Dirac operator

Let $D$ be a first-order differential operator acting on a vector bundle $V$ over a Riemannian manifold $M$.

If

$D^{2}=\triangle ,$

$\triangle$ 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 $D^{2}$ must equal the Laplacian.

Examples

1: $-i\partial _{x}$ 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 ${\begin{matrix}{\frac {1}{2}}\end{matrix}}$ confined to a plane, which is also the base manifold. Physicists generally think of wavefunctions $\psi :\mathbb {R} ^{2}\to \mathbb {C} ^{2}$ which they write

${\begin{pmatrix}\chi (x,y)\\\eta (x,y)\end{pmatrix}}.$

$x$ and $y$ are the usual coordinate functions on $\mathbb {R} ^{2}$. $\chi$ specifies the probability amplitude for the particle to be in the spin-up state, similarly for $\eta$. The so-called spin-Dirac operator can then be written

$D=-i\sigma _{x}\partial _{x}-i\sigma _{y}\partial _{y},$

where $\sigma _{i}$ are the Pauli matrices. Note that the anticommutation relations for the Pauli matrices make the proof of the above defining property trivial. Those commutation 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 $D=\gamma ^{\mu }\partial _{\mu }$ using Einstein's summation convention.

See also

Clifford algebra, Connection, Dolbeault operator, manifold, Riemannian geometry, Heat kernel.