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The one-particle states as well as the fields in quantum field theory are regarded as representations of Poincare group, e.g. scalar, spinor, and vector representations.

Is there any systematical procedure that one starts from the Dynkin label for a given representation, to construct a Lagrangian of that field theory? If yes, where can I find that procedure?

I don't care adding interaction by gauge invariance from these Lagrangians will cause non-renormalizability or not. I can live with effective theories.

user26143
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2 Answers2

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Not all irreducible representations (irrep's for short) of the Poincaré group lead to a Lagrangian. One example (see my comment to Julio Parra's answer) are the zero-mass, "continuous-helicity" (sometimes called "infinite-helicity") representations.

There is, however, a way to begin from a positive energy irrep of the Poincaré group (i.e. a 1-particle space) and construct the algebras of free (i.e. non-interacting) local observables directly, without recoursing to a Lagrangian. It is based on methods coming from operator algebras - see, for instance, R. Brunetti, D. Guido and R. Longo, Modular Localization and Wigner Particles, Rev. Math. Phys. 14 (2002) 759-786, arXiv:math-ph/0203021.

Pedro Lauridsen Ribeiro
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I don't think such a thing exists. Usually reps only help you classify the kind of objects you have (i.e the quantum numbers that identify them) and how they transform under the corresponding group. The only thing similar I know about is that some of the Poincare group reps, or actually the vector spaces that carry them, have a correspondence with the Hilbert space of solutions of some wave equation

  • spin 0 : Klein-Gordon equation

  • spin 1/2 : Dirac equation

  • spin 3/2 : Rarita-Schwinger

  • etc

and can construct a Lagrangian/Action which gives these as the dynamics.

jpm
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