Proof of Atiyah-Singer Index Theorem by Canonical Quantum Mechanics
Mathematical Physics
2018-07-05 v2 math.MP
Abstract
We show that the Atiyah-Singer index theorem of Dirac operator can be directly proved in the canonical formulation of quantum mechanics, without using the path-integral technique. This proof takes advantage of an algebraic isomorphism between Clifford algebra and exterior algebra in small (high temperature) limit, together with simple properties of quantum mechanics of harmonic oscillator. Compared to the proof given by heat kernel, we try to prove this theorem more quantum mechanically.
Keywords
Cite
@article{arxiv.1703.05508,
title = {Proof of Atiyah-Singer Index Theorem by Canonical Quantum Mechanics},
author = {Zixian Zhou and Xiuqing Duan and Kai-Jia Sun},
journal= {arXiv preprint arXiv:1703.05508},
year = {2018}
}