Abstract
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.
Highlights
Since 1973 [1], the present authors have been publishing papers on the harmonic oscillator wave functions which can be Lorentz-boosted, leading to a number of books [2,3,4,5]
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
It was noted that the harmonic oscillator functions exhibit the contraction and orthogonality properties quite consistent with known rules of special relativity and quantum mechanics [6,7]
Summary
Since 1973 [1], the present authors have been publishing papers on the harmonic oscillator wave functions which can be Lorentz-boosted, leading to a number of books [2,3,4,5]. In the fourth paper published in 1963, Dirac considered two coupled oscillators using step-up and step-down operators He constructed the Lie algebra (closed set of commutation relations for the generators) of the de Sitter group, known as O(3, 2), using ten quadratic forms of those step-up and step-down operators. In 1971 [25], Feynman, Kislinger, and Ravndal published a paper saying that quantum field theory works for scattering problems with running waves, harmonic oscillators may be useful for studying bound states in the relativistic world They formulated a Lorentz-invariant differential equation separable into a Klein–Gordon equation for a free hadron, and a Lorentz-invariant oscillator equation for the bound state of the quarks. Based on Dirac’s four papers listed we venture to say that this was Dirac’s ultimate purpose
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