# Proof of Atiyah-Singer Index Theorem by Canonical Quantum Mechanics

## TL;DR

This paper presents a novel quantum mechanical proof of the Atiyah-Singer index theorem for Dirac operators, avoiding path-integral methods by leveraging algebraic isomorphisms and harmonic oscillator properties.

## Contribution

It introduces a direct quantum mechanical proof of the index theorem, differing from traditional heat kernel approaches, using algebraic and harmonic oscillator techniques.

## Key findings

- Proof avoids path-integral techniques
- Utilizes algebraic isomorphism between Clifford and exterior algebra
- Demonstrates quantum mechanical approach to index theorem

## 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.