Dirac operator
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.[1] 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.[2] 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, the Sobolev space of smooth, square-integrable functions.It can be extended to a self-adjoint operator on that domain.) Another Dirac operator arises in Clifford analysis.In euclidean n-space this is where {ej: j = 1, ..., n} is an orthonormal basis for euclidean n-space, and Rn 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.For a spin manifold, M, the Atiyah–Singer–Dirac operator is locally defined as follows: For x ∈ M and e1(x), ..., ej(x) a local orthonormal basis for the tangent space of M at x, the Atiyah–Singer–Dirac operator is whereis the spin connection, a lifting of the Levi-Civita connection on M to the spinor bundle over M. The square in this case is not the Laplacian, but instead, 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.[4][5][6] In Clifford analysis, the operator D : C∞(Rk ⊗ Rn, S) → C∞(Rk ⊗ Rn, Ck ⊗ S) acting on spinor valued functions defined by is sometimes called Dirac operator in k Clifford variables.is the Dirac operator in the i-th variable.