Dirac and Squeezed States

Modern physics is largely the physics of harmonic oscillators and two-by-two matrices, since otherwise problems are not soluble. Richard Feynman was not interested in writing science fictions. He was interested in solving problems.

Paul A. M. Dirac spent most of his professional time on harmonic oscillators. I was able to talk to Eugene Wigner because I can speak his secret language: two-by-two matrices. I knew two-by-two matrices are somewhat more than Pauli matrices.

Dirac and Wigner were not able to talk to each other because Dirac was only interested in four-by-four matrices, not Wigner's two-by-two. Wigner knew what harmonic oscillators are, but he seldom used them as research tools.

Thus, the best way to bridge the gap between these two great brothers-in-law is to study both harmonic oscillators and two-by-two matrices. It is incredibly easy to study both of them. It is thus a profitable business to study them and look for the gap between Wigner and Dirac. I happen to say this too often, and you probably heard this from me before.

    • There are many different ways of doing harmonic oscillators. Heisenberg constructed a non-commuting algebra. Schroedinger solved a second-order differential equation with the localization boundary condition. He then constructed step-up and step-down operators. Fock used this mathematical operation to invent his Fock space.

    Then, coupled harmonic oscillators. The problem becomes that of diagonalization of two-by-two matrices. In 1963, Paul A. M. Dirac published a paper on this subject. He of course used the method of step-up and step-down operators to deal with harmonic oscillators, but ignored the property of two-by-two matrices. Dirac ended up with the symmetry of the O(3,2) group, which is regarded as too complicated by physicists.

    In either case, Dirac's two-oscillator formalism later became the mathematical foundation for two-mode squeezed states of light in quantum optics.

    Many of you have seen the circle/ellipse logo used for the International Conference on Squeezed States and Uncertainty relations, widely known as ICSSUR. Before internet webpages became available, this conference used to circulate posters. You are invited to look at those ICSSUR posters.

    The 9th meeting of the ICSSUR series was held in Bradford (England) in 2007. The 10th meeting will take place at one of the most interesting places in quantum optics.

    In 1949, Dirac published a paper on forms of relativistic dynamics. There he invented the light-cone coordinate system. There also, Dirac did not see the two-by-two-matrix content of his light-cone coordinate system.

    It should be clear that the Lorentz boost and the squeezed state have the same mathematical content. It is not uncommon for different branches of physics to share the same mathematics. We are quite familiar with the second-order differential equation taking care of resonances in mechanical and electrical systems.

    Likewise, the mathematics of the Lorentz group (two-by-two matrices) can take care of both special relativity in high-energy physics and modern optics. At this point, ray optics is also a physics of two-by-two matrices. Click here for details.

    This paper is still under preparation. Please come again.

    Y.S.Kim