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Categories: Differential operators | Quantum mechanics | Quantum field theory

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 $M$ .

$D^2=\triangle,$

$\triangle$ 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 $\mathbb{R}^2$ . $χ$ 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

$D=-i\sigma_x\partial_x-i\sigma_y\partial_y,$

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.

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.

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