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

$-i\partial _{x}$ is a Dirac operator on the tangent bundle over a line.

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].

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.

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.

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 $D:C^{\infty }(\mathbb {R} ^{k}\otimes \mathbb {R} ^{n},S)\to C^{\infty }(\mathbb {R} ^{k}\otimes \mathbb {R} ^{n},\mathbb {C} ^{k}\otimes S)$ 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.

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

Examples

Generalisations

See also

References