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arXiv · 1703.05508

Proof of Atiyah-Singer Index Theorem by Canonical Quantum Mechanics

Zixian Zhou·Xiuqing Duan·Kai-Jia Sun
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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 $\tau$ (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.

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Zixian Zhou, Xiuqing Duan, Kai-Jia Sun. 2017-03-16. Proof of Atiyah-Singer Index Theorem by Canonical Quantum Mechanics. https://doi.org/10.1088/1751-8121/aac546

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