手征共形场论中区间的模能量范围
The modular energy range of an interval in chiral conformal field theory
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中文总结 AI 辅助
该论文精确确定了手征共形场论中区间模哈密顿量在有界能量态下的取值范围，发现上界为线性、下界为对数，并给出尖锐的Bekenstein界。
中文摘要 AI 辅助
我们精确确定了有界能量态下手征共形场论中区间的模哈密顿量的取值范围。对于中心荷为 $c$ 的理论中长度为 $R$ 的区间，以及平均能量至多为 $E$ 的态，模能量的上确界为 $M_{\max} = \frac{\pi R}{2}E + \frac{c}{6} - \frac{\pi c^{2}}{288\\,RE} + O((RE)^{-2})$，且这两个常数均不可改进。下确界是对数形式的，$M_{\min} \simeq -\frac{c}{12}[\ln(24\pi RE/c)-1]$，因此范围在上方是线性的，在下方是对数的，两者在 $RE \to 0$ 时相遇；它在每个能量下都严格更宽，超过 $\frac{2}{5}\pi RE$，这经过 Stieltjes 替换后归结为方差的非负性。两个界均可达到，证明它们的单参数权重族正是答案的 Legendre 结构。\n 这两个组成部分清晰地分离。系数 $\pi R/2$ 仅由 Möbius 表示论给出：互补权重是特殊共形生成元的平移，因此是正的，所以对于完整的模哈密顿量，该界不需要附加常数。常数 $c/6$ 正是将该权重在端点处截断的代价，由 Fewster 和 Hollands 的量子能量不等式提供；饱和态将其负能量恰好放置在那里。因此该常数在纠缠点上可加，对于 $n$ 个等长不相交区间为 $nc/6$。与相对熵的非负性复合后，得到尖锐的 Bekenstein 界，$\Delta S_I \leq \frac{\pi R}{2}E + \frac{c}{6}$，从本身不施加约束的模形式恢复了该不等式的原始形式。
英文摘要
We determine exactly the range of the modular Hamiltonian of an interval in a chiral conformal field theory over states of bounded energy. For an interval of length $R$ in a theory of central charge $c$, and states of mean energy at most $E$, the supremum of the modular energy is $M_{\max} = \frac{πR}{2}E + \frac{c}{6} - \frac{πc^{2}}{288\,RE} + O((RE)^{-2})$, and neither constant can be improved. The infimum is logarithmic, $M_{\min} \simeq -\frac{c}{12}[\ln(24πRE/c)-1]$, so the range is linear above and logarithmic below, the two meeting as $RE \to 0$; it is strictly wider above, by more than $\frac{2}{5}πRE$ at every energy, which reduces after a Stieltjes substitution to the positivity of a variance. Both bounds are attained, the one-parameter family of weights proving them being the Legendre structure of the answer. The two ingredients separate cleanly. The coefficient $πR/2$ is Möbius representation theory alone: the complementary weight is a translate of the special conformal generator, hence positive, so for the full modular Hamiltonian the bound needs no additive constant. The constant $c/6$ is precisely the price of truncating that weight at the endpoints, supplied by the quantum energy inequality of Fewster and Hollands; the saturating state places its negative energy exactly there. The constant is accordingly additive over entangling points, being $nc/6$ for $n$ disjoint intervals of equal length. Composing with positivity of relative entropy gives a sharp Bekenstein bound, $ΔS_I \leq \frac{πR}{2}E + \frac{c}{6}$, recovering the original form of that inequality from the modular form that by itself imposes no constraint.
发表机构
- Department of Computer Science, University of York（约克大学计算机科学系）
- College of Physics, Jilin University（吉林大学物理学院）
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