2024/10/16 by B. Sazdović, Sazdović, B. · 1 citation
Physics and Astronomy · #Black Holes and Theoretical Physics #Cosmology and Gravitation Theories #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Mathematical Physics (math-ph) #Quantum Chromodynamics and Particle Interactions
paper · pdf · doi:10.48550/arxiv.2410.12549
openalex publication_date 2024/10/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
There is ambitious pretension formulated by Weinberg \citeW 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 \citeWi, BW. To achieve that Ref.\citeW 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 Pa → i ∂a and introduce representation for arbitrary spin operator Sa 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 ΨA 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.