Quantum Mechanics for Mathematicians

The four dimensional Fourier transform of a solution is of the form

ψ ̃(p) =^1

(2π)^2

∫

R^4

e−i(−p^0 x^0 +p·x)ψ(x)d^4 x

=θ(p 0 )δ(−p^20 +|p|^2 +m^2 )ψ ̃+(p) +θ(−p 0 )δ(−p^20 +|p|^2 +m^2 )ψ ̃−(p)

The Poincar ́e group acts on solutions to the Dirac equation by

ψ(x)→u(a,Λ)ψ(x) =S(Λ)ψ(Λ−^1 ·(x−a)) (47.9)

or, in terms of Fourier transforms, by

ψ ̃(p)→u ̃(a,Λ)ψ ̃(p) =e−i(−p^0 a^0 +p·a)S(Λ)ψ ̃(Λ−^1 ·p) (47.10)

Here Λ is inSpin(3,1), the double cover of the Lorentz group, and Λ·xmeans
the action ofSpin(3,1) on Minkowski space vectors. S(Λ) is the spin repre-
sentation, realized explicitly as 4 by 4 matrices by exponentiating quadratic
combinations of the Clifford algebra generators (using a chosen identification of
theγjwith 4 by 4 matrices). Spinor fieldsψcan be interpreted as elements of
the tensor product of the spinor representation space (R^4 orC^4 ) and functions
on Minkowski space. Then equation 47.9 means thatS(Λ) acts on the spinor
factor, and the action on functions is the one induced from the Poincar ́e action
on Minkowski space.
Recall that (equation 29.4) conjugation byS(Λ) takes vectorsvto their
Lorentz transformv′= Λ·v, in the sense that

S(Λ)−^1 /vS(Λ) =/v′

so

/pS(Λ)ψ ̃(Λ

− (^1) ·p) =S(Λ)
/p
′S− (^1) (Λ)S(Λ)ψ ̃(p′) =S(Λ)
/p
′ψ ̃(p′)
wherep′= Λ−^1 ·p. As a result the action 47.10 takes solutions of the Dirac
equation 47.3 to solutions, since
(i/p−m)u ̃(a,Λ)ψ ̃(p) =(i/p−m)e−i(−p^0 a^0 +p·a)S(Λ)ψ ̃(Λ−^1 ·p)
=e−i(−p^0 a^0 +p·a)S(Λ)(i/p′−m)ψ ̃(p′) = 0

47.2 Majorana spinors and the Majorana field

The analog for spin^12 of the real scalar field is known as the Majorana spinor
field, and can be constructed using a choice of real-valued matrices for the
generatorsγ 0 ,γj, acting on a four-component real-valued fieldψ. Such a choice
was given explicitly in section 41.2, and can be rewritten in terms of 2 by 2
block matrices, using the real matrices

σ 1 =

(

0 1

1 0

)

, iσ 2 =

(

0 1

−1 0

)

, σ 3 =

(

1 0

0 − 1

)