Dirac operator

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.

Formal definitionEdit

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.


Example 1Edit

D = −ix is a Dirac operator on the tangent bundle over a line.

Example 2Edit

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 : R2C2


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.[1]

Example 3Edit

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. It can be extended to a self-adjoint operator on that domain. The square, in this case, is not the Laplacian, but instead   (after setting  )

Example 4Edit

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.

Example 5Edit

For a spin manifold, M, the Atiyah–Singer–Dirac operator is locally defined as follows: For xM and e1(x), ..., ej(x) a local orthonormal basis for the tangent space of M at x, the Atiyah–Singer–Dirac operator is


where   is 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   where   is the scalar curvature of the connection.[2]


In Clifford analysis, the operator D : C(RkRn, S) → C(RkRn, CkS) 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 i-th variable. This is a common generalization of the Dirac operator (k = 1) and the Dolbeault operator (n = 2, k arbitrary). It is an invariant differential operator, invariant under the action of the group SL(k) × Spin(n). The resolution of D is known only in some special cases.

See alsoEdit


  1. ^ "Spinor structure", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
  2. ^ Jurgen Jost, (2002) "Riemannian Geometry ang Geometric Analysis (3rd edition)", Springer. See section 3.4 pages 142 ff.