Dirac's Poem for Squeezed States and Entangled Oscillators

According to Karl Marx, the point is to change the world. From the ancient times, many people changed the world by writing poems. The Book of Psalms from the Old Testament makes us happy and give us inspirations.


    Dirac's poem on two entangled oscillators (above),
    Dirac in 1962 while he was writing this poem.

  • In 1963 paper in J. Math. Phys. 4 , 901, Dirac considered two oscillators. Again, he started writing a poem with step-up and step-down operators. In so doing, he ended up with ten bilinear operators stupefying a closed set of commutation relations. Click here for their expressions.

  • Dirac observed that these operators satisfy the set of commutation relations for the O(3,2) deSitter group applicable to the space consisting of three space-like and two time-like components.

  • He also observed that it is also possible to construct a set of generators for four-by-four matrices performing canonical transformations for two sets of position and momentum variables.

  • Dirac did not mention whether this formalism can be applied to the photon number states. However, we can forgive him in view of his contribution to quantization procedure for electromagnetic field, as Pauli explained in his 1943 review paper.

  • It is thus quite safe to say that Dirac wrote the first paper leading to two-photon coherent states, or squeezed states.

  • In addition, this set of operators leads to two harmonic oscillators entangled to each other. What is so great about engangled oscillators? The answer is very simple. Quantum physics starts from two-level systems and harmonic oscillators. In order to understand the mechanism of entanglement, it is essential to work out the entangled oscillator system.

    Photo of Dirac by Bulent Atalay.


    The phase spce to the first oscillator expands, and while that for
    the second oscillator shrinks.

  • His ten oscillator-based generators can be converted to ten four-by-four matrices applicable to four dimensional phase space consisting two position and two momentum variables. They can also be written as ten four-by-four matrices with imaginary elements. Click here for detailed computations to produce these matrices from Dirac's oscillator-based generators.

    Let ug go back to the Dirac equation. There are fifteen Dirac matrices in the Majorana representation. This set includes all ten matrices needed for squeezed states or entanglement. How about the remaining five matrices?

    If we take them into account, the result is that the phase space for the first oscillator can expand like a thermal expansion, while the other shrinks without a lower limit. This result raises the following possibilities.

    1. The phase space expands for the first oscillator, while it is shrinking for the second oscillator. If the second oscillator is not observed, the first oscillator goes through a thermal excitation. This point was discussed in terms of Feynman's rest of the universe in my recent paper with Marilyn Noz.

    2. For the two-oscillator system, the uncertainty relation should be defined in terms of the four-dimensional phase space. The minimum uncertainty could be defined in terms of the minimum volume.

    These questions could be debated in the future. Indeed, Dirac's matrix poems and oscillator poems lead to these intriguing questions.

Dirac met people, but he seldom communicated with them. Why?


Who is this young man?
copyright@2013 by Y. S. Kim, unless otherwise specified. The photo of Dirac and Feynman is from the Caltech Photo Archives. This photo was taken by Marek Holzman during the International Conference on Relativitic Theory of Gravitation in Warsaw (Poland) on July 25-31 1962, organized by Leopold Infeld.