In mathematics and quantum mechanics, a Dirac operator is a differential operator that is a formal square root, or half-iterate, of a second order operator such as a Laplacian. The original case which concerned Dirac was to factorise formally an operator for Minkowski space, to get a form of quantum theory compatible with special relativity; to get the relevant Laplacian as a product of first order operators he introduced spinors.
In general, let D be a first-order differential operator acting on a
vector bundle V over a Riemannian manifold M.
If
being the Laplacian of V, D is
called a Dirac operator.
In high-energy physics, this requirement is often relaxed: only the second-order part
of D2 must equal the Laplacian.
Examples
1: 
is a Dirac operator on the tangential bundle over a line.
2: We now consider a simple bundle of importance in physics: The configuration space of a particle
with spin 
confined to a plane, which is also the base manifold.
Physicists generally think of wavefunctions 
which they write
x and y are the usual coordinate functions on 
.
χ specifies the probability amplitude for the particle to be in the
spin-up state, similarly for η. The so-called spin-Dirac operator
can then be written
where σi are the Pauli matrices. Note that the anticommutation relations
for the Pauli matrices make the proof of the above defining property trivial. Those
relations define the notion of a Clifford algebra.
3: The most famous Dirac operator describes the propagation of a free electron in
three dimensions and is elegantly written
using Einstein's summation convention.
See also
