黎曼假设与短程费米格林函数
Riemann Hypothesis and Short Distance Fermionic Green's Functions
摘要：
我们证明了具有修正色散关系和短程参数$a$的二维费米子的格林函数由Lerch zeta函数给出。格林函数被定义在半径为R的圆柱体上，我们证明了条件$R = a$产生的黎曼zeta函数作为量子跃迁振幅的费米子。我们制定的黎曼假设物理上作为一个非零条件的过渡振幅之间的两个特殊的状态与原点和一个点的一半周围的圆柱体，其中每一个是固定点的$Z_2$变换。通过研究部分和，我们表明，过渡振幅制定类似于中微子混合在低维背景下。我们还导出了费米子理论的热配分函数和在温度$1/a.$下的热发散。在一个交替谐振子形式中，我们讨论了二维弦论和矩阵模型与费米子描述的关系。最后，我们推导出各种表示的格林函数使用能量动量积分，点粒子路径积分，和字符串传播。
英文摘要：
We show that the Green's function of a two dimensional fermion with a modified dispersion relation and short distance parameter $a$ is given by the Lerch zeta function. The Green's function is defined on a cylinder of radius R and we show that the condition $R = a$ yields the Riemann zeta function as a quantum transition amplitude for the fermion. We formulate the Riemann hypothesis physically as a nonzero condition on the transition amplitude between two special states associated with the point of origin and a point half way around the cylinder each of which are fixed points of a $Z_2$ transformation. By studying partial sums we show that that the transition amplitude formulation is analogous to neutrino mixing in a low dimensional context. We also derive the thermal partition function of the fermionic theory and the thermal divergence at temperature $1/a.$ In an alternative harmonic oscillator formalism we discuss the relation to the fermionic description of two dimensional string theory and matrix models. Finally we derive various representations of the Green's function using energy momentum integrals, point particle path integrals, and string propagators.
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