# Relation between representations of the Poincaré group and linear PDEs

What is the relation between finite dimensional representations of the Poincaré group and Poincaré-invariant linear homogenous partial differential equations?

(It that simplifies the task or provides additional insights, I would also be interested in the analagous question for the connected component (of the identity) of the Poincaré group)

To provide some details: Any linear homogenous partial differential equation for some assortment of "fields" can be expressed as $A[\phi](x)=0$ where $A: C^{\infty}(\mathbb{R}^4,V)\to C^{\infty}(\mathbb{R}^4,V)$ is a linear differential operator and V is some finite dimensional, real or complex vector space equiped with a representation $\rho$ of the Poincaré group. Such an equation is called Poincaré invariant if $$A[\phi]=0 \Leftrightarrow A[\phi']=0 \text{ for all } \phi\in C^{\infty}(\mathbb{R}^4,V) \text{ and all Poincaré transformations},$$ where, given a Poincaré transformation $T:\mathbb{R}^4\to\mathbb{R}^4$, the transformed field is as usual $$\phi'(x):=\rho(T)[\phi(T^{-1}(x))].$$

Some of the more common such representations are the "Spin zero representation" which "corresponds" to the Klein-Gordon equation, or the "Dirac Spinor Representation" corresponding to the Dirac equation. I do NOT, however, JUST want "the PDE" for each (n,m) Lorentz-representation (although I must admit I do not know the general pattern). My question asks first for a complete classification of Poincaré invariant lin. hom. PDEs and then if/how you can break them up into some kind of "irreducible" decoupled equations, which then maybe relates to Wigner's classification. Certainly most interesting are equations of order at most 2. I'm also interested in PDEs of order higher than 2 if some classification can be given, however.

• FINITE dimensional representations of the Poincaré group? – Qmechanic Jun 14 '17 at 21:49
• Yes, $(V,\rho)$ is supposed to be a finite-dimensional representation. I only consider infinite-dimensional representations of the form specified in the question (the "usual" one). Of course the general case might also be interesting. – Adomas Baliuka Jun 14 '17 at 22:25