130_notes.dvi

Classification Covariant Form no. of Components

Scalar ψψ ̄ 1

Pseudoscalar ψγ ̄ 5 ψ 1

Vector ψγ ̄μψ 4

Axial Vector ψγ ̄ 5 γμψ 4

Rank 2 antisymmetric tensor ψσ ̄μνψ 6

Total 16

Products of moreγmatrices turn out to repeat the same quantities because the square of anyγ
matrix is 1.

For many purposes, it is useful to write the Dirac equation in the traditional formHψ=Eψ. To do
this, we must separate the space and time derivatives, making the equation less covariant looking.
(
γμ

∂

∂xμ

+

mc

̄h

)

ψ= 0

(

icγ 4 γjpj+mc^2 γ 4

)

ψ=− ̄h

∂

∂t

ψ

Thus we can identify the operator below as the Hamiltonian.

H=icγ 4 γjpj+mc^2 γ 4

The Hamiltonian helps us identify constants of the motion. If an operator commutes withH, it
represents a conserved quantity.

Its easy to see thepkcommutes with the Hamiltonian for a free particle so thatmomentum will
be conserved. The components of orbital angular momentum do not commute withH.

[H,Lz] =icγ 4 [γjpj,xpy−ypx] = ̄hcγ 4 (γ 1 py−γ 2 px)

The components of spin also do not commute withH.

[H,Sz] = ̄hcγ 4 [γ 2 px−γ 1 py]

But, from the above, thecomponents of total angular momentum do commutewithH.

[H,Jz] = [H,Lz] + [H,Sz] = ̄hcγ 4 (γ 1 py−γ 2 px) + ̄hcγ 4 [γ 2 px−γ 1 py] = 0

The Dirac equation naturallyconserves total angular momentumbut not the orbital or spin
parts of it.

We can also see that thehelicity, or spin along the direction of motion does commute.

[H,S~·~p] = [H,S~]·~p= 0

For any calculation, we need to know the interaction term with the Electromagnetic field. Based on
the interaction of field with a current

Hint=−

1

c

jμAμ