u−( 0 )e−imt+u−( 0 )eimt=
0
cos(mt)
−sin(mt)
0
correspond physically to a relativistic spin^12 particle of massmat rest, with
the first having spin “up” in the 3-direction, the second spin “down”.
The Majorana spinor field theory comes with a significant complication with
respect to the case of scalar fields. The complex four-componentα(p) provide
twice as many basis elements as one needs to describe the solutions of the Dirac
equation (put differently, they are not independent, but satisfy the relation
47.13). Quantizing using four sets of annihilation and creation operators (one
for each component ofα) would produce a quantum field theory with too many
degrees of freedom by a factor of two. The standard solution to this problem is
to make a choice of basis elements of the space of solutions for each value ofp
by defining polarization vectors
u+(p) =L(p)u+( 0 ), u−(p) =L(p)u−( 0 )HereL(p) is an element ofSL(2,C) chosen so that, acting by a Lorentz
transformation on energy-momentum vectors it takes (m, 0 ) to (ωp,p). More
explicitly, using equation 40.4, one has
L(p)(
m 0
0 m)
L†(p) =(
ωp+p 3 p 1 −ip 2
p 1 +ip 2 ωp−p 3)
Such a choice is not unique and is a matter of convention. Explicit choices are
discussed in most quantum field theory textbooks (although in a different repre-
sentation of theγ-matrices), see for instance chapter 3.3 of [67]. Note that these
polarization vectors are not the same as the Bloch sphere polarization vectors
used in earlier chapters. They are defined on the positive mass hyperboloid, not
on the sphere, and for these there is no topological obstruction to a continuous
definition.
Solutions are then written as
ψ(t,x) =
1
(2π)^3 /^2∫
R^3∑
s=±(αs(p)us(p)e−iωpteip·x+αs(p)us(p)eiωpte−ip·x)d^3 p
√
2 ωp
(47.14)
One now has the correct number of functions to parametrize pseudo-classical
complexified dual phase spaceV⊗C. These are the single-component complex
functionsα+(p),α−(p), providing elements ofVJ+r=H 1 and their conjugates
α+(p),α−(p), which provide elements ofVJ−r.
To quantize the Majorana field in a way that allows a simple understanding
of the action of the full Poincar ́e group, we need a positive-definite Poincar ́e
invariant inner product on the space of solutions of the Dirac equation. We
have already seen what the right inner product is (see equation 47.11), but
unfortunately this is not written in a way that makes Lorentz invariance man-
ifest. Unlike the case of the Klein-Gordon equation, working in momentum