Dirac operator

from Wikipedia, the free encyclopedia

The Dirac operator is a differential operator that is a square root of the Laplace operator . The original case Paul Dirac dealt with was the formal factorization of an operator for the Minkowski space , which makes quantum theory compatible with special relativity .

definition

Let it be a geometrical differential operator of the first order that acts on a vector bundle over a Riemannian manifold . If then

holds, where a generalized Laplace operator is on , it is called a Dirac operator.

history

Paul Dirac originally considered the root of the D'Alembert operator and wanted to establish the relativistic quantum field theory of an electron with it.

Dirac considered the differential operator for n = 4

where are the Dirac matrices . However, according to today's understanding, this is no longer a Dirac operator.

In the 1960s Michael Francis Atiyah and Isadore M. Singer took up this differential operator defined by Dirac and developed the (generalized) Dirac operator mainly described here in this article. The name Dirac operator was coined by Atiyah and Singer. The operator greatly influenced mathematics and mathematical physics in the 20th century.

The Dirac operator of a Dirac bundle

Let it 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

the Dirac operator associated with the Dirac bundle . He has the representation in local coordinates

Examples

Elementary example

The operator is a Dirac operator over the tangent bundle of .

Spin Dirac operator

Am considered, the configuration space of a particle with spin 1 / 2 , that the level is restricted, which forms the manifold base. The state is described by a wave function ψ G with two complex components, which should therefore apply in each case , whereby overall states that differ by only one complex factor are identified. So the overall condition is:

Here, and the usual Cartesian coordinates : defines the probability amplitude for the uplink component spin (spin-up), and similarly for the spin-down component. The so-called Spin-Dirac operator can then be written as

where σ x and σ x are the Pauli matrices . Note that the anti-commutative relationships of the Pauli matrices make a proof of the above definition trivial. These relationships define the concept of Clifford algebra # Examples using the example of quaternion algebra . Solutions of the Dirac equation for spinor fields are often called harmonic spinors .

Hodge De Rham Operator

Let be an orientable Riemannian manifold and let be the outer derivative and the operator adjoint to the outer derivative with respect to the L²-metric . Then

a Dirac operator.

Atiyah-Singer-Dirac operator

There is also a Dirac operator in Clifford analysis . In n-dimensional Euclidean space, i.e. H. for is where is an orthonormal basis of Euclidean space and is embedded in a Clifford algebra . This is a special case of the Atiyah-Singer-Dirac operator that acts on the sections of a spinor bundle .
    

    

For a spin manifold , the Atiyah-Singer-Dirac operator is locally defined as follows: For and a local orthonormal basis for the tangent space of in is the Atiyah-Singer-Dirac operator , with a parallel transport of the Levi-Civita relation on for the spinor bundle is over .

    

properties

The main symbol of a generalized Laplace operator is . Correspondingly, the main symbol of a Dirac operator is and thus both classes of differential operators are elliptic .

Generalizations

The operator that acts on the spinor-valued functions defined below,

is often mentioned in Clifford analysis as a Dirac operator in k Clifford variables. In this notation, S the space of spinors, are n -dimensional variables and is the Dirac operator in th variables. This is a common generalization of the Dirac operator ( k = 1 ) and the Dolbeault cohomology ( n = 2 , k arbitrary). It is a differential operator that is invariant to the operation of the group . The injective resolution of D is only known for a few special cases.

See also

literature

credentials

  1. Liviu I. Nicolaescu: Lectures on the geometry of manifolds. 2nd edition. World Scientific Pub Co., Singapore et al. 2007, ISBN 978-981-270853-3 , p. 498
  2. Herbert Schröder: Functional Analysis . 2. corr. Edition. Harri Deutsch, 2000, ISBN 3-8171-1623-3 , pp. 364 .
  3. Yanlin Yu: The index theorem and the heat equation method . 1st edition. World Scientify, Singapore 2001, ISBN 981-02-4610-2 , pp. 195 .
  4. DV Alekseevskii (originator): Spinor structure . Encyclopedia of Mathematics
  5. Liviu I. Nicolaescu: Lectures on the geometry of manifolds. 2nd edition. World Scientific Pub Co., Singapore et al. 2007, ISBN 978-981-270853-3 , p. 499