An Analogue of the Kac—Wakimoto Formula¶and Black Hole Conditional Entropy
An Analogue of the Kac—Wakimoto Formula¶and Black Hole Conditional Entropy
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DOI：
10.1007/s002200050116
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发表时间：
1996-05
影响因子：
2.4
通讯作者：
R. Longo
中科院分区：
文献类型：
--
作者：
R. Longo
给出了量子场论中超选择扇区维数的局部公式，即固有哈密顿量指数的真空期望值。在手性共形理论的特殊情况下，这提供了Kac和Wakimoto在某些仿射李代数表示范围内得到的全局公式的局部模拟。我们的公式与模型无关，它在一般量子场论中的版本适用于黑洞热力学。与黑洞相关的两个热平衡态之间的相对自由能与不同超选择扇区中条件熵的变化成正比，其中条件熵定义为与代表该扇区的DHR局域自同态相关的cones - st ørmer熵。比例常数是霍金温度的一半。因此，相对自由能与有理数的对数成比例量化，特别是一旦初始状态或最终状态固定，它等于整数对数的线性函数。
A local formula for the dimension of a superselection sector in Quantum Field Theory is obtained as vacuum expectation value of the exponential of the proper Hamiltonian. In the particular case of a chiral conformal theory, this provides a local analogue of a global formula obtained by Kac and Wakimoto within the context of representations of certain affine Lie algebras. Our formula is model independent and its version in general Quantum Field Theory applies to black hole thermodynamics. The relative free energy between two thermal equilibrium states associated with a black hole turns out to be proportional to the variation of the conditional entropy in different superselection sectors, where the conditional entropy is defined as the Connes—Størmer entropy associated with the DHR localized endomorphism representing the sector. The constant of proportionality is half of the Hawking temperature. As a consequence the relative free energy is quantized proportionally to the logarithm of a rational number, in particular it is equal to a linear function of the logarithm of an integer once the initial state or the final state is taken fixed.