Why Are Two Indices Used for the Generators of Lorentz and Poincare Groups?

[Bobhawke]

Just a quick question here: I was going through my notes and I noticed that the generators of both these groups are labeled two indices. I was wondering if there is any particular reason for this, since it seems to me that one index would work perfectly well.

Thanks

 

[humanino]

There is no more reason to attach 2 indices to a Lorentz transformation than to an ordinary 3D rotation, or a matrix in general. A rotation will be specified by angles between one axis (first index) and another axis (second index). If it pleases you, you can store them in a vector, but that is clumsy.

 

[Entropia]

for your question! The reason for using two indices for the generators of the Lorentz and Poincare groups is because these groups are defined in terms of vector and tensor transformations in four-dimensional spacetime. The two indices represent the four components of a Lorentz or Poincare transformation, with one index for the spacetime position and one for the momentum or energy. This allows for a more complete and accurate description of these transformations, as they involve both spatial and temporal components. Additionally, using two indices helps to maintain the symmetry of the group and allows for easier manipulation of the generators in calculations. I hope this helps clarify the use of two indices for the generators in these groups.