𝐩 j⁆ = (I * ℏ) • δ[i,j] • ContinuousLinearMap.id ℂ 𝓢(Space d, ℂ) := by ext ψ x show 𝐱 i (𝐩 j ψ) x - 𝐩 j (𝐱 i ψ) x = _ trans (I * ℏ) * (-x i * ∂[j] ψ x + ∂[j] ((fun x : Space d ↦ x i) • ⇑ψ) x) · simp only [positionCLM_apply, momentumCLM_apply,...

⁅𝐇(ε), 𝐀(ε)ᵢ⁆ = iℏk·ε²𝐫(ε)⁻³𝐩ᵢ - 3ℏ²k/2·ε²𝐫(ε)⁻⁵𝐱ᵢ