34.4 The Dirac operator

34.4 The Dirac operator

Concept links · terms present in this machine draft; source roles are unverified: vector space

The above construction can be generalized to the case of any dimension $d$ as follows. Recall from chapter 29 that associated to ${\mathbf{R}}^d$ with a standard inner product, but of a general signature $(r,s)$ (where $r+s=d,r$ is the number of + signs, $s$ the number of $-\mathrm{signs})$ we have a Cliford algebra $\mathrm{Cliff}(r,s)$ with generators ${\gamma }_j$ satisfying

$$\begin{array}{rl}{\gamma }_j{\gamma }_k& =-{\gamma }_k{\gamma }_j,j\ne k\\ {\gamma }_j^2& =+1\text{for}j=1,\dots ,r{\gamma }_j^2=-1,\text{for}j=r+1,\dots ,d\end{array}$$

To any vector $\text{ v }\in {\text{ R }}^d$ with components $v_j.$ recall that we can associate a corresponding element $v$/ in the Cliford algebra by

$$\mathbf{v}\in {\mathbf{R}}^d\to \mathbf{y}=\sum _{j=1}^d{\gamma }_j{v}_j\in \mathrm{Cliff}(r,s)$$

Multiplying this Cliford algebra element by itself and using the relations above, we get a scalar, the length-squared of the vector

$${\mathbf{v}}^2=v_1^2+v_2^2\cdots +v_r^2-v_{r+1}^2-\cdots -v_d^2=|\mathbf{v}{|}^2$$

This shows that by introducing a Cliford algebra, we can find an interesting new sort of square root for expressions like $|\mathbf{v}{|}^2$ . We can define:

Definition (Dirac operator)

The Dirac operator is the operator

$$\partial /=\sum _{j=1}^d{\gamma }_j\frac{\partial }{\partial q_j}$$

This will be a first-order diferential operator with the property that its square is the Laplacian

$${\partial }^2=\frac{{\partial }^2}{\partial q_1^2}+\cdots +\frac{{\partial }^2}{\partial q_r^2}-\frac{{\partial }^2}{\partial q_{r+1}^2}-\cdots -\frac{{\partial }^2}{\partial q_d^2}$$

The Dirac operator $\partial$ acts not on functions but on functions taking values in the spinor vector space $S$ that the Cliford algebra acts on. Picking a matrix representation of the ${\gamma }_j,$ the Dirac operator will be a constant coeficient firstorder diferential operator acting on wavefunctions with dim $S$ components. In chapter 47 we will study in detail what happens for the case of $r=3,s=1$ and see how the Dirac operator there provides an appropriate wave equation with the symmetries of special relativistic space-time.

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来源版本：2025-10-20

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