# The Lorentz Group

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These new operators define an algebra on <math>SU(2)</math> (ie are an <math>su(2)</math> algebra): | These new operators define an algebra on <math>SU(2)</math> (ie are an <math>su(2)</math> algebra): | ||

<math display="block">\left[N_i,N_j^\dagger\right]=0</math><math display="block">\left[N_i,N_j\right]=i\epsilon_{ijk}N_k,\;\left[N^\dagger_i, N^\dagger_j\right] = i\epsilon_{ijk}N^\dagger_k</math> | <math display="block">\left[N_i,N_j^\dagger\right]=0</math><math display="block">\left[N_i,N_j\right]=i\epsilon_{ijk}N_k,\;\left[N^\dagger_i, N^\dagger_j\right] = i\epsilon_{ijk}N^\dagger_k</math> | ||

− | Hence, these three operators define a set of three independent operators satisfying an <math> | + | Hence, these three operators define a set of three independent operators satisfying an <math>SU(2)</math> algebra. |

Now we are interested in understanding how to classify <math>SU(2)</math> representations. | Now we are interested in understanding how to classify <math>SU(2)</math> representations. |

## Latest revision as of 11:03, 23 December 2016

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:

*g*acting on :

**generators**of the Lorentz group, satisfying the following commutation rule:

Starting from we can construct the following operators:

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.