Entropy and the Spectral Action
Entropy and the Spectral Action
复制标题
DOI：
10.1007/s00220-019-03297-8
复制
发表时间：
2020-01-01
影响因子：
2.4
通讯作者：
van Suijlekom, Walter D.
中科院分区：
文献类型：
--
作者：
Chamseddine, Ali H.;Connes, Alain;van Suijlekom, Walter D.
我们计算的信息论冯诺依曼熵的状态相关联的费米子第二量子化的频谱三元组。我们表明，这个熵是由一个特定的通用功能的频谱三元组的频谱行动。我们的论文的主要结果是这个函数和黎曼zeta函数之间的令人惊讶的关系。它特别地通过系数c（d）的值来表现自己，系数c（d）乘以谱三元组的热膨胀中的d维项。我们发现c（d）是黎曼xi函数在-d处的初等表达式的乘积。特别是c（4）是zeta（5）的有理倍数，c（2）是zeta（3）的有理倍数。该函数方程给出了正维系数和负维系数之间的对偶性，正维系数决定了高能膨胀，负维系数将偶数维与奇数维交换。
We compute the information theoretic von Neumann entropy of the state associated to the fermionic second quantization of a spectral triple. We show that this entropy is given by the spectral action of the spectral triple for a specific universal function. The main result of our paper is the surprising relation between this function and the Riemann zeta function. It manifests itself in particular by the values of the coefficients c(d) by which it multiplies the d dimensional terms in the heat expansion of the spectral triple. We find that c(d) is the product of the Riemann xi function evaluated at -d by an elementary expression. In particular c(4) is a rational multiple of zeta(5) and c(2) a rational multiple of zeta(3). The functional equation gives a duality between the coefficients in positive dimension, which govern the high energy expansion, and the coefficients in negative dimension, exchanging even dimension with odd dimension.