We then have in particular
as well as
For a pure rotation
The case
For the case of j = 1/2 it is easy to give an explicit expression for the matrices
Using
we have
or recalling (2.14), we may write this also as
We thus conclude that
Defining
we may thus write the matrices (2.34) in the compact form
These matrices will play a central role in our discussion of the Dirac equation in Chapter 4. Their explicit form is most easily obtained by returning to (2.35) and noting that
Now,
Hence
Similarly
These explicit expressions will prove useful in our discussion of the Dirac equation in Chapter 4.
For the rest of these lectures we set c = 1.
2.5Transformation properties of massive 1-particle states
In analogy to the Galilei transformations discussed in Chapter 1, we take U[L(
where
with normalization
Note that the spin of a particle at rest is a well defined quantity, whereas for a moving relativistic particle this is not the case. The kinematical factor introduced in (2.39) compensates for the non-covariant normalization of the 1-particle states:
The form of this kinematical factor can be motivated in the following way: The normalization (2.42) of the 1-particle states corresponds to the completeness relation
Now, d3p is not a relativistically invariant integration measure, whereas d3p/ω(
The delta-function insures the proper energy momentum relation for a free particle,
while the theta-function insures that the vector pμ is time-like, that is, the particle has positive energy. Both properties are preserved by Lorentz transformations in
and
We thus conclude that
This explains roughly the origin of the kinematical factor in (2.39).10 Now let U[Λ] be the unitary operator inducing a Lorentz transformation on the 1-particle state |
