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 definition

In general, let D be a first-order differential operator acting on a vector bundle V over a Riemannian manifold M. If

$D^{2}=\Delta ,\,$

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.

Examples

Example 1

D = −i ∂_x is a Dirac operator on the tangent bundle over a line.

Example 2

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

$\psi (x,y)={\begin{bmatrix}\chi (x,y)\\\eta (x,y)\end{bmatrix}}$

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

$D=-i\sigma _{x}\partial _{x}-i\sigma _{y}\partial _{y},$

where σ_i 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.

Example 3

Feynman's Dirac operator describes the propagation of a free fermion in three dimensions and is elegantly written

$D=\gamma ^{\mu }\partial _{\mu }\ \equiv \partial \!\!\!/,$

using the Feynman slash notation. In introductory textbooks to quantum field theory, this will appear in the form

$D=c{\vec {\alpha }}\cdot (-i\hbar \nabla _{x})+mc^{2}\beta$

where ${\vec {\alpha }}=(\alpha _{1},\alpha _{2},\alpha _{3})$ are the off-diagonal Dirac matrices $\alpha _{i}=\beta \gamma _{i}$, with $\beta =\gamma _{0}$ and the remaining constants are $c$ the speed of light, $\hbar$ being Planck's constant, and $m$ the mass of a fermion (for example, an electron). It acts on a four-component wave function $\psi (x,t)\in L^{2}(\mathbb {R} ^{3},\mathbb {C} ^{4})$, 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 $D^{2}=\Delta +m^{2}$ (after setting $\hbar =c=1.$)

Example 4

Another Dirac operator arises in Clifford analysis. In euclidean n-space this is

$D=\sum _{j=1}^{n}e_{j}{\frac {\partial }{\partial x_{j}}}$

where {e_j: 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 5

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

$\sum _{j=1}^{n}e_{j}(x){\tilde {\Gamma }}_{e_{j}(x)},$

where ${\tilde {\Gamma }}$ is a lifting of the Levi-Civita connection on M to the spinor bundle over M.

Generalisations

In Clifford analysis, the operator D : C^∞(R^k ⊗ Rⁿ, S) → C^∞(R^k ⊗ Rⁿ, C^k ⊗ S) acting on spinor valued functions defined by

$f(x_{1},\ldots ,x_{k})\mapsto {\begin{pmatrix}\partial _{\underline {x_{1}}}f\\\partial _{\underline {x_{2}}}f\\\ldots \\\partial _{\underline {x_{k}}}f\\\end{pmatrix}}$

is sometimes called Dirac operator in k Clifford variables. In the notation, S is the space of spinors, $x_{i}=(x_{i1},x_{i2},\ldots ,x_{in})$ are n-dimensional variables and $\partial _{\underline {x_{i}}}=\sum _{j}e_{j}\cdot \partial _{x_{ij}}$ 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 also

• Dirac equation
• Clifford algebra
• Clifford analysis
• Connection
• Dolbeault operator
• Heat kernel
• Spinor bundle

References

• Friedrich, Thomas (2000), Dirac Operators in Riemannian Geometry, American Mathematical Society, ISBN 978-0-8218-2055-1
• Colombo, F., I.; Sabadini, I. (2004), Analysis of Dirac Systems and Computational Algebra, Birkhauser Verlag AG, ISBN 978-3-7643-4255-5