130_notes.dvi

cosh^2

χ

2

−sinh^2

χ

2

=

1

4

((e

χ 2

+e

− 2 χ

)^2 −(e

χ 2

−e

− 2 χ

)^2 )

=

1

4

(eχ+ 2 +e−χ−eχ+ 2−e−χ) = 1

γ 1 coshχ+iγ 4 sinhχ = a 1 νγν

aμν=







coshχ 0 0 isinhχ

0 1 0 0

0 0 1 0

−isinhχ 0 0 coshχ







γ 1 coshχ+iγ 4 sinhχ=γ 1 coshχ+iγ 4 sinhχ

That checks forγ 1. Now, tryγ 4.

γ 4 cosh^2

χ

2

+iγ 4 γ 1 γ 4 cosh

χ

2

sinh

χ

2

−iγ 1 γ 4 γ 4 cosh

χ

2

sinh

χ

2

+γ 1 γ 4 γ 4 γ 1 γ 4 sinh^2

χ

2

= a 4 νγν

γ 4 cosh^2

χ

2

−iγ 1 cosh

χ

2

sinh

χ

2

−iγ 1 cosh

χ

2

sinh

χ

2

+γ 4 sinh^2

χ

2

= −isinhχγ 1 + coshχγ 4

γ 4 cosh^2

χ

2

− 2 iγ 1 cosh

χ

2

sinh

χ

2

+γ 4 sinh^2

χ

2

= −isinhχγ 1 + coshχγ 4

γ 4 coshχ−iγ 1 sinhχ = −isinhχγ 1 + coshχγ 4

That one also checks. As a last test, tryγ 2.

γ 2 cosh^2

χ

2

+iγ 2 γ 1 γ 4 cosh

χ

2

sinh

χ

2

−iγ 1 γ 4 γ 2 cosh

χ

2

sinh

χ

2

+γ 1 γ 4 γ 2 γ 1 γ 4 sinh^2

χ

2

= γ 2

γ 2 cosh^2

χ

2

+iγ 2 γ 1 γ 4 cosh

χ

2

sinh

χ

2

−iγ 2 γ 1 γ 4 cosh

χ

2

sinh

χ

2

−γ 2 sinh^2

χ

2

= a 2 νγν

γ 2 = γ 2

The Dirac equation is therefore shown to beinvariant under boosts along thexidirection if
we transform the Dirac spinor according toψ′=Sboostψwith the matrix

Sboost= cosh

χ

2

+iγiγ 4 sinh

χ

2

and tanhχ=β.

The pure rotation about the z axis should also be verified.

(

cos

θ

2

+γ 1 γ 2 sin

θ

2

)− 1

γμ

(

cos

θ

2

+γ 1 γ 2 sin

θ

2

)

= aμνγν

(

cos

θ

2

−γ 1 γ 2 sin

θ

2

)

γμ

(

cos

θ

2

+γ 1 γ 2 sin

θ

2

)

= aμνγν

γμcos^2

θ

2

+γμγ 1 γ 2 cos

θ

2

sin

θ

2

−γ 1 γ 2 γμcos

θ

2

sin

θ

2

−γ 1 γ 2 γμγ 1 γ 2 sin^2

θ

2

= aμνγν