# Systematically constructing a Lagrangian with given Poincare representations for particles

In one approach to constructing field theories, we begin with some desired particle content (e.g. a massive spin-1 particle), and then we construct a corresponding classical Lagrangian (e.g. the Proca Lagrangian). Upon quantizing the theory, we should obtain the desired particle contents.

How do we systematically construct a Lagrangian, given the desired particle contents (i.e. Poincare irreps)? How do we do this in general spacetime dimensions? When must we use a gauge theory?

In the above approach, we usually begin by specifying how the space of 1-particle states should transform as a unitary irreducible representation of the Poincare group. For instance, we might specify a massive spin-1 particle. Then we write down a classical Lagrangian, e.g. the Proca Lagrangian. The classical fields over spacetime transform under the Poincare group, and these representations are actually specified by giving a representation of the Lorentz group, which transforms the field labels. The resulting Poincare representation acting on spacetime field configurations is not a unitary irreducible representation, nor is it directly related to the unitary representation we hope to obtain on the Hilbert space. After canonical quantization, we do ultimately obtain a unitary irrep of Poincare on the space of 1-particle states. Roughly speaking, the original Poincare representation (acting on spacetime field configurations in the Lagrangian) is "reduced" by restricting to solutions of the equation of motion, and also sometimes by gauge-fixing. The space of classical solutions to the EOM then coincides with the space of 1-particle states in the quantized theory.

I would like to see a systematic, general procedure for carrying out the above story. The treatments I have seen generally say something like: Make sure you choose fields with enough degrees of freedom (but there are some choices here), then write down some Lorentz-invariant quadratic interaction (again some choices), then calculate the number of DOF left after implementing the EOM, perhaps implementing some gauge symmetry, and hope it works. But what is the general recipe, for general representations in general dimensions?

I understand there may not be a unique recipe; I am interested in any recipe. Also, we can focus on building free theories. If the representation-theoretic details for general spins (or spin-like parameters) in general spacetime dimensions are too difficult, I would be happy to focus first on $$d=4$$, or a schematic overview.

Related: In Weinberg, Vol. 1, Chapter 5, he describes a general procedure for defining a free field operator in the Hamiltonian formalism, in terms of ladder operators for a particle in a given Poincare irrep. Perhaps a Lagrangian may then be found via Legendre transform of the free Hamiltonian, but I'm not sure about possible subtleties.

Related questions, which do not fully answer the above:

Can Poincare representations be embedded in non-standard Lorentz representations?

From representations to field theories

Embedding particles (with spin) into fields (Schwartz qft)

Do the Wightman axioms uniquely fix the representation of the Poincaré group on the one-particle states given the representation on the fields? (asks related question in reverse direction)

• Are the massless higher spin actions of Fang and Fronsdal, and the massive higher spin actions of Singh and Hagen not what you're looking for? Mar 11, 2021 at 2:39
• SigmaAlpha, thank you -- I do think those essentially answer the question, at least for free theories, which is what I asked. I see the many problems with higher spin interactions are a whole different question. Mar 11, 2021 at 3:10