Dirac-Operator
Dirac operator is a differential operator, which is a square root of Laplace's operator. The original case, with which Paul Dirac employed, was the formal factorization of an operator for the Minkowski space, which makes the quantum theory compatible with the special theory of relativity.
Definition
Be a geometric first order differential operator acting on a vector bundle over a Riemannian manifold.
Applies
Wherein a generalized Laplace operator is on, it means Dirac operator.
History
Originally Paul Dirac had considered the root of the D' Alembert operator and thus justified the relativistic quantum field theory of an electron. So he looked at the operator
Where the Dirac matrices. However, this is according to current understanding, no Dirac operator more.
The Dirac operator of a Dirac bundle
Let be a Riemannian manifold and a Dirac bundle, consisting of a Clifford module of a Hermitian metric on and a Clifford connection on. Then, the operator is
Of the Dirac bundle associated Dirac operator. In local coordinates it has the representation
Examples
Elementary example
The operator is a Dirac operator on the tangent bundle of.
Spin - Dirac operator
The configuration space of a particle with spin 1 /2, which is restricted to a plane, which forms the base manifold. It is ψ by a wave function: R2 → C2 described
Where x and y coordinates of the usual functions of R2: χ defines the probability amplitude of an upward spin state ( spin-up ), and η analogous to the spin -down. The so-called spin- Dirac operator can then be written as
Where? i the Pauli matrices. Note that the antikommutativen relations of the Pauli matrices make a proof of the above definition trivial. These relationships define the notion of Clifford algebra. Solutions of the Dirac equation for spinor fields are often called harmonic spinors.
Hodge -de Rham operator
Be an orientable Riemannian manifold and is the exterior derivative and the adjoint to the exterior derivative with respect to L ² metric operator. Then
A Dirac operator.
Atiyah -Singer - Dirac operator
There is also a Dirac operator in Clifford analysis. In n-dimensional Euclidean space, the in which is an orthonormal basis of the Euclidean space and embedded as in a Clifford algebra is considered. This is a special case of ATIYAH -Singer -Dirac operator, acting on the sections of a spinor bundle.
For a spin manifold, the Atiyah-Singer Dirac operator is defined locally as follows: For and a local orthonormal basis for the tangent space of in the Atiyah -Singer - Dirac operator, with a lifts to the Levi -Civita connection on the spinor bundle over is.
Properties
The main symbol of a generalized Laplacian is. Accordingly, the main symbol of the Dirac operator and thus both classes of differential operators are elliptic.
Generalizations
The operator, acting on the spinorwertige functions defined below,
Is in Clifford analysis often referred to as the Dirac operator in k CliffordVariablen. In this notation, S is the space of spinors are n- dimensional variables, and is the Dirac operator in the -th variable. This is a generalization of the common Dirac operator (k = 1) and the Dolbeault cohomology (n = 2, k is arbitrary). It is a differential operator, which is invariant to the operation of the group. The Injective resolution of D is known only for some special cases.