47.4 Dirac spinors

47.4 Dirac spinors

Concept links · terms present in this machine draft; source roles are unverified: eigenvalue · charge operator

In section 47.2 we saw that four-component real Majorana spinors could be used to describe neutral massive spin $\frac{1}{2}$ relativistic particles, while in section $47.3$ we saw that with two-component complex Weyl spinors one could describe charged massless spin $\frac{1}{2}$ particles. To get a theory of charged massive spin $\frac{1}{2}$ particles, one needs to double the number of degrees of freedom, which one can do in two diferent ways, with equivalent results:

• In section 44.1.2 we saw that one could get a theory of charged scalar relativistic particles, by taking scalar fields valued in ${\mathbf{R}}^2$ . One can do much the same thing for Majorana fields, having them take values in ${\mathbf{R}}^4\otimes {\mathbf{R}}^2$ rather than ${\mathbf{R}}^4$ . The theory will then, as in the scalar case, have an internal $SO(2)=U(1)$ symmetry by rotations of the ${\mathbf{R}}^2$ factor, a charge operator, and potential coupling to an electromagnetic field (using the covariant derivative). It will describe charged massive spin $\frac{1}{2}$ particles, with antiparticle states that are now distinguishable from particle states.

• Instead of a pair of Majorana spinor fields, one can take a pair ${\psi }_L,{\psi }_R$ of Weyl fields, with opposite signs of eigenvalue for $J_W$ . This will describe massive spin $\begin{array}{l}\frac{1}{2}\end{array}$ particles and antiparticles. The $U(1)$ symmetry acts in the same way on the ${\psi }_L$ and the ${\psi }_R.$ , so they have the same charge. One can also consider the $U(1)$ action that is the inverse on ${\psi }_L$ of that on ${\psi }_R,$ but it is only in the case $m=0$ that the corresponding charge commutes with the Hamiltonian (in which case the theory is said to have an “axial symmetry”).

The conventional starting point in physics textbooks is that of ${\mathbf{C}}^4-\mathrm{valued}$ spinor fields, a point of view we have avoided in order to keep straight the various ways in which complex numbers enter the theory. One should consult any quantum field theory textbook for an extensive discussion of this case, something we will not try and reproduce here. A standard topic in such textbooks is to show how the Pauli-Schr¨odinger theory of section 34 is recovered in the nonrelativistic limit, where $|\mathbf{p}|/m$ goes to zero.

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