can also consider theU(1) action that is the inverse onψLof that onψR,
but it is only in the casem= 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 ofC^4 -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 non-
relativistic limit, where|p|/mgoes to zero.
47.5 For further reading
The material of this chapter is discussed in detail in every textbook on relativis-
tic quantum field theory. These discussions usually start with the massive Dirac
spinor case, only later restricting to the massless case and Weyl spinors. They
may or may not contain some discussion of the Majorana spinor case, usually
described by imposing a condition on the Dirac case removing half the degrees
of freedom (see for instance chapter 48 of [53]).