# Generators of the Poincaré group

I am specifically interested in constructing the generators of an Poincaré group for a 2+1 dimensional Euclidean field theory. But I am pretty new to the subject, so I would like to ask some basic questions.

I know the form of the Poincaré Algebra and I know how to proceed from the algebra to obtain Casimir invariants, but I have not found a book that explains which specific representations the generators (that obtain the commutation relations) have for the common scalar, spinor or vector field. I guess I have to start from infinitesimal field transformations, could someone help me?

• Euclidean 2+1 dimensions? 2 space + 1 space? Or did you mean Minkowski?
– Kosm
Jun 24, 2019 at 17:52
• I mean Euclidean with metric $\eta_{\mu\nu} = \delta_{\mu\nu}$ and 2 spatial dimensions + 1 time dimension. But i am happy with answers for the standard, Minkowski 4d case Jun 24, 2019 at 18:12
• If a flat metric is just the Kronecker delta, it is called Euclidean. In contrast, if there is one time dimension then it is Minkowski, in any dimension (greater than two)
– Kosm
Jun 24, 2019 at 18:23
• I think we both mean the same. I guess my formulation of my 3d space as 2+1 is misleading. I really mean 3 spacetime dimensions with a flat metric. Jun 24, 2019 at 18:28
• How Poincare generators act on scalar, spinor or vector field is pretty much the definition of the scalar, spinor or vector field. Jun 24, 2019 at 22:21