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 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=\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

-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 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.

Dirac Operator

Examples

See also