Stepping Stones from Heisenberg to Einstein
Among the more than 1,000 webpages stored in my computer, the most popular page is about Heisenberg talking about Einstein. Click here for the page. I constructed this page in 2005. This pages gets more than 30 hits every day. The question is why?
- The answer is very simple. The physics world is still meandering around
the open space between Einstein and Heisenberg. The question is whether this
gap can be bridged. I am not the first person to ask this question. It was
Paul A. M. Dirac's life-time business. However, Dirac's problem was that
he did not talk with too many people. Have you met anyone who talked to him?
Dirac and Wigner met occasionally, but it is not clear whether they ever discussed physics. I was fortunate enough talk with Dirac twice (in 1962 and 1978). Click here for my 1962 meeting.
I met Wigner while I was a graduate student at Princeton (1968-61). During the period from 1985 to 1990, I had enough resources to tell Wigner what he really wanted to hear. As a consequence, I published six papers with him. I had to explain to him what Dirac did in his papers.
- In his own papers, Dirac occasionally quoted Wigner's 1939 paper on the
Lorentz groups, but he made no attempts to interpret his physics in terms
of Wigner's little groups. It is thus fun to fill in this gap, and I talked
about this aspect of physics exetnsively in my earlier publications.
First of all, let me point out where Wigner's work is located in the physics map. We all know it took Newton 20 years to extend his law of gravity between the sun and earth. They are not point particles.
You also know Einstein formulated the law of Lorentz transformations for point particles. Do you know how the hydrogen's electron orbit looks to a moving observer? We can illustrate this aspect in the following figure.
Should we then invent a new mathematics to deal with the internal space-time symmetries. If you do not know the answer to this question, you are quite OK. Dirac did not know. Look at the following figure.
- Let us go back to the hydrogen atom. When this atom moves, it should satisfy
Einstein's energy-momentum relation. How would the electron orbit look?
This question can be addressed as a Dirac issue, as shown in this diagram.
- If the moving hydrogen is too strange to you, we can talk about the Dirac
equation. We all know this equation is for spin-half particles. If the
particle is at rest, we use three two-by-two Pauli spin matrices for the internal
- If we change the sign of Ji, the commutation relations
Eqs. (2) and (5) do not remain invariant. This leads to Wigner's anti-untary
transformation, which we use in dealing with the time-reversal problems.
On the other hand, those commutation relations are invariant under the sign change of Ki . Thus, the generators Ki and -Ki can be accommodated in the same equation. This is what the Dirac equation is all about. Indeed, it is possible to write the rotation generators and and the boost generators as
- Wigner was looking for physical applications of his little groups. The question is
why he did not mention the Dirac equation as an example. There are two possible
- While worrying so much about changing the sign of Ji, he
did not pay much attention to the sign of Ki. The reason could
be that there are no measurable dynamical variables associated with these non-Hermitian
- Wigner did not like Hermann Weyl. When I mentioned Weyl's name, he became quite emotional. I seem to know the reason, but choose not to elaborate on it. Thus, the Dirac equation in the Weyl representation was not in Wigner's scope.
Gell-Mann, Bardeenm, and Wigner,
College Park, Maryland (1988).
If Wigner did not see this aspect of the Dirac equation, should we complain? No, he created a job for us. We can study this problem and publish papers. This is what I am doing these days. Click here for one of the latest published papers on this problem.
As for physical applications of Wigner's little groups, there is another pressing issue: photons and Maxwell waves. This issue has also been discussed extensively in the literature. Click here and here for details.
- While worrying so much about changing the sign of Ji, he did not pay much attention to the sign of Ki. The reason could be that there are no measurable dynamical variables associated with these non-Hermitian operators.
- The main point of this webpage is that the three anti-Hermitian matrices given in
Eq.(4) act as the step stones from Heisenberg to Einstein.
Click here for more about filling in the gap between Einstein and
Y.S.Kim (November 2017)
- On the question of the Einstein-Heisenberg gap, you may also be interested in the following pages.