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?

  • 2
    $\begingroup$ Euclidean 2+1 dimensions? 2 space + 1 space? Or did you mean Minkowski? $\endgroup$
    – Kosm
    Jun 24, 2019 at 17:52
  • $\begingroup$ 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 $\endgroup$ Jun 24, 2019 at 18:12
  • $\begingroup$ 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) $\endgroup$
    – Kosm
    Jun 24, 2019 at 18:23
  • $\begingroup$ 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. $\endgroup$ Jun 24, 2019 at 18:28
  • 2
    $\begingroup$ How Poincare generators act on scalar, spinor or vector field is pretty much the definition of the scalar, spinor or vector field. $\endgroup$ Jun 24, 2019 at 22:21


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