Proof of Atiyah-Singer Index Theorem by Canonical Quantum Mechanics

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.