130_notes.dvi

The transformed equation will be the same as the Dirac equation ifS−^1 γμSaμν=γν. Multiply by
the inverse Lorentz transformation.

aμν(a)−νλ^1 = δμλ

aμνaλν = δμλ

S−^1 γμSaμνaλν = γνaλν

S−^1 γλS = γνaλν

S−^1 γμS=γνaμν

This is therequirement onSfor covariance of the Dirac equation.

Rotations and boosts are symmetry transformations of the coordinates in 4 dimensions. Consider
the cases of rotations about the z axis and boosts along the x direction, as examples.

a(μνrot) =







cosθ sinθ 0 0

−sinθ cosθ 0 0

0 0 1 0

0 0 0 1







a(μνboost) =







γ 0 0 iβγ

0 1 0 0

0 0 1 0

−iβγ 0 0 γ





=







coshχ 0 0 isinhχ

0 1 0 0

0 0 1 0

−isinhχ 0 0 coshχ







Theboost is just another rotation in Minkowski spacethrough and angleiχ=itanh−^1 β.
For example a boost with velocityβin the x direction is like a rotation in the 1-4 plane by an angle
iχ. Let us review theLorentz transformation for boosts in terms of hyperbolic functions.
We define tanhχ=β.

tanhχ =

eχ−e−χ

eχ+e−χ

=β

coshχ =

eχ+e−χ

2

sinhχ =

eχ−e−χ

2

cos(iχ) =

ei(iχ)+e−i(iχ)

2

=

e−χ+eχ

2

= coshχ

sin(iχ) =

ei(iχ)−e−i(iχ)

2 i

=

e−χ−eχ

2 i

=i

eχ−e−χ

2

=isinhχ

γ =

1

√

1 −β^2

=

1

√

1 −tanh^2 χ

=

1

√

(1 + tanhχ)(1−tanhχ)

=

1

√

2 eχ

eχ+e−χ

2 e−χ

eχ+e−χ

=

eχ+e−χ

2

= coshχ

βγ = tanhχcoshχ= sinhχ

a(μνboost) =







coshχ 0 0 isinhχ

0 1 0 0

0 0 1 0

−isinhχ 0 0 coshχ





