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:

$$x^{\prime \mu }=\Lambda _{\nu }^{\mu }x^{\nu },\;\Lambda \eta \Lambda ^{T}=\eta$$

$$x^{\prime \mu }=\Lambda _{\nu }^{\mu }x^{\nu },\;\Lambda \eta \Lambda ^{T}=\eta$$

For a generic element of the Lorentz group g acting on $f(x)$$f(x)$:

$$g=e^{i\epsilon ^{\mu \nu }M_{\mu \nu }}$$

$$g=e^{i\epsilon ^{\mu \nu }M_{\mu \nu }}$$

$$f(x)\mapsto f^{\prime }(x^{\prime })=e^{i\epsilon ^{\mu \nu }M_{\mu \nu }}f(x)$$

$$f(x)\mapsto f^{\prime }(x^{\prime })=e^{i\epsilon ^{\mu \nu }M_{\mu \nu }}f(x)$$

Where $M_{\mu \nu }$$M_{\mu \nu }$ are the generators of the Lorentz group, satisfying the following commutation rule:

$$\left[M_{\mu \nu },M_{\rho \sigma }\right]=i\eta _{\nu \rho }M_{\mu \sigma }-i\eta _{\mu \rho }M_{\nu \sigma }-i\eta _{\nu \sigma }M_{\mu \rho }+i\eta _{\mu \sigma }M_{\nu \rho }$$

$$\left[M_{\mu \nu },M_{\rho \sigma }\right]=i\eta _{\nu \rho }M_{\mu \sigma }-i\eta _{\mu \rho }M_{\nu \sigma }-i\eta _{\nu \sigma }M_{\mu \rho }+i\eta _{\mu \sigma }M_{\nu \rho }$$

The most general realization of $M_{\mu \nu }$$M_{\mu \nu }$ is:

$$M_{\mu \nu }=i\left(x_{\mu }\partial _{\nu }-x_{\nu }\partial _{\mu }\right)+S_{\mu \nu }$$

$$M_{\mu \nu }=i\left(x_{\mu }\partial _{\nu }-x_{\nu }\partial _{\mu }\right)+S_{\mu \nu }$$

The first part of the generator defines spatial rotations, while the second part defines the "spin" part of $M_{\mu \nu }$$M_{\mu \nu }$ when we are considering its action on fields. The Lorentz group generators are antisymmetric $M_{\mu \nu }=-M_{\nu \mu }$$M_{\mu \nu }=-M_{\nu \mu }$ and together with the commutation rule define an algebra.

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

$$J_{i}={\frac {1}{2}}\epsilon _{ijk}M_{jk}$$

$$J_{i}={\frac {1}{2}}\epsilon _{ijk}M_{jk}$$

$$K_{i}=M_{0i}$$

$$K_{i}=M_{0i}$$

$$N_{i}={\frac {1}{2}}\left(J_{i}+iK_{i}\right),\;N_{i}^{\dagger }={\frac {1}{2}}\left(J_{i}-iK_{i}\right)$$

$$N_{i}={\frac {1}{2}}\left(J_{i}+iK_{i}\right),\;N_{i}^{\dagger }={\frac {1}{2}}\left(J_{i}-iK_{i}\right)$$

These new operators define an algebra on $SU(2)$$SU(2)$ (ie are an $su(2)$$su(2)$ algebra):

$$\left[N_{i},N_{j}^{\dagger }\right]=0$$

$$\left[N_{i},N_{j}^{\dagger }\right]=0$$

$$\left[N_{i},N_{j}\right]=i\epsilon _{ijk}N_{k},\;\left[N_{i}^{\dagger },N_{j}^{\dagger }\right]=i\epsilon _{ijk}N_{k}^{\dagger }$$

$$\left[N_{i},N_{j}\right]=i\epsilon _{ijk}N_{k},\;\left[N_{i}^{\dagger },N_{j}^{\dagger }\right]=i\epsilon _{ijk}N_{k}^{\dagger }$$

Hence, these three operators define a set of three independent operators satisfying an $SU(2)$$SU(2)$ algebra.

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

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