  The purpose of this article is to study Ezra Getzler's approach to the
Atiyah-Singer index theorem from the perspective of Alain Connes' tangent
groupoid. We shall construct a "rescaled" spinor bundle on the tangent
groupoid, define a convolution operation on its smooth, compactly supported
sections, and explain how the algebra so-obtained incorporates Getzler's symbol
calculus.
