Dirac operator

From Wikipedia, the free encyclopedia

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 = Δ,

with $Δ$ 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.

[edit] 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, and $\mathbb{R}^{n}$ is considered to be embedded in a Clifford algebra.

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

5: For a spin manifold, M, the Atiyah-Singer-Dirac operator is locally defined as follows: For $x\in M$ and $e_{1}(x),\ldots,e_{j}(x)$ a local orthonormal basis for the tangent space of M at x, the Atiyah-Singer-Dirac operator is $\Sigma_{j=1}^{n}e_{j}(x)\tilde{\Gamma}_{e_{j}(x)}$ , where $\tilde{\Gamma}$ is a lifting of the Levi-Civita connection on M to the spinor bundle over M.

[edit] See also

Dirac operator

From Wikipedia, the free encyclopedia

[edit] Examples

[edit] See also

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