130_notes.dvi

S−boost^1 = cosh

χ

2

−iγiγ 4 sinh

χ

2

S†rot= cos

θ

2

+γjγisin

θ

2

= cos

θ

2

−γiγjsin

θ

2

Sboost† = cosh

χ

2

−iγ 4 γisinh

χ

2

= cosh

χ

2

+iγiγ 4 sinh

χ

2

γ 4 Srot† γ 4 = cos

θ

2

−γiγjsin

θ

2

=Srot−^1

γ 4 Sboost† γ 4 = cosh

χ

2

−iγiγ 4 sinh

χ

2

=S−boost^1

γ 4 S†γ 4 =S−^1

ψ ̄′= (Sψ)†γ 4 =ψ†γ 4 γ 4 S†γ 4 =ψ†γ 4 S−^1 =ψS ̄ −^1

This also holds forSP.

SP=γ 4

SP†=γ 4

S−P^1 =γ 4

γ 4 SP†γ 4 =γ 4 γ 4 γ 4 =γ 4 =S−P^1

From this we can quickly get thatψψ ̄ is invariant under Lorentz transformations and hence is a
scalar.
ψ ̄′ψ′=ψS ̄ −^1 Sψ=ψψ ̄

Repeating the argument forψγ ̄μψwe have

ψ ̄′γμψ′=ψS ̄ −^1 γμSψ=aμνψγ ̄νψ

according to our derivation of the transformationsS. Under the parity transformation

ψ ̄′γμψ′=ψS ̄ −^1 γμSψ=ψγ ̄ 4 γμγ 4 ψ

the spacial components of the vector change sign and the fourthcomponent doesn’t. It transforms
like aLorentz vectorunder parity.

Similarly, forμ 6 =ν,
ψσ ̄μνψ≡ψiγ ̄ μγνψ

forms arank 2 (antisymmetric) tensor.

We now have 1+4+6 components for the scalar, vector and rank 2 antisymmetric tensor. To get an
axial vector and a pseudoscalar, wedefine the product of all gamma matrices.

γ 5 =γ 1 γ 2 γ 3 γ 4

which obviouslyanticommuteswith all the gamma matrices.

{γμ,γ 5 }= 0