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Concept
Dirac operator
Summary
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. It was first published in 1928.
In general, let D be a first-order differential operator acting on a vector bundle V over a Riemannian manifold M. If
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.
D = −i ∂x is a Dirac operator on the tangent bundle over a line.
Consider a simple bundle of notable importance in physics: the configuration space of a particle with spin 1/2 confined to a plane, which is also the base manifold. It is represented by a wavefunction ψ : R2 → C2
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
where σi 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.
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
where are the off-diagonal Dirac matrices , with and the remaining constants are the speed of light, being Planck's constant, and the mass of a fermion (for example, an electron). It acts on a four-component wave function , the Sobolev space of smooth, square-integrable functions.
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Related courses (2)
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Introduction to Quantum Mechanics with examples related to chemistry
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Presentation of particle properties, their symmetries and interactions.
Introduction to quantum electrodynamics and to the Feynman rules.
Related publications (9)
Related concepts (8)
Gamma matrices
In mathematical physics, the gamma matrices, also called the Dirac matrices, are a set of conventional matrices with specific anticommutation relations that ensure they generate a matrix representation of the Clifford algebra It is also possible to define higher-dimensional gamma matrices. When interpreted as the matrices of the action of a set of orthogonal basis vectors for contravariant vectors in Minkowski space, the column vectors on which the matrices act become a space of spinors, on which the Clifford algebra of spacetime acts.
Spin structure
In differential geometry, a spin structure on an orientable Riemannian manifold (M, g) allows one to define associated spinor bundles, giving rise to the notion of a spinor in differential geometry. Spin structures have wide applications to mathematical physics, in particular to quantum field theory where they are an essential ingredient in the definition of any theory with uncharged fermions. They are also of purely mathematical interest in differential geometry, algebraic topology, and K theory.
Atiyah–Singer index theorem
In differential geometry, the Atiyah–Singer index theorem, proved by Michael Atiyah and Isadore Singer (1963), states that for an elliptic differential operator on a compact manifold, the analytical index (related to the dimension of the space of solutions) is equal to the topological index (defined in terms of some topological data). It includes many other theorems, such as the Chern–Gauss–Bonnet theorem and Riemann–Roch theorem, as special cases, and has applications to theoretical physics.