Soft Spots in Dirac's Papers
and Feynman's Papers

On the issue of the Poincar� group

Photo from the AIP E. Segre Visual Archives. Dirac and Wigner were brothers-in-law. Click here for a story. These two brothers-in-law seldom talked about physics.

• Dirac was mostly interested in bound-state problems and localized probability distributions. He used harmonic oscillators whenever appropriate. In his 1949 paper on the Forms of Relativistic Dynamics in Rev. Mod. Phys. 21 , 392 (1949), Dirac stated that the task of formulating relativistic quantum mechanics is equivalent to constructing a suitable representation of the Poincar� group.

  If this is so, why did he not use harmonic oscillators to construct the desired representation? Why did he not talk with his brother-in-law about the problem? His brother-in-law was Eugene Wigner who started constructing representations of the Poincar� group in 1939.

  Yes, this creates an interesting homework problem for us. Click here for the solution of this problem.

  Feynman giving his talk at the 1970 APS Spring Meeting in Washington. His talk was completely crazy to most of the audience, but not to me.

• Feynman invented Feynman diagrams, but he realized that his diagrams are applicable only to scattering processes where the particles are free in the remote past and remote future. In his 1970 talk, for bound states, Feynman told us to use harmonic oscillators instead of Feynman diagrams.

  He then published his talk with his students in Phys. Rev. D 3, 2706 (1971). However, this paper is thoroughly self-inconsistent from the mathematical point of view, and they even make apologies. However, their mathematics can be fixed up. If this is done, the mathematics is a representation of the Poincar� group based on harmonic oscillators, given in the paper that fixes up Dirac's soft spot mentioned above. Click here again.

• Here is a graphical illustration of where Wigner and Feynman stand in Einstein's world.

On the issue of the time-separation variable

The Bohr radius is a very important quantity in quantum mechanics. It is spacial separation between the proton and electron in the hydrogen atom. If the hydrogen atom starts to fly, we have to worry about it time separation. This concept does not exist in the present Copenhagen interpretation of quantum mechanics. Indeed, this time-separation is one of the hidden variables in the present form of quantum mechanics.
• Paul A. M. Dirac worried about the time-energy uncertainty as in 1927 when Heisenberg's uncertainty relation was inaugurated. [Proc. Roy. Soc. (London) A114, 243 (1927)]. He talked about the relation between the life time and line broadening.

  In 1945, Dirac attempted to construct a representation of the Lorentz group with four-dimensional oscillators [Proc. Roy. Soc. (London) A183 , 284 (1945)]. His wave functions are localized in both space and time variables. Presumable, his space variables are separation variables in the hydrogen atom, but Dirac never mentioned the time separation for the time variable.

• Richard Feynman, in his 1971 paper with his students, noted the existence of the time-separation, but he did not do anything about it. His reason was very simple. He does not know what to do about it.

• Indeed, the time-separation variable is one of my major research subjects for many years. According to Einstein, this is clearly a physical variable, but it does not exist in the present Copenhagen formulation of quantum mechanics. Thus is the time separation is qualified to become a hidden variable. The failure to observe this variable leads to an increase in entropy. Among the many papers I wrote on this subject, you may look at
  1. Kim and Wigner in Phys. Lett. A 147, 343 (1990).

  2. Kim and Noz in Symmetry 3, 16 (2011).

Who is this young man?

copyright@2011,2015 by Y. S. Kim, unless otherwise specified.

1. Click here for his home page.

2. His Einstein page.
3. Princeton page.

4. His style page.