黎曼流形的数算子
Number operators for Riemannian manifolds
摘要：
黎曼流形的Hodge复形上的Dirac算子d+ δ被认为是湮灭算子A。在加权空间L_mu^2 Ω上，[A，A*]与激发态上的正常数相乘当且仅当μ的测度密度的对数满足一对方程。这些方程等价于M上调和距离函数的存在性。在这些条件下，N=A*A有包含非负整数的谱。非平坦，非产品的例子。结果被概括为一个量子版本的Cheeger-Gromoll分裂定理。
英文摘要：
The Dirac operator d+delta on the Hodge complex of a Riemannian manifold is regarded as an annihilation operator A. On a weighted space L_mu^2 Omega, [A,A*] acts as multiplication by a positive constant on excited states if and only if the logarithm of the measure density of mu satisfies a pair of equations. The equations are equivalent to the existence of a harmonic distance function on M. Under these conditions N=A*A has spectrum containing the nonnegative integers. Nonflat, nonproduct examples are given. The results are summarized as a quantum version of the Cheeger--Gromoll splitting theorem.
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