47.2.1 Majorana spinor fields in momentum space
Recall that in the case of the real relativistic scalar field studied in chapter 43 we
had the following expression (equation 43.8) for a solution to the Klein-Gordon
equation
φ(t,x) =1
(2π)^3 /^2∫
R^3(α(p)e−iωpteip·x+α(p)eiωpte−ip·x)d^3 p
√
2 ωpwithα(p) andα(p) parametrizing positive and negative energy subspaces of the
complexified phase spaceM⊗C. This was quantized by an infinite dimensional
version of the Bargmann-Fock quantization described in chapter 26, with dual
phase spaceMthe space of real-valued solutions of the Klein-Gordon equation,
and the complex structureJrthe relativistic one discussed in section 43.2.
For the Majorana theory, one can write four-component Majorana spinor
solutions to the Dirac equation as
ψ(t,x) =1
(2π)^3 /^2∫
R^3(α(p)e−iωpteip·x+α(p)eiωpte−ip·x)d^3 p
√
2 ωp(47.12)
whereα(p) is a four-component complex vector, satisfying
−γ 0 M(γM·p+im)α(p) =ωpα(p) (47.13)Theseα(p) are the positive energy solutionsψ ̃+(p) of 47.6, with the conjugate
equation forα(p) giving the negative energy solutions of 47.7 (with the sign of
pinterchanged,α(p) =ψ ̃−(−p)).
For eachp, there is a two dimensional space of solutions to equation 47.13.
One way to choose a basisu+(p),u−(p) of this space is by first considering the
casep= 0. The equation 47.13 becomes
γ 0 Mα( 0 ) =iα( 0 )which will have a basis of solutions
u+( 0 ) =1
2
1
0
0
i
, u−(^0 ) =1
2
0
1
−i
0
These will satisfy
−
i
2γM 1 γ 2 Mu+( 0 ) =1
2
u+( 0 ), −i
2γ 1 Mγ 2 Mu−( 0 ) =−1
2
u−( 0 )The two solutions
u+( 0 )e−imt+u+( 0 )eimt=
cos(mt)
0
0
sin(mt)