We consider previously derived upper and lower bounds on the number of operators in a window of scaling dimensions [\Delta - \delta,\Delta + \delta] [ Δ − δ , Δ + δ ] at asymptotically large \Delta Δ in 2d unitary modular invariant CFTs. These bounds depend on a choice of functions that majorize and minorize the characteristic function of the interval [\Delta - \delta,\Delta + \delta] [ Δ − δ , Δ + δ ] and have Fourier transforms of finite support. The optimization of the bounds over this choice turns out to be exactly the Beurling-Selberg extremization problem, widely known in analytic number theory. We review solutions of this problem and present the corresponding bounds on the number of operators for any \delta \geq 0 δ ≥ 0 . When 2\delta \in \mathbb Z_{\geq 0} 2 δ ∈ ℤ ≥ 0 the bounds are saturated by known partition functions with integer-spaced spectra. Similar results apply to operators of fixed spin and Virasoro primaries in c>1 c > 1 theories.
摘要本文系统建立三维数空间 𝕋 上的 δ-连续泛函分析理论。通过引入以双曲参数 δ 为连续谱系参数的泛函族 F[δ],本文证明了:δ→0 时 F[δ] 退化为纯波动泛函 e^{iωt},δ→+∞ 时 F[δ] 趋向纯粒子(局域)泛函,而中间 δ 值给出波粒混合态的连续过渡谱系。理论还证明了范数守恒泛函与中道偏差守恒泛函两条守恒律,并给出离散情形的黄金衰减不动点条件 δ_{n+1}=φ^{-1}δ_n。全部结论均通过数值实验严格验证。关键词:三维数;δ-连续泛函;波粒二象性;黄金分割;中道积分;守恒泛函 AbstractThis paper establishes the theory of δ-continuous functional analysis on the 3D number space 𝕋. By introducing a functional family F[δ] parameterized by the hyperbolic parameter δ, we prove: as δ→0, F[δ] reduces to the pure-wave functional e^{iωt}; as δ→+∞, F[δ] tends to a pure-particle (localized) functional; intermediate δ values give a continuous spectrum of wave-particle mixed states. Two conservation laws—norm conservation and middle-deviation conservation—are proved, and the golden-ratio discrete fixed-point condition δ_{n+1}=φ^{-1}δ_n is derived. All conclusions are rigorously verified by numerical experiments.Keywords: 3D numbers; δ-continuous functional; wave-particle duality; golden ratio; middle integration; conservation functional.
For a system with canonical variables x and p, [x,p]=i\ensuremath{\Elzxh}, we associate with each density operator \ensuremath{\rho} an x coherence \ensuremath{\delta}x and a p coherence \ensuremath{\delta}p which specify, respectively, the characteristic distance and momentum range over which the underlying quantum-mechanical nature of the ensemble cannot be neglected. These coherences are less than or equal to the associated uncertainties \ensuremath{\delta}x\ensuremath{\le}\ensuremath{\Delta}x, \ensuremath{\delta}p\ensuremath{\le}\ensuremath{\Delta}p; so the ratios
cx\ensuremath{\equiv}\ensuremath{\delta}x/\ensuremath{\Delta}x,
cp\ensuremath{\equiv}\ensuremath{\delta}p/\ensuremath{\Delta}p give measures of the quantum character of the ensemble with respect to x and p. Examples are presented.