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. It's represented by a a wavefunction ψ: R² → C²

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

   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 fermion in three dimensions and is elegantly written

   $D=\gamma ^{\mu }\partial _{\mu }\ \equiv \partial \!\!\!/,$

   using the Feynman slash notation.


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

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

   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

   $\sum _{j=1}^{n}e_{j}(x){\tilde {\Gamma }}_{e_{j}(x)}$,

   where ${\tilde {\Gamma }}$ is a lifting of the Levi-Civita connection on M to the spinor bundle over M.

See also

• Clifford algebra
• Clifford analysis
• Connection
• Dolbeault operator
• Heat kernel
• Spinor bundle