130_notes.dvi

the non-relativistic limit, thenormalization condition is a bit unnaturalin the two component
theory. The normalization correction would be very large for the “negative energy” states if we
continued to useψA.

Even though it is a second order differential equation, we only need to specify the wave function and
whether it is negative or positive energy to do the time development.The Dirac theory has many
advantages in terms of notation and ease of forming Lorentz covariant objects. A decision must be
made when we determine how many independent fields there are.

36.9 Relativistic Covariance

It is important to show that the Dirac equation, with its constantγmatrices, can becovariant.
This will come down to finding the righttransformation of the Dirac spinorψ. Remember that
spinors transform under rotations in a way quite different from normal vectors. The four components
if the Dirac spinor do not represent x, y, z, and t. We have already solved a similar problem. We
derived therotation matrices for spin^12 states, finding that they are quite different than rotation
matrices for vectors. For a rotation about thejaxis, the result was.

R(θ) = cos

θ

2

+iσjsin

θ

2

We can think ofrotations and boosts as the two basic symmetry transformationsthat we
can make in 4 dimensions. We wish to find the transformation matricesfor the equations.

ψ′ = Srot(θ)ψ

ψ′ = Sboost(β)ψ

We will work with theDirac equationand its transformation. We know how the Lorentz vectors
transform so we can derive a requirement on the spinor transformation. (Remember thataμνworks
in an entirely different space than doγμandS.)

γμ

∂

∂xμ

ψ(x) +

mc

̄h

ψ(x) = 0

γμ

∂

∂x′μ

ψ′(x′) +

mc

̄h

ψ′(x′) = 0

ψ′(x′) = Sψ(x)

∂

∂x′μ

= aμν

∂

∂xν

γμaμν

∂

∂xν

Sψ+

mc

̄h

Sψ = 0

S−^1

(

γμaμν

∂

∂xν

Sψ+

mc

̄h

Sψ

)

= 0

S−^1 γμSaμν

∂

∂xν

ψ+

mc

̄h

S−^1 Sψ = 0

(

S−^1 γμSaμν

) ∂

∂xν

ψ+

mc

̄h

ψ = 0