# The Poincarré Group

The Poincarré group is obtained from the Lorentz one, adding the requirement of invariance under spacetime translations, that is:

${\displaystyle x^{\mu }\mapsto x^{\prime \mu }=x^{\mu }+a^{\mu }}$
Where ${\displaystyle a^{\mu }}$ is a constant spacetime vector and it depends on four parameters. Merging the Lorentz group with this new requirement we get:
${\displaystyle x^{\mu }\mapsto x^{\prime \mu }=\Lambda _{\nu }^{\mu }x^{\nu }+a^{\mu },\;\Lambda \eta \Lambda ^{T}=\eta }$
One should prove that this is a group.

Now we have to find generators of this transformation. To do so we have to find new generators inducing translations, that is:

${\displaystyle a^{\prime \mu }=x^{\mu }+\epsilon ^{\mu }\equiv x^{\mu }+i\epsilon ^{\mu }P_{\rho }x^{\mu }}$
Requiring the last identity to hold we get:
${\displaystyle i\epsilon ^{\mu }P_{\rho }x^{\mu }=i\epsilon ^{\mu }\left(-i\partial _{\rho }\right)x^{\mu }=\epsilon ^{\mu }\delta _{\rho \mu }=\epsilon ^{\mu }}$
Meaning that:
${\displaystyle P_{\rho }=-i\partial _{\rho }}$

Poincarré algebra is defined by:

${\displaystyle \left[M_{\mu \nu },M_{\rho \sigma }\right]==i\eta _{\nu \rho }M_{\mu \sigma }-i\eta _{\mu \rho }M_{\nu \sigma }-i\eta _{\nu \sigma }M_{\mu \rho }+i\eta _{\mu \sigma }M_{\nu \rho }}$
${\displaystyle \left[P_{\mu },P_{\rho }\right]=0}$
${\displaystyle \left[M_{\mu \nu },P_{\rho }\right]=-i\eta _{\mu \rho }P_{\nu }+i\eta _{\nu \rho }P_{\mu }}$
Now we want to classify representations of Poincarré algebra using suitable quantum numbers. From the discussion on Lorentz group, we know that these quantum numbers must be Casimir operators' eigenvalues. This means that we have to look for suitable Casimir operators.

• ${\displaystyle P^{2}=P^{\mu }P_{\mu }}$, which is a scalar and commutes with all the generators;
• We can no longer use Lorentz group's Casimir operator, since it does not commute with ${\displaystyle P_{\mu }}$ (${\displaystyle \left[M_{\mu \nu },P_{\rho }\right]\neq 0}$). We define the Pauli-Lubanski vector${\displaystyle W_{\mu }}$:
${\displaystyle W_{\mu }={\frac {1}{2}}\epsilon _{\mu \nu \rho \sigma }P^{\nu }M^{\rho \sigma }}$
Moreover ${\displaystyle W^{2}=W^{\mu }W_{\mu }}$ commutes with all the generators of the Poincarré group.

One can write explicitly ${\displaystyle W^{\mu }}$ as:

${\displaystyle W^{\mu }={\frac {1}{2}}\epsilon ^{\mu \nu \rho \sigma }\left(ix_{\rho }\partial _{\sigma }-ix_{\sigma }\partial _{\rho }+S_{\rho \sigma }\right)\left(-i\partial _{\nu }\right)}$
It follows:
${\displaystyle W^{\mu }=-{\frac {i}{2}}\epsilon ^{\mu \nu \rho \sigma }S_{\rho \sigma }\partial _{\nu }}$
And:
${\displaystyle W^{2}=-{\frac {1}{4}}\epsilon ^{\mu \nu \rho \sigma }\epsilon _{\mu }^{\tau \eta \xi }S_{\rho \sigma }S_{\eta \xi }\partial _{\nu }\partial _{\tau }}$
Let us now understand why these two operators has suitable eigenvalues.

• ${\displaystyle P^{2}=m^{2}>0}$, it follows that ${\displaystyle W^{2}=-m^{2}s(s+1),\;s=0,1/2,1,\dots }$. Thus, representations are now classified in terms of the couple of quantum numbers ${\displaystyle (m,s)}$. More precisely ${\displaystyle m}$ is what we identify with the particle mass and ${\displaystyle s}$ is what we identify with particle spin. Different states of given ${\displaystyle (m,s)}$ are labeled by the eigenvalues of ${\displaystyle P_{i},\,i=1,2,3}$ (giving the mass) and possible ${\displaystyle S_{3}}$ projections ${\displaystyle -s,-s+1,\dots ,s}$ (total of ${\displaystyle 2s+1}$ degrees of freedom, identify the spin).
• ${\displaystyle P^{2}=0}$, it follows that ${\displaystyle W^{2}=0}$, moreover ${\displaystyle P^{\mu }W_{\mu }=0}$. This means that ${\displaystyle W_{\mu }\propto P_{\mu }}$, and the proportional constant is ${\displaystyle \pm s}$. In this case ${\displaystyle s=0,1/2,1,\dots }$ is called helicity and this situation describes massless particles with 2 degrees of freedom (spin orientation along the direction of motion).
• ${\displaystyle P^{2}=0,\;W^{2}=-c<0}$ or ${\displaystyle P^{2}=-m^{2}<0}$, this situation does not describe any physical state.

Using these concepts of group theory, the classification of fields can be made in terms of different representations of Poincarré group. In the following we are giving a description of the different type of fields and we are developing classical field theory to understand how to interpret them as particles.