From representations to field theories

Question

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.

This post imported from StackExchange Physics at 2014-04-25 09:28 (UCT), posted by SE-user user26143

Answer 1

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.

This post imported from StackExchange Physics at 2014-04-25 09:28 (UCT), posted by SE-user Pedro Lauridsen Ribeiro

Comments

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.

This post imported from StackExchange Physics at 2014-04-25 09:28 (UCT), posted by SE-user user1504

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@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).

This post imported from StackExchange Physics at 2014-04-25 09:28 (UCT), posted by SE-user Pedro Lauridsen Ribeiro

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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-tw‌​o-point-function/46578

This post imported from StackExchange Physics at 2014-04-25 09:28 (UCT), posted by SE-user Pedro Lauridsen Ribeiro

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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...

This post imported from StackExchange Physics at 2014-04-25 09:28 (UCT), posted by SE-user user26143

Answer 2

I'm not sure such a thing exists. Usually reps only helps you classify the kind of particles you have (i.e the quantum numbers that identify them) and how they transform under the corresponding group. I believe how to represent this particles mathematically and what is their dynamics is a different matter.

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 you may be able to construct a Lagrangian/Action which gives these as the dynamics. But this is just the usual problem of finding a Hilbert space isomorphic to the Hilbert space of quantum states.

If the solutions can be properly quantised and interpreted as quantum fields is another issue and usually problems appear. For instance, if you try to couple Rarita-Schwinger fields to electromagnetism you encounter superluminal propagation.

Anyone else have any ideas?

This post imported from StackExchange Physics at 2014-04-25 09:28 (UCT), posted by SE-user Julio Parra

Comments

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.

This post imported from StackExchange Physics at 2014-04-25 09:28 (UCT), posted by SE-user Pedro Lauridsen Ribeiro

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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.

This post imported from StackExchange Physics at 2014-04-25 09:28 (UCT), posted by SE-user user1504

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@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.

This post imported from StackExchange Physics at 2014-04-25 09:28 (UCT), posted by SE-user Pedro Lauridsen Ribeiro

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@PedroLauridsenRibeiro: My comment was directed at Julio. I'm not arguing that representations determine a Lagrangian. This fails even for the 2d chiral boson.

This post imported from StackExchange Physics at 2014-04-25 09:28 (UCT), posted by SE-user user1504

Answer 3

If you don't ask for Lagrangians but for more indirectly defined field theories you can find the required information in a series of papers by Weinberg listed in the entry ``Representations of the Poincare group, spin and gauge invariance''  http://arnold-neumaier.at/physfaq/topics/poincareRep.html of my Theoretical Physics FAQ.

Weinberg derives free field theories for all particles with any nonnegative real mass and any finite spin, and the low order interaction terms. Vol. I of his quantum field theory book contains the former but not the latter.

Infinite spin positive energy representations cannot be realized by local field theories.

Comments

The last sentence is interesting and is probably among the reasons why Lumo always says that string theory (which can for example represent Regge trajectories that can go in principle up to infinite spins while energy is positive) is not a quantum field theory ...

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This may be a misunderstanding. The infinite spin representation (also referred to as continuous spin representation) I was talking of is an irreducible representation of the Poincare group. Whereas Regge trajectories correspond to a direct sum of possibly infinitely many finite spin representations.