130_notes.dvi

For rotations and boosts,γ 5 commutes withSsince it commutes with the pair of gamma matrices.
For a parity inversion, it anticommutes withSP =γ 4. Therefore its easy to show thatψγ ̄ 5 ψ
transforms like apseudoscalarandψiγ ̄ 5 γμψtransforms like anaxial vector. This now brings
our total to 16 components of bilinear (in the spinor) covariants. Note that things likeγ 5 σ 12 =
iγ 1 γ 2 γ 3 γ 4 γ 1 γ 2 =−iγ 3 γ 4 is just a constant times another antisymmetric tensor element, soits nothing
new.

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

Theγmatrices can be used along with Dirac spinors to make a Lorentz scalar, pseudoscalar, vector,
axial vector and rank 2 tensor. This is thecomplete set of covariants, which of course could be
used together to make up Lagrangians for physical quantities. Allsixteen quantities defined satisfy
Γ^2 = 1.

36.12Constants of the Motion for a Free Particle

We know that operators representingconstants of the motion commute with the Hamilto-
nian. The form of the Dirac equation we have been using does not have a clear Hamiltonian. This
is true essentially because of the covariant form we have been using. For a Hamiltonian formulation,
we need to separate the space and time derivatives. Lets find theHamiltonian in the Dirac
equation.
(
γμ

∂

∂xμ

+

mc

̄h

)

ψ= 0

(

γj

∂

∂xj

+γ 4

∂

∂ict

+

mc

̄h

)

ψ= 0

(

γjpj−γ 4

̄h

c

∂

∂t

−imc

)

ψ= 0

(γjpj−imc)ψ=γ 4

̄h

c

∂ψ

∂t

(γ 4 γjpj−imcγ 4 )ψ=

̄h

c

∂ψ

( ∂t

icγ 4 γjpj+mc^2 γ 4

)

ψ=Eψ

H=icγ 4 γjpj+mc^2 γ 4

Its easy to see thepkcommutes with the Hamiltonian for a free particle so thatmomentum will
be conserved.