Dirac Oscillators

Dirac's Harmonic Oscillators

Paul A. M. Dirac was quite fond of harmonic oscillators, and attempted to combine quantum mechanics with special relativity by constructing unitary representations of the Lorentz group using harmonic oscillators, in his paper.

P. A. M. Dirac, Unitary representations of the Lorentz group, Proc. Roy. Soc. (London) A183, 284 (1945).

What dirac did not tell in these 1945 paper is that his harmonic oscillator can represent the c-number time-energy uncertainty relation discussed in his earlier papers published in 1927. In these papers, Dirac's main concern was the space-time asymmetry. Why there are no excitations along the time-like direction.

P. A. M. Dirac, The quantum theory of radiation, Proc. Roy. Soc. (London), A114 , 243 (1927).
P. A. M. Dirac, The quantum theory of dispersion, Proc. Roy. Soc. (London) A114 , 710 (1927).

In 1949, the Reviews of Modern Physics published a special issue to celebrate Einstein's 70th birthday. This issue contains Dirac paper entitled "Forms of Relativistic Dynamics" [Rev. Mod. Phys. 21 , 392 (1949). In this paper, he introduced his light-cone coordinate system, in which Lorentz-boost becomes a squeeze transformation.

This light-cone paper defines space-time localization, and its Lorentz covariance. However, Dirac did not consider combining this localization property with the time-energy uncertainty relation his 1945 paper on harmonic oscillators.

But Dirac later completes this job by publishing a paper in the journal of Mathematical physics entitled "A remarkable representation of the O(3,2) de Sitter group" [J. Math. Phys. 4, 901 (1963)].

It is fun to combine these papers to construct a covariant model of quantum mechanics where harmonic oscillator wave functions can be Lorentz-boosted. Click here for a detailed explanation with explicit wave functions.