Dirac's Efforts to synthesize Quantum Mechanics and Special Relativity

You may magnify each figure by clicking on it.

Einstein formulated his theory of special relativity in 1905. Many people say this is an outdated theory. These days, you have to talk about Einstein's general relativity having to do with the universe and happenings in far-away places. On the other hand, is the formula E = mc² outdated?

Paul A. M. Dirac is the name familiar to all of us. He wrote many papers in a number of different directions. What was his main interest?

It took many agonizing years for physicists to formulate quantum mechanics, but we say quantum mechanics was born in 1927 when Heisenberg gave his interpretation to the Poisson brackets. They act as the bridges between classical mechanics and quantum mechanics, and also between the particle nature and wave nature of matter.
The question is whether the uncertainty brackets serve as the bridge between quantum mechanics and special relativity.
In 1927, Driac started worrying about whether this new theory is consistent with special relativity. He then made continued efforts to make quantum mechanics consistent with Einstein's special relativity or Lorentz covariant. Among the many papers he wrote on this subject, he made his original contributions in the following four papers.

Unlike Heisenberg, Dirac was interested in whether Heisenberg's brackets are consistent with special relativity.
1. Dirac (1927). The Quantum Theory of the Emission and Absorption of Radiation, Proc. Roy. Soc. (London) A [114], 243 - 265 (1927).
2. Dirac (1945). The Quantum Theory of the Emission and Absorption of Radiation, Proc. Roy. Soc. (London) A [A183], 284 - 295 (1945).
3. Dirac (1949). Forms of Relativistic Dynamics, Rev. Mod. Phys. [21] 392 - 399 (1949).
4. Dirac (1963). A Remarkable Representation of the 3 + 2 deSitter Group, J. Math. Phys. [4], 901 - 909 (1963).
Let us consider these four papers. They are original and deep-penetrating. No ordinary physicists could write papers like these. This does not necessarily mean that there is nothing to add to them.
1. Dirac's papers are like poems. It is a pleasure to read them. However, there are no figures, unlike Feynman who was a cartoonist (Feynman diagrams are cartoons). Let us compare Dirac and Feynman. Click here. It is thus fun to translate Dirac's poems into cartoons.
2. In his papers, Dirac never mentioned his earlier papers on the same subject. In his paper of 1945, Dirac introduced the time variable in his harmonic oscillator wave functions, but he did not mention the difficulty of incorporating the c-number time-energy uncertainty in the Lorentz-covariant world which he pointed out in his earlier paper of 1925.
  By talking about oscillator wave function, Dirac was talking about localized waves for bound states. For this system, Wignr's little group is O(3), namely the three-dimensional rotation group. This takes care of the c-number nature of the time-energy uncertainty relation. Click here for an explanation.
  
  Integration of Dirac's 1927, 1845, and 1949 papers.
  
  Ten generators of the O(3,2) group. Is it possible to convert these to the ten generators of the inhomogeneous Lorentz group?
3. In his 1949 paper, Dirac introduced his light-cone coordinate system. This paper tells that the Lorentz boost is a squeeze transformation.
  It is thus fun to combine this squeeze transformation with the oscillator wave function he introduced in his earlier paper of 1945.
  Under this squeeze transformation, the circle gets deformed into an ellipse. It is even more fun to see whether this elliptic deformation is observable in the real world. Click here for a story.
4. In his 1963 paper, Dirac constructed the O(3,2) deSitter group from two coupled oscillators. This group is Lorentz group applicable to three space coordinates and two time variables. Compared with the Lorentz group of O(3,1), this group has an extra time variable.
  Earlier in 1949, Dirac said that the task of constructing the quantum mechanics in the Lorentz-covariant world is to construct a representation of the inhomogeneous Lorentz group, consisting of three rotations, three boosts, and four translations.
Thus, it is fun to see whether it is possible to obtain the inhomogeneous Lorentz group mentioned in his 1949 paper from the O(3,2) group he obtained from two coupled oscillators. Indeed, this is possible. The key is to flatten one of the time variables to make four translation variables, separating O(3,2) into O(3,1) and four translations. This process is called the group contraction introduced by Inonu and Wigner in 1953.
We thus arrrive at a serious conclusion. The special relativity, with Einstein's E = mc², contained in the inhomogeneous Lorentz group is derivable from the Heisenberg brackets. Have you heard this story before? Click here for a published article on this subject.
We discuss coupled harmonic oscillators in undergraduate physics classes. We never expected this toy could play a role in combining quantum mechanics and special relativity into one scientific discipline. According to philosophers, this is a Hegelian synthesis.

In 2019, I published the following papers telling that Einstein's E = mc² can be derived from the Heisenberg brackets.
1. Einstein's E = mc^2 derivable from Heisenberg's Uncertainty Relations,
  with Sibel Baskal and Marilyn Noz,
  Quantum Reports [1] (2), 236 - 251 (2019),
  doi:10.3390/quantum1020021.
  ArXiv. For pdf with sharper images, click here.
2. Role of Quantum Optics in Synthesizing Quantum Mechanics and Relativity,
  Invited paper presented at the 26th International Conference on Quantum Optics and Quantum Information (Minsk, Belarus, May 2019).
  ArXiv. For pdf with sharper images, click here.
3. Poincaré Symmetry from Heisenberg's Uncertainty Relations,
  with S. Baskal and M. E. Noz,
  Symmetry [11](3), 236 - 267 (2019),
  doi:10.3390/sym11030409.
  ArXiv.

Why is he with Einstein?

Click here for Y. S. Kim's home page.