  A new supersymmetric proof of the Atiyah-Singer index theorem is presented.
The Peierls bracket quantization scheme is used to quantize the supersymmetric
classical system corresponding to the index problem for the twisted Dirac
operator. The problem of factor ordering is addressed and the unique quantum
system that is relevant to the index theorem is analyzed in detail. The
Hamiltonian operator is shown to include a scalar curvature factor,
$\hbar^2R/8$. The path integral formulation of quantum mechanics is then used
to obtain a formula for the index. For the first time, the path integral
"measure" and the Feynman propagator of the system are exactly computed. The
derivation of the index formula relies solely on the definition of a Gaussian
superdeterminant. The two-loop analysis of the path integral is also carried
out. The results of the loop and heat kernel expansions of the path interal are
in complete agreement. This confirms the existence of the scalar curvature
factor in the Schr\"odinger equation and validates the supersymmetric proof of
the index theorem. Many other related issues are addressed. Finally, reviews of
the index theorem and the supersymmetric quantum mechanics are presented.
