Poincare field theory for massive particles

TL;DR

This paper introduces a novel approach to formulating relativistic quantum field theories by directly deriving field equations as Casimir eigenvalue problems, emphasizing the role of group representations and invariants.

Contribution

It proposes a method to derive field equations from Casimir eigenvalue problems, providing a group-theoretic foundation for relativistic quantum fields, including fermions.

Findings

Derived field equations as Casimir eigenvalue problems.

Constructed projection operators for specific spins.

Obtained Dirac equation for fermions.

Abstract

There is ambitious pretension formulated by Weinberg \cite{W} that {\it any relativistic quantum theory will look at sufficiently low energy like a quantum field theory.} It is based on the observation that for formulation of quantum field theory {\it ... much better starting point is Wigner's definition of particles as representations of inhomogeneous Lorentz group} \cite{Wi, BW}. To achieve that Ref.\cite{W} starts with particles and get to the field equations later. Here we propose a complementary approach and directly introduce field equations as Casimir eigenvalue problem. Note that Casimir invariants commute with all group elements and therefore commute between each other. So, they have common eigenvalues (for Poincare group mass and spin ) and common eigenstates (here irreducible representation of Poincare group). We use derivatives as standard representation for momenta $P_a \to i \partial_a$ and introduce representation for arbitrary spin operator $ S_{a b} $ with the help of recurrence relations. To solve eigenvalue problem for Casimir operators we will go to the formulation with standard momentum, where differential equations turn to algebraic ones. Then for arbitrary field $\Psi^A $ we construct projection operators on particular spins. The irreducible representations, have a role of equations of motion. For fermions we can go to linear form of equations of motion and obtain Dirac equation.