Dirac operator on arvau

Wikipedia / Dirac operator

In mathematics and in quantum mechanics, a Dirac operator is a first-order differential operator that is a formal square root, or half-iterate, of a second-order differential operator such as a Laplacian. It was introduced in 1847 by William Hamilton and in 1928 by Paul Dirac. The question which concerned Dirac was to factorise formally the Laplace operator of the Minkowski space, to get an equation for the wave function which would be compatible with special relativity.

Formal definition

be a first-order differential operator acting on a vector bundle over a Riemannian manifold is the (positive, or geometric) Laplacian is called a Dirac operator.

Note that there are two different conventions as to how the Laplace operator is defined:

which is negative-definite for any smooth compactly supported which is not identically zero, and
the "geometric", positive-definite

History

W.R. Hamilton defined "the square root of the Laplacian" in 1847 in his series of articles about quaternions: this new characteristic will have the negative of its symbolic square expressed by the following formula of which it is clear that the applications to analytical physics must be extensive in a high degree.

Examples

Example 1

is a Dirac operator on the tangent bundle over a line.

Example 2

Consider a simple bundle of notable importance in physics: the configuration space of a particle with spin–

specifies the probability amplitude. The so-called spin-Dirac operator can then be written

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

Feynman's Dirac operator describes the propagation of a free fermion in three dimensions and is elegantly written

using the Feynman slash notation. In introductory textbooks to quantum field theory, this will appear in the form

are the off-diagonal Dirac matrices the speed of light being the Planck constant the mass of a fermion (for example, an electron, the Sobolev space of smooth, square-integrable functions. It can be extended to a self-adjoint operator)

Example 4

Another Dirac operator arises in Clifford analysis-space this is

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 and e1(x), ..., ej(x) a local orthonormal basis for the tangent space of M at x, the Atiyah–Singer–Dirac operator is

is the spin connection, a lifting of the Levi-Civita connection on M to the spinor bundle over M is the scalar curvature of the connection.

Example 6

On Riemannian manifold with Levi-Civita connection and an orthonormal basis we can define exterior derivative and coderivative as

Then we can define a Dirac-Kähler operator, as follows

The operator acts on sections of Clifford bundle in general, and it can be restricted to spinor bundle, an ideal of Clifford bundle, only if the projection operator on the ideal is parallel.

Generalisations

In Clifford analysis, the operator acting on spinor valued functions defined by

is sometimes called Dirac operator in k Clifford variables. In the notation, S are n is the Dirac operator in the i-th variable. This is a common generalization of the Dirac operator () and the Dolbeault operator (, k arbitrary). It is an invariant differential operator, invariant under the action of the group . The resolution of D is known only in some special cases.