Dirac Operators

Dirac operator is a differential operator that is formally a square root of a second order operator like the Laplacian.
For example, consider the Laplacian on \mathbb R given by \Delta =-\dfrac{d^2}{dx^2} . The Dirac operator here is given by D=i \dfrac{d}{d x} which satisfies D^2=\Delta .

Consider the operator that acts pairs of functions on the Euclidean plane given by \displaystyle D\left(\begin{array}{l}f \\ g\end{array}\right)=2 i\left(\begin{array}{l} \frac{\partial g}{\partial z} \\ \frac{\partial f}{\partial \bar{z}} \end{array}\right) . Using \displaystyle \frac{\partial}{\partial \bar{z}}=\frac{1}{2}\left(\frac{\partial}{\partial x}+i \frac{\partial}{\partial y}\right) and \displaystyle\frac{\partial}{\partial z}=\frac{1}{2}\left(\frac{\partial}{\partial x}-i \frac{\partial}{\partial y}\right) , we get that

\displaystyle D = \left(\begin{array}{ll}0 & i \\ i & 0\end{array}\right) \frac{\partial}{\partial x}+\left(\begin{array}{cc}0 & 1 \\ -1 & 0\end{array}\right) \frac{\partial}{\partial y} .

We can see that \displaystyle D^2 = -\frac{\partial^{2}}{\partial x^{2}}-\frac{\partial^{2}}{\partial y^{2}}=\Delta .
In general, we see that the operator \displaystyle D=e_1 \frac{\partial}{\partial x} +e_2 \frac{\partial}{\partial y} with

\displaystyle e_1^2 =-I, e_2^2 =-I, e_1e_2+e_2e_1=0

satisfies \displaystyle D^2 = \Delta .
More generally we see that \displaystyle D=\sum_{i=1}^{n} e_{i} \frac{\partial}{\partial x_{i}} with

\displaystyle e_{i}^{2}=-I, \quad i=1, \ldots, n ; \quad e_{i} e_{j}+e_{j} e_{i}= O, \quad i \neq j

satisfies \displaystyle D^2 =-\sum_{i=1}^{n} \frac{\partial^2}{\partial x_{i}^2} .

These relations \displaystyle e_{i}^{2}=-I, \quad i=1, \ldots, n ; \quad e_{i} e_{j}+e_{j} e_{i}= O define a Clifford algebra corresponding to the quadratic form -x_{1}^{2}-\ldots-x_{n}^{2} .

For n=3 , the algebra is the standard quaternion algebra generated I, J, K . A representation of the algebra is the natural action on itself or we can think the algebra as \mathbb C + J \mathbb C and consider the action on the two copies of the complex numbers. This gives the Pauli matrices

\displaystyle I=\left(\begin{array}{cc}i & 0 \\ 0 & -i\end{array}\right), \quad J=\left(\begin{array}{cc}0 & -1 \\ 1 & 0\end{array}\right), \quad K=\left(\begin{array}{ll}0 & i \\ i & 0\end{array}\right)

Note that for n=2, 3 , we obtained representations with 2 \times 2 matrices and hence the operator acts on pairs of functions. More generally the smallest representation of the algebra is of dimension 2^{\left[\frac{n}{2}\right]} called the space of spinors, and hence the operators acts on functions which are spinor valued.

Clifford Algeba: A general Clifford algebra on a quadratic space (equipped with a quadratic form) is defined as the algebra generated by the orthogonal elements e_i , scalar \bf 1 with the relations e_i^2 = Q(e_i){\bf 1}, e_ie_j+e_je_i=0. . For example starting with \mathbb R and Q(x) =-x^2 , we get e^2=-1 and hence the algebra of complex numbers. If we start with Q(x, y)=-x^2-y^2 , we get the quaternion algebra seen before.

Feynman’s Dirac Operator: Dirac was trying to find the square root of the operator p^2+m^2 =-\Delta +m^2 =-\frac{\partial^{2}}{\partial x^{2}}-\frac{\partial^{2}}{\partial y^{2}}-\frac{\partial^{2}}{\partial z^{2}} +m^2. Trying a solution of the form e_1 \frac{\partial}{\partial x}+e_2 \frac{\partial}{\partial y}+e_3 \frac{\partial}{\partial z}+e_0m also gives us a some Clifford algebra relations on the anti commutators \{e_i, e_i\} = e_ie_j+e_je_i (But here the corresponding quadratic form will have a different signature, in fact we can see that e_1^2=e_2^2=e_3^2=-I, e_0^2=I). Similarly we can try to find the square root of relativistic energy relation E^2-p^2-m^2=-\frac{\partial^{2}}{\partial t^{2}}+\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y^{2}}+\frac{\partial^{2}}{\partial z^{2}} -m^2 .

Generalization to Spin Riemannian manifolds: We can try to define the Dirac operator which in local coordinates looks like

\displaystyle \sum_{i=1}^{n} e_i(x) \nabla_{e_i}

where \nabla_{e_i} are covariant derivatives with respect to Levi-Civita connection and e_i(x) is a local orthonormal frame. But to make sense of this operator on the spinor field globally, we need some topological conditions on the manifold which is called the Spin structure.

Example: Sphere (with the standard metric) satisfies the topological condition ( second Stiefel-Whitney class has to vanish).

\displaystyle D=\Gamma_1\left(\frac{1}{r} \partial_{\phi}+\frac{1}{2} \frac{\cot \phi}{r}\right)+\Gamma_2({r \sin \phi}) \partial_{\theta} , with Clifford relations

\Gamma_1^2=\Gamma_2^2 =-I, \Gamma_1\Gamma_2+\Gamma_2\Gamma_1 =0 satisfies D^2 =\Delta + \frac{1}{2r^2}. The constant term is the scalar curvature ( Lichnerowicz formula) 

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