标题： 黎曼流形的数量算子 显示英文标题

标题： Number operators for Riemannian manifolds

摘要： 狄拉克算子 d+delta 在黎曼流形的霍奇复形上被视为湮灭算子 A。 在加权空间 L_mu^2 Omega 上，[A,A*] 在激发态上作为乘以正常数作用当且仅当测度密度的对数满足一对方程。 这些方程等价于 M 上存在一个调和距离函数。 在这些条件下 N=A*A 的谱包含非负整数。 给出了非平坦、非乘积的例子。 结果被总结为切赫-格罗莫尔分裂定理的量子版本。 显示英文摘要

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.