Dirac operator - AbsoluteAstronomy.com

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

and quantum mechanics

Quantum mechanics

Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

, a Dirac operator is a differential operator

Differential operator

In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation, accepting a function and returning another .This article considers only linear operators,...

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

Paul Dirac

Paul Adrien Maurice Dirac, OM, FRS was an English theoretical physicist who made fundamental contributions to the early development of both quantum mechanics and quantum electrodynamics...

was to factorise formally an operator for Minkowski space

Minkowski space

In physics and mathematics, Minkowski space or Minkowski spacetime is the mathematical setting in which Einstein's theory of special relativity is most conveniently formulated...

, to get a form of quantum theory compatible with special relativity

Special relativity

Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...

; to get the relevant Laplacian as a product of first-order operators he introduced spinor

Spinor

In mathematics and physics, in particular in the theory of the orthogonal groups , spinors are elements of a complex vector space introduced to expand the notion of spatial vector. Unlike tensors, the space of spinors cannot be built up in a unique and natural way from spatial vectors...

s.

In general, let

be a first-order differential operator acting on a
vector bundle

Vector bundle

In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X : to every point x of the space X we associate a vector space V in such a way that these vector spaces fit together...

over a Riemannian manifold

Riemannian manifold

In Riemannian geometry and the differential geometry of surfaces, a Riemannian manifold or Riemannian space is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, which varies smoothly from point to point...

.

If

with

being the Laplacian of

is called a Dirac operator.

In high-energy physics, this requirement is often relaxed: only the second-order part
of

must equal the Laplacian.

Examples

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,

where x and y are the usual coordinate functions on R². χ specifies the probability amplitude
Probability amplitude
In quantum mechanics, a probability amplitude is a complex number whose modulus squared represents a probability or probability density.For example, if the probability amplitude of a quantum state is \alpha, the probability of measuring that state is |\alpha|^2...

for the particle to be in the spin-up state, and similarly for η. The so-called spin-Dirac operator can then be written
where σ_i are the Pauli matrices
Pauli matrices
The Pauli matrices are a set of three 2 × 2 complex matrices which are Hermitian and unitary. Usually indicated by the Greek letter "sigma" , they are occasionally denoted with a "tau" when used in connection with isospin symmetries...

. 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
Clifford algebra
In mathematics, Clifford algebras are a type of associative algebra. As K-algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercomplex number systems. The theory of Clifford algebras is intimately connected with the theory of quadratic forms and orthogonal...

.

Solutions to the Dirac equation for spinor fields are often called harmonic spinorshttp://eom.springer.de/S/s086780.htm.

The most famous Dirac operator describes the propagation of a free fermion
Fermion
In particle physics, a fermion is any particle which obeys the Fermi–Dirac statistics . Fermions contrast with bosons which obey Bose–Einstein statistics....

in three dimensions and is elegantly written
using the Feynman slash notation
Feynman slash notation
In the study of Dirac fields in quantum field theory, Richard Feynman invented the convenient Feynman slash notation...

.

There is also the Dirac operator arising in Clifford analysis
Clifford analysis
Clifford analysis, using Clifford algebras named after William Kingdon Clifford, is the study of Dirac operators, and Dirac type operators in analysis and geometry, together with their applications...

. In euclidean n-space this is
where
is an orthonormal basis for euclidean n-space, and is considered to be embedded in a Clifford algebra
Clifford algebra
In mathematics, Clifford algebras are a type of associative algebra. As K-algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercomplex number systems. The theory of Clifford algebras is intimately connected with the theory of quadratic forms and orthogonal...

.

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 and a local orthonormal basis for the tangent space of M at x, the Atiyah-Singer-Dirac operator is,
where is a lifting of the Levi-Civita connection on M to the spinor bundle over M.

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 is the space of spinors,

are n-dimensional variables and

is the Dirac operator in the

-th variable. This is a common generalization of the Dirac operator (k=1) and the Dolbeault operator

Dolbeault cohomology

In mathematics, in particular in algebraic geometry and differential geometry, Dolbeault cohomology is an analog of de Rham cohomology for complex manifolds. Let M be a complex manifold...

(n=2, k arbitrary). It is an invariant differential operator, invariant to the action of the group

. The resolution of D is known only in some special cases.

Examples

Generalisations

See also