Quantum Mechanics for Mathematicians

as follows

γ 0 M=

(

0 −iσ 2

−iσ 2 0

)

, γ 1 M=

(

σ 3 0

0 σ 3

)

γM 2 =

(

0 iσ 2

−iσ 2 0

)

, γM 3 =

(

−σ 1 0

0 −σ 1

)

Quadratic combinations of Clifford generators have a basis

γM 0 γ 1 M=

(

0 σ 1

σ 1 0

)

, γ 0 Mγ 2 M=

(

−1 0

0 1

)

, γM 0 γ 3 M=

(

0 σ 3

σ 3 0

)

γ 1 Mγ 2 M=

(

0 σ 1

−σ 1 0

)

, γM 2 γ 3 M=

(

0 −σ 3

σ 3 0

)

, γ 1 Mγ 3 M=

(

−iσ 2 0

0 −iσ 2

)

and one has

γ 5 M=iγ 0 MγM 1 γ 2 Mγ 3 M=

(

σ 2 0

0 −σ 2

)

The quantized Majorana field can be understood as an example of a quanti-
zation of a pseudo-classical fermionic oscillator system (as described in section
30.3.2), by the fermionic analog of the Bargmann-Fock quantization method
(as described in section 31.3). We take as dual pseudo-classical phase spaceV
the real-valued solutions of the Dirac equation in the Majorana representation.
Using values of the solutions att= 0, continuous basis elements ofVare given
by the four component distributional field Ψ(x), with components Ψa(x) for
a= 1, 2 , 3 ,4.
This spaceVcomes with an inner product

(ψ,φ) =

∫

R^3

ψT(x)φ(x)d^3 x (47.11)

In this form the invariance under translations and under spatial rotations is
manifest, with theS(Λ) acting by orthogonal transformations on the Majorana
spinors when Λ is a rotation. One way to see this is to note that theS(Λ) in
this case are exponentials of linear combinations of the antisymmetric matrices
γM 1 γ 2 M,γ 2 Mγ 3 M,γ 1 MγM 3 , and thus are orthogonal matrices.
As we have seen in chapter 30, the fermionic Poisson bracket of a pseudo-
classical system is determined by an inner product, with the one above giving
in this case
{Ψa(x),Ψb(x′)}+=δ^3 (x−x′)δab

The Hamiltonian that will give a pseudo-classical system evolving according to
the Dirac equation is

h=

1

2

∫

R^3

ΨT(x)γ 0 (γ·∇−m)Ψ(x)d^3 x

One can see this by noting that the operatorγ 0 (γ·∇−m) is minus its adjoint
with respect to the inner product 47.11, sinceγ 0 is an antisymmetric matrix,