The Lorentz Group

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:

${\displaystyle x^{\prime \mu }=\Lambda _{\nu }^{\mu }x^{\nu },\;\Lambda \eta \Lambda ^{T}=\eta }$
For a generic element of the Lorentz group g acting on ${\displaystyle f(x)}$:
${\displaystyle g=e^{i\epsilon ^{\mu \nu }M_{\mu \nu }}}$
${\displaystyle f(x)\mapsto f^{\prime }(x^{\prime })=e^{i\epsilon ^{\mu \nu }M_{\mu \nu }}f(x)}$
Where ${\displaystyle M_{\mu \nu }}$ are the generators of the Lorentz group, satisfying the following commutation rule:
${\displaystyle \left[M_{\mu \nu },M_{\rho \sigma }\right]=i\eta _{\nu \rho }M_{\mu \sigma }-i\eta _{\mu \rho }M_{\nu \sigma }-i\eta _{\nu \sigma }M_{\mu \rho }+i\eta _{\mu \sigma }M_{\nu \rho }}$
The most general realization of ${\displaystyle M_{\mu \nu }}$ is:
${\displaystyle M_{\mu \nu }=i\left(x_{\mu }\partial _{\nu }-x_{\nu }\partial _{\mu }\right)+S_{\mu \nu }}$
The first part of the generator defines spatial rotations, while the second part defines the "spin" part of ${\displaystyle M_{\mu \nu }}$ when we are considering its action on fields. The Lorentz group generators are antisymmetric ${\displaystyle M_{\mu \nu }=-M_{\nu \mu }}$ and together with the commutation rule define an algebra.

Starting from ${\displaystyle M_{\mu \nu }}$ we can construct the following operators:

${\displaystyle J_{i}={\frac {1}{2}}\epsilon _{ijk}M_{jk}}$
${\displaystyle K_{i}=M_{0i}}$
${\displaystyle N_{i}={\frac {1}{2}}\left(J_{i}+iK_{i}\right),\;N_{i}^{\dagger }={\frac {1}{2}}\left(J_{i}-iK_{i}\right)}$
These new operators define an algebra on ${\displaystyle SU(2)}$ (ie are an ${\displaystyle su(2)}$ algebra):
${\displaystyle \left[N_{i},N_{j}^{\dagger }\right]=0}$
${\displaystyle \left[N_{i},N_{j}\right]=i\epsilon _{ijk}N_{k},\;\left[N_{i}^{\dagger },N_{j}^{\dagger }\right]=i\epsilon _{ijk}N_{k}^{\dagger }}$
Hence, these three operators define a set of three independent operators satisfying an ${\displaystyle su(2)}$ algebra.

Now we are interested in understanding how to classify ${\displaystyle SU(2)}$ representations. First of all we have to define a Casimir Operator:

In our case one can prove that ${\displaystyle N^{2}}$ and ${\displaystyle (N^{\dagger })^{2}}$ are Casimir operators, with quantum numbers respectively ${\displaystyle n=0,1/2,1,\dots }$ and ${\displaystyle m=0,1/2,1,\dots }$ We use these operators to classify Lorentz representation in terms of the couple ${\displaystyle (n,m)}$. In analogy to quantum mechanics and the representation of orbital momentum and spin, being ${\displaystyle J_{i}=N_{i}+N_{i}^{\dagger }}$ we can define ${\displaystyle n+m}$ as the spin of the representation.