Dirac operator: Difference between revisions
No edit summary |
|||
Line 16: | Line 16: | ||
2: We now consider a simple bundle of importance in physics: The configuration space of a particle |
2: We now consider a simple bundle of importance in physics: The configuration space of a particle |
||
with spin <sup>1</sup>⁄<sub>2</sub> confined to a plane, which is also the base manifold. |
with spin <sup>1</sup>⁄<sub>2</sub> confined to a plane, which is also the base manifold. It's represented by a wavefunction ψ: '''R'''<sup>2</sup> → '''C'''<sup>2</sup> |
||
:<math>\begin{bmatrix}\chi(x,y) \\ \eta(x,y)\end{bmatrix}</math> |
:<math>\begin{bmatrix}\chi(x,y) \\ \eta(x,y)\end{bmatrix},</math> |
||
where ''x'' and ''y'' are the usual coordinate functions on '''R'''<sup>2</sup>. |
where ''x'' and ''y'' are the usual coordinate functions on '''R'''<sup>2</sup>. |
||
Line 31: | Line 31: | ||
Solutions to the Dirac equation for spinor fields are often called ''harmonic spinors''[http://eom.springer.de/S/s086780.htm]. |
Solutions to the Dirac equation for spinor fields are often called ''harmonic spinors''[http://eom.springer.de/S/s086780.htm]. |
||
3: The most famous Dirac operator describes the propagation of a free |
3: The most famous Dirac operator describes the propagation of a free [[fermion]] in |
||
three dimensions and is elegantly written |
three dimensions and is elegantly written |
||
:<math>D=\gamma^\mu\partial_\mu\,</math> |
:<math>D=\gamma^\mu\partial_\mu\ \equiv \partial\!\!\!/,</math> |
||
using [[Einstein's summation convention]] and even more elegantly as |
|||
:<math>D=\partial\!\!\!/</math> |
|||
using the [[Feynman slash notation]]. |
using the [[Feynman slash notation]]. |
Revision as of 17:38, 31 March 2011
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 be a first-order differential operator acting on a vector bundle over a Riemannian manifold .
If
with being the Laplacian of , is called a Dirac operator.
In high-energy physics, this requirement is often relaxed: only the second-order part of must equal the Laplacian.
Examples
1. 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. It's represented by a wavefunction ψ: R2 → C2
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
where σi 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 fermion in three dimensions and is elegantly written
using the Feynman slash notation.
4: There is also the Dirac operator arising in Clifford analysis. In euclidean n-space this is
where
is an orthonormal basis for euclidean n-space, and 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 and a local orthonormal basis for the tangent space of M at x, the Atiyah-Singer-Dirac operator is , where is a lifting of the Levi-Civita connection on M to the spinor bundle over M.