Editing Atiyah–Singer index theorem (section)

==Proof techniques==

===Pseudodifferential operators===
{{main|pseudodifferential operator}}
Pseudodifferential operators can be explained easily in the case of constant coefficient operators on Euclidean space. In this case, constant coefficient differential operators are just the [[Fourier transform]]s of multiplication by polynomials, and constant coefficient pseudodifferential operators are just the Fourier transforms of multiplication by more general functions.

Many proofs of the index theorem use pseudodifferential operators rather than differential operators. The reason for this is that for many purposes there are not enough differential operators. For example, a pseudoinverse of an elliptic differential operator of positive order is not a differential operator, but is a pseudodifferential operator. 
Also, there is a direct correspondence between data representing elements of K(B(''X''), ''S''(''X'')) (clutching functions) and symbols of elliptic pseudodifferential operators.

Pseudodifferential operators have an order, which can be any real number or even −∞, and have symbols (which are no longer polynomials on the cotangent space), and elliptic differential operators are those whose symbols are invertible for sufficiently large cotangent vectors. Most versions of the index theorem can be extended from elliptic differential operators to elliptic pseudodifferential  operators.

===Cobordism===
The initial proof was based on that of the [[Hirzebruch–Riemann–Roch theorem]] (1954), and involved [[cobordism theory]] and [[pseudo-differential operator|pseudodifferential operator]]s.

The idea of this first proof is roughly as follows. Consider the ring generated by pairs (''X'', ''V'') where ''V'' is a smooth vector bundle on the compact smooth oriented manifold ''X'', with relations that  the sum and product of the ring on these generators are given by disjoint union and product of manifolds (with the obvious operations on the vector bundles), and any boundary of a manifold with vector bundle is 0. This is similar to the cobordism ring of oriented manifolds, except that the manifolds also have a vector bundle. The topological and analytical indices are both reinterpreted as functions from this ring to the integers. Then one checks that these two functions  are in fact both ring homomorphisms. In order to prove they are the same, it is then only necessary to check they are the same on a set of generators of this ring. Thom's cobordism theory gives a set of generators; for example, complex vector spaces with the trivial bundle together with certain bundles over even dimensional spheres. So the index theorem can be proved by checking it on these particularly simple cases.

===K-theory===
Atiyah and Singer's first published proof used [[K-theory]] rather than cobordism. If ''i'' is any inclusion of compact manifolds from ''X'' to ''Y'', they defined a 'pushforward' operation ''i''<sub>!</sub> on elliptic operators of ''X'' to elliptic operators of ''Y'' that preserves the index. By taking ''Y'' to be some sphere that ''X'' embeds in, this reduces the index theorem to the case of spheres. If ''Y'' is a sphere and ''X'' is some point embedded in ''Y'', then any elliptic operator on ''Y'' is the image under ''i''<sub>!</sub> of some elliptic operator on the point. This reduces the index theorem to the case of a point, where it is trivial.

===Heat equation===
{{harvs|txt=yes|last=Atiyah |author2-link=Raoul Bott|last2=Bott|author3-link=Vijay Kumar Patodi|last3=Patodi|year=1973}} gave a new proof of the index theorem using the [[heat equation]], see e.g. {{harvtxt|Berline|Getzler|Vergne|1992}}. 
The proof is also published in {{harv|Melrose|1993}} and {{harv|Gilkey|1994}}.

If ''D'' is a differential operator with adjoint ''D*'', then ''D*D'' and ''DD*'' are self adjoint operators whose non-zero eigenvalues have the same  multiplicities. However their zero eigenspaces may have different multiplicities, as these multiplicities are the dimensions of the kernels of ''D'' and ''D*''. Therefore, the index of ''D'' is given by 
:<math>\operatorname{index}(D) = \dim \operatorname{Ker}(D) - \dim \operatorname{Ker}(D^*) = \dim \operatorname{Ker}(D^*D) - \dim \operatorname{Ker}(DD^*) = \operatorname{Tr}\left(e^{-t D^* D}\right) - \operatorname{Tr}\left(e^{-t DD^*}\right)</math>

for any positive ''t''. The right hand side is given by the trace of the difference of the kernels of two heat operators. These have an asymptotic expansion for small positive ''t'', which can be used to evaluate the limit as ''t'' tends to 0, giving a proof of the Atiyah–Singer index theorem.  The asymptotic expansions for small ''t'' appear very complicated, but invariant theory shows that there are huge cancellations between the terms, which makes it possible to find the leading terms explicitly. These cancellations were later explained using supersymmetry.