We will classify particles and fields in terms of quantum numbers of representations of groups. More precisely we will use the Poincarré group. Now we are giving a description of the Lorentz groups (i.e. boosts and rotations) and then, after a quick review on groups, we will give a brief description of the Poicarré group.
As we have seen in the first chapter, Lorentz transformations are described by:
For a generic element of the Lorentz group g
are the generators
of the Lorentz group, satisfying the following commutation rule:
The most general realization of
The first part of the generator defines spatial rotations, while the second part defines the "spin" part of
when we are considering its action on fields. The Lorentz group generators are antisymmetric
and together with the commutation rule define an algebra.
Starting from we can construct the following operators:
These new operators define an algebra on
(ie are an
Hence, these three operators define a set of three independent operators satisfying an
Now we are interested in understanding how to classify representations.
First of all we have to define a Casimir Operator:
In our case one can prove that and are Casimir operators, with quantum numbers respectively and
We use these operators to classify Lorentz representation in terms of the couple . In analogy to quantum mechanics and the representation of orbital momentum and spin, being we can define as the spin of the representation.