Dirac operator

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 Paul 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

$D^{2}=\Delta ,$

with $\Delta$ 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 tangent bundle over a line.

2: We now consider a simple bundle of importance in physics: The configuration space of a particle with spin ¹⁄₂ confined to a plane, which is also the base manifold. Physicists generally think of wavefunctions ψ: R² → C² which they write

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

where x and y are the usual coordinate functions on R². χ specifies the probability amplitude for the particle to be in the spin-up state, and similarly for η. The so-called spin-Dirac operator can then be written

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

where σ_i 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.

Solutions to the Dirac equation for spinor fields are often called harmonic spinors[1].

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 and even more elegantly as

$D=\partial \!\!\!/$

using the Feynman slash notation.

4: There is also the Dirac operator arising in Clifford analysis. In euclidean n-space this is

$D=\Sigma _{j=1}^{n}e_{j}{\frac {\partial }{\partial x_{j}}}$

where

$\{e_{j}:j=1,\ldots ,n\}$

is an orthonormal basis for euclidean n-space.

This is a special case of the Atiyah-Singer-Dirac operator acting on sections of a spinor bundle.

See also

• Clifford algebra
• Connection
• Dolbeault operator
• Heat kernel.
• spinor bundle