Dirac operator

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

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

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

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

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.[1]

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

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

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

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

where ${\displaystyle {\vec {\alpha }}=(\alpha _{1},\alpha _{2},\alpha _{3})}$ are the off-diagonal Dirac matrices ${\displaystyle \alpha _{i}=\beta \gamma _{i}}$, with ${\displaystyle \beta =\gamma _{0}}$ and the remaining constants are ${\displaystyle c}$ the speed of light, ${\displaystyle \hbar }$ being Planck's constant, and ${\displaystyle m}$ the mass of a fermion (for example, an electron). It acts on a four-component wave function ${\displaystyle \psi (x)\in L^{2}(\mathbb {R} ^{3},\mathbb {C} ^{4})}$, 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 ${\displaystyle D^{2}=\Delta +m^{2}}$ (after setting ${\displaystyle \hbar =c=1.}$)

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

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

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.

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

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

where ${\displaystyle {\tilde {\Gamma }}}$ 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 ${\displaystyle D^{2}=\Delta +R/4}$ where ${\displaystyle R}$ is the scalar curvature of the connection.[2]