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 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 D² 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.

Generalisations

In Clifford analysis, the operator $D: C^\infty(\R^k\otimes \R^n,S)\to C^\infty(\R^k\otimes\R^n,\C^k\otimes S)$ acting on spinor valued functions defined by

$f(x_1,\ldots,x_k)\mapsto \begin{pmatrix} \partial_{\underline{x_1}}f\\ \partial_{\underline{x_2}}f\\ \ldots\\ \partial_{\underline{x_k}}f\\ \end{pmatrix}$

is sometimes called Dirac operator in k Clifford variables. In the notation, S is the space of spinors, $x_i=(x_{i1},x_{i2},\ldots,x_{in})$ are n-dimensional variables and $\partial_{\underline{x_i}}=\sum_j e_j\cdot \partial_{x_{ij}}$ is the Dirac operator in the i-th variable. This is a common generalization of the Dirac operator (k=1) and the Dolbeault operator (n=2, k arbitrary). It is an invariant differential operator, invariant to the action of the group $SL(k)\times Spin(n)$. The resolution of D is known only in some special cases.

See also

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

References

• Friedrich, Thomas (2000), Dirac Operators in Riemannian Geometry, American Mathematical Society, ISBN 978-0-8218-2055-1 
• Colombo, F., I.; Sabadini, I. (2004), Analysis of Dirac Systems and Computational Algebra, Birkhauser Verlag AG, ISBN 978-3764342555