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 [...]

Integration of Dirac’s Efforts to Construct a Quantum Mechanics Which is Lorentz-Covariant

The lifelong efforts of Paul A. M. Dirac were to construct localized quantum systems in the Lorentz covariant world. In 1927, he noted that the time-energy uncertainty should be included in the Lorentz-covariant picture. In 1945, he attempted to construct a representation of the Lorentz group using a normalizable Gaussian function localized both in the space and time variables. In 1949, he introduced his instant form to exclude time-like oscillations. He also introduced the light-cone coordinate system for Lorentz boosts. Also in 1949, he stated the Lie algebra of the inhomogeneous Lorentz group can serve as the uncertainty relations in the Lorentz-covariant world. It is possible to integrate these three papers to produce the harmonic oscillator wave function which can be Lorentz-transformed. In addition, Dirac, in 1963, considered two coupled oscillators to derive the Lie algebra for the generators of the O(3,2) de Sitter group, which has ten generators. It is proven possible to contract this group to the inhomogeneous Lorentz group with ten generators, which constitute the fundamental symmetry of quantum mechanics in Einstein’s Lorentz-covariant world.

Poincaré Symmetry from Heisenberg’s Uncertainty Relations

It is noted that the single-variable Heisenberg commutation relation contains the symmetry of the S p ( 2 ) group which is isomorphic to the Lorentz group applicable to one time-like dimension and two space-like dimensions, known as the S O ( 2 , 1 ) group. According to Paul A. M. Dirac, from the uncertainty commutation relations for two variables, it possible to construct the de Sitter group S O ( 3 , 2 ) , namely the Lorentz group applicable to three space-like variables and two time-like variables. By contracting one of the time-like variables in S O ( 3 , 2 ) , it is possible to construct the inhomogeneous Lorentz group I S O ( 3 , 1 ) which serves as the fundamental symmetry group for quantum mechanics and quantum field theory in the Lorentz-covariant world. This I S O ( 3 , 1 ) group is commonly known as the Poincaré group.

DERIVATION OF THE SOURCE-FREE MAXWELL AND GRAVITATIONAL RADIATION EQUATIONS BY GROUP THEORETICAL METHODS

We derive source-free Maxwell-like equations in flat space–time for any helicity j by comparing the transformation properties of the 2(2j+1) states that carry the manifestly covariant representations of the inhomogeneous Lorentz group with the transformation properties of the two helicity j states that carry the irreducible representations of this group. The set of constraints so derived involves a pair of curl equations and a pair of divergence equations. These reduce to the free-field Maxwell equations for j = 1 and the analogous equations coupling the gravito-electric and the gravito-magnetic fields for j = 2.