## 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).

- 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.