# From representations to field theories

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.

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.

• Seems worth mentioning that Brunetti et al only construct free fields, as it's not clear if the OP thinks that the representations uniquely determine the dynamics. Mar 7 '14 at 20:31
• @user1504 - Yes, I should have mentioned that. I'll amend my answer accordingly. On the other hand, if you look at the last paragraph of the question, it seemed to me that the OP wanted to add the interaction term at a later stage by minimal coupling, so it seemed reasonable to me to assume that he wanted to get the appropriate "free" part first (please user26143, do correct me if I'm wrong). Mar 7 '14 at 22:21
• In due time: I raised related points in my answer to the following related physics.SE question: physics.stackexchange.com/questions/13488/to-construct-an-action-from-a-given-two-point-function/46578 Mar 7 '14 at 22:34
• Thank you so much for your answer. Yes. I want to get the Lagrangian for the free field first. Excuse me, would you provide a reference for the non-existance of the Lagrangian for the "continuous-helicity" representations? I looked at arXiv:math-ph/0203021, I don't have the access for the ref[30] "G.J. Iverson, G. Mack, Quantum fields and interaction of massless particles: the continuous spin case, Ann. of Phys. 64 (1971) 211-253" at this moment... Mar 8 '14 at 5:43
• User26143, try this... sciencedirect.com/science/article/pii/0003491671902843 May 7 '14 at 4:51

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.

• There is a subfamily of irreps of the Poincaré group - namely, the zero-mass, "continuous-helicity" representations - which does not admit any Lagrangian formulation whatsoever. Mar 7 '14 at 15:49
• You can make a much stronger statement: The representations do not uniquely determine a Lagrangian. You have to add other assumptions to get dynamical laws. Mar 7 '14 at 20:33
• @user1504 - Sorry if I'm picky, but that would be a(n important!) "non-uniqueness" statement, whereas my counter-example is rather a "non-existence" one, so both statements deal with different issues. Mar 7 '14 at 22:26
• @PedroLauridsenRibeiro: My comment was directed at Julio. I'm not arguing that representations determine a Lagrangian. This fails even for the 2d chiral boson. Mar 8 '14 at 2:39