Symmetries of Dirac’s Coupled Oscillators and Dirac’s Matrices

In his 1963 paper, Dirac started from two annihilation and creation operators, and constructed four operators which resulted in sixteen linear independent operators, organized to represent two coupled harmonic oscillators. Therefore, we discuss the mathematics of two coupled harmonic oscillators as these are used in many physics models. We show the Lorentz covariance of this system. From the Schrödinger equation Schrödinger equation for two harmonic oscillators, Dirac observed that unitary transformations applicable to the ground-state function could be generated by ten combinations of his sixteen bilinear forms. He then observed that these ten combinations form the Lie algebra for the \((3 + 2)\)  de Sitter, or O(3, 2) Lorentz group. In 1976, Yuen used these bilinear forms to generate the two-photon coherent state and in 1986, Yurke et al., used them to develop the symmetries of the two photon problem or the squeezed states of light. We give a discussion of this and note that only a subset of Dirac’s operators was used. A discussion of the symmetries of the Dirac matrices and of the fifteen generators derivable from them is included. These fifteen generators form the O(3, 3) Lorentz group. Additionally, we discuss the symmetries associated with the O(3, 2) Lorentz group and detail the procedure for contracting this group to the Poincaré group.