where the polarization vectoruL(p)∈C^2 satisfies
σ·puL(p) =−p 0 uL(p)Note that theuL(p) are the same basis elements of a specificp-dependentC⊂
C^2 subspace first seen in the case of the Bloch sphere in section 7.5, and later in
the case of solutions to the Pauli equation (theu−(p) of equation 34.8 isuL(p)
for positive energy, theu+(p) of equation 34.7 isuL(p) for negative energy). The
positive energy (p 0 =|p|) solutions have negative helicity, while the negative
energy (p 0 =−|p|) solutions have positive helicity. After quantization, this
wave equation leads to a quantum field theory describing massless left-handed
helicity particles and right-handed helicity antiparticles. Unlike the case of the
Majorana field, the theory of the Weyl field comes with a non-trivial internal
symmetry, due to the action of the groupU(1) on solutions by multiplication
by a phase, and this allows the introduction of a charge operator.
Recall that our general analysis of irreducible representations of the Poincar ́e
group in chapter 42 showed that we expected to find such representations by
looking at functions on the positive and negative energy null-cones, with values
in representations ofSO(2), the group of rotations preserving the vectorp.
Acting on solutions to the Weyl equations, the generator of this group is given by
the helicity operator (equation 47.18). The solution space to the Weyl equations
provides the expected irreducible representations of helicity±^12 and of either
positive or negative energy.
47.4 Dirac spinors
In section 47.2 we saw that four-component real Majorana spinors could be used
to describe neutral massive spin^12 relativistic particles, while in section 47.3 we
saw that with two-component complex Weyl spinors one could describe charged
massless spin^12 particles. To get a theory of charged massive spin^12 particles,
one needs to double the number of degrees of freedom, which one can do in two
different 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 inR^2. One can do
much the same thing for Majorana fields, having them take values in
R^4 ⊗R^2 rather thanR^4. The theory will then, as in the scalar case,
have an internalSO(2) =U(1) symmetry by rotations of theR^2 factor, a
charge operator, and potential coupling to an electromagnetic field (using
the covariant derivative). It will describe charged massive spin^12 particles,
with antiparticle states that are now distinguishable from particle states. - Instead of a pair of Majorana spinor fields, one can take a pairψL,ψRof
Weyl fields, with opposite signs of eigenvalue forJW. This will describe
massive spin^12 particles and antiparticles. TheU(1) symmetry acts in
the same way on theψLand theψR, so they have the same charge. One