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.

Formal definition

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, \,

where ∆ is the Laplacian of V, then 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

Example 1: D=-i ∂_x is a Dirac operator on the tangent bundle over a line.

Example 2: We now consider a simple bundle of importance in physics: The configuration space of a particle with spin ½ confined to a plane, which is also the base manifold. It's represented by a wavefunction ψ: R² → C²

\psi(x,y) = \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.

Example 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.

Example 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, ..., n} is an orthonormal basis for euclidean n-space, and Rⁿ 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.

Example 5: For a spin manifold, M, the Atiyah-Singer-Dirac operator is locally defined as follows: For x ∈ M and e₁(x), ..., 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^∞(R^k ⊗ Rⁿ, S) → C^∞(R^k ⊗ Rⁿ, C^k ⊗ 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 under the action of the group SL(k) × 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 978-3-7643-4255-5

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Dirac operator

Formal definition

Examples

Generalisations

See also

References