Elements of Group Theory

The mathematical language which encodes the symmetry properties in physics is group theory. In this chapter we recall the main results. We introduce the concepts of finite and infinite groups, that of group representations and the Clebsch–Gordan decomposition. We study, in particular, Lie groups and Lie algebras and give the Cartan classification. Some simple examples include the groups U(1), SU(2) – and its connection to O(3) – and SU(3). We use the method of Young tableaux in order to find the properties of products of irreducible representations. Among the non-compact groups we focus on the Lorentz group, its relation with O(4) and SL(2,C), and its representations. We construct the space of physical states using the infinite-dimensional unitary representations of the Poincaré group.

Gauging the Higher-Spin-Like Symmetries by the Moyal Product. II

Continuing the study of the Moyal Higher Spin Yang–Mills theory started in our previous paper we provide a detailed discussion of matter coupling and the corresponding tree-level amplitudes. We also start the investigation of the spectrum by expanding the master fields in terms of ordinary spacetime fields. We note that the spectrum can be consistent with unitarity while still preserving Lorentz covariance, albeit not in the usual way, but by employing an infinite-dimensional unitary representation of the Lorentz group.

Editorial to the Special Issue “80 Years of Professor Wigner’s Seminal Work: On Unitary Representations of the Inhomogeneous Lorentz Group”

The present Special Issue is dedicated to celebrate 80 years of the Professor Eugene Paul Wigner paper “On Unitary Representations of the Inhomogeneous Lorentz Group”, published in 1939 [...]

CPTM Symmetry for the Dirac Equation and Its Extended Version Based on the Vector Representation of the Lorentz Group

We revisit the CPT theorem for the Dirac equation and its extended version based on the vector representation of the Lorentz group. Then it is proposed that CPTM may apply to this fundamental equation for a massive fermion a s a singlet or a doublet with isospin. The symbol M stands here for reversing the sign of the mass in the Dirac equation, which can be accomplished by operation on it with the so-called gamma-five matrix that plays an essential role for the chirality in the Standard Model. We define the CPTM symmetry for the standard and extended Dirac equation and discuss its physical implications and some possible consequences for general relativity.

Spin-Valued Four Bosons Electrodynamics

Electromagnetism is based on electric charge and spin. The study here corresponds to understand on spin effects at a vectorial electrodynamics. Its scenario is a non-linear abelian electromagnetism where the electric charge is transmitted through a four bosons quadruplet, constituted by the usual photon, massive photon and charged massive photons. These four bosons intermediate the charge exchange ΔQ = 0, ±1.The spin is introduced at first principles. A spintronics Lagrangian for four vector fields is performed. Considering that spin is a space-time physical entity derived from Lorentz Group, these vector fields are associated to Lorentz Group, as Lie algebra valued. Similarly to non-abelian gauge theories where Aμ≡ Aμ,ata, one introduces the relationship Aμ≡ Aμ,κλΣκλ where (Σκλ)αβ is the Lorentz Group generator. Thus, based on three fundamentals which are light invariance, electric charge conservation law and vector fields Lie algebra valued through Lorentz Group generators, one derives a spin-valued four vectorial electrodynamics. It is given by the fields quadruplet Aμ1 ≡ {Aμ, Uμ, Vμ±} where Aμ means the usual photon, Uμ a massive photon and Vμ± massive charged photons. Two novelties appear. The first one is that, new terms are developed into usual four bosons electromagnetism. They contribute to Lagrangian, equations of motion, Noether theorem. The second one is that the equations of motion derive a renormalizable spin coupling with the electric and magnetic fields.There is a spin-1 electrodynamics to be investigated. A neutral electromagnetism is mandatory to be analyzed. Something beyond dipole, quadrupole and so on. Understand the role of spin in the electrical and magnetic properties of particles. A spin vectorial expression S--> is obtained. It adds EM interactions not depending on electric charge and with spin interactions through electric dipole and magnetic moments.