Heisenberg's Gift to Einstein
E = mc^{2} derivable from the Poisson brackets

is most frequently visited. Since this page is so popular, you are invited to visit this page again. For the same reason, I keep adding new stories about Heisenberg's meeting with Einstein.
 In 1954, Heisenberg went to Princeton to talk with Einstein, and
the meeting lasted longer than scheduled, according to Heisenberg.
However, it was not a successful meeting. After the meeting,
Einstein expressed his displeasure to his personal friend named "Johanna Fantova."
Click here for the story.
 Why did Heisenberg fail to make Einstein happy? The answer is
very simple. He was not able to tell a story Einstein wanted to hear.
Heisenberg was aware that Einstein did not like his interpretation
of the Poisson brackets. However, he could have made Einstein happy
by telling him that the the Poisson brackets lead to the symmetry of
Einstein's special relativity which leads to E = mc^{2}.
He could not tell this story because he was not aware of this in 1954.
It is now possible.
How did this young man become so close to Wigner? He told the stories Wigner wanted to hear.  In 1987, I surprised many people by publishing a paper with Eugene Wigner.
Wigner was known as one of the most difficult persons to approach, and
he was totally isolated from his colleagues at Princeton's physics department.
However, I was able to tell him the story he wanted to hear. The story goes like this.
 Einstein got his Nobel prize in 1921, but not for his mc^{2}.
 It is generally agreed that Einstein deserved one full Nobel for his mc^{2}.
Likewise, Wigner deserved one full prize for his 1939 paper, and I showed him
the following table.
 Wigner got his prize in 1963, but not for
his 1939 paper on internal spacetime
symmertries
 I showed Wigner the following table in one of my papers published in 1986.
This table is from one of my papers published in 1986.Contents of Einstein's E = mc^{2}
Particle Massive/Slow between Massless/Fast Einstein Energy
MomentumE = p^{2}/2m E =
[m^{2}c^{4} + (cp)^{2}]^{1/2 }E = cp Wigner Helicity
spin, GaugeS_{3}
S_{1} S_{2}Winner's
Little GroupsHelicity
Gauge Trans.This table made Wigner happy enough to invite me to his office regularly to hear the stories he wanted to hear. Based on those stories, I published a number of papers with Wigner.
 Einstein got his Nobel prize in 1921, but not for his mc^{2}.
 Let us go back to the Heisenberg issue. Heisenberg could have
made Einstein by telling his E = mc^{2} is a conseqeunce
of the symmetries derivable from his uncertainty commutation relations.
He did not know this in 1954.
This fact was not known until I published a paper in 2019, with Sibel Baskak and Marilyn Noz. You are welcome to read this paper, but I can summarize the result using this webpage.
You may click here for the rotation generator and two squeeze generators for the Sp(2) group corresponding to the limited Lorentz group. You may cliche Click here for detailed calculations.The symmetry of the Poisson bracket leads to the the Lorentz group appliable to x, z, and t coordinate. How about the y direction?  Thus, in order to get the Lorentz group apllicable to the threedimensional
space, we can bring in another Heisenberg relation. We noted that the best way to
study was to use the Gaussian form for twodimensional space of x and p. The Gaussisn
form is of course that for the harmonic oscillator.
Thus, the best way is to increase the dimensision is to use another osicllator. For this twooscillator system, we need fourdimensional phase space for two pairs of x and p variables. We also need some physics guidelines.
The oscillator form of the Heisenberg relation was most useful in quantum optics. We all know how the coherent state works for the singlephoton coherent systems.
More recently, in 1976 (Yuen), twophoton coherent state was introduced, with this generator.
Ten years lalter, in 1986, Yurke, McCall, and Klauder introduced twophoton interferometers. They lead to six more generators.
 We now have seven operators. It is of mathematical interest to see whether they can
form a closed set of commutation relations (called Lie algebra). These operators generate
canonical transformations in the fourdimensional phase space. They can be represented by
fourbyfour matrices.
In addition, there are ten fivebyfive matrices which can generate Lorentz transformations applicable to three space dimensions (x,y,z) and two timelike dimensions (t,s), or the group O(3,2). This aspect was noticed by Paul A. M. Dirac in 1963.You click on this figure to enlarge it.  In the language of Dirac's O(3,2), there are three rotation generators
for the threedimensional
space of (x, y, z). In addition, there is one rotation generator for the two timelike
coordinates (t, s). There are thus
four rotation generators. These generators are fivebyfive matrices
applicable to (x,y,z,t,s).
There are three boost generators, K_{1}, K_{2}, and K_{3} with respect to the timelike variable t, and three more: Q_{1}, Q_{2}, and Q_{3} for boosts with respect to the timelike variable s. These six generators are tabulated in this table.
 Again, according to Dirac,
the fundamental spacetime symmetry consist of three rotation generators, three boost
generators, and four translation generators. This symmetry is often called the
Poincaré symmetry or the symmetry of the inhomogeneous Lorentz group IO(3,1). The best
way to accomplish this purpose is to convert Q_{1}, Q_{2}, Q_{3},
and S_{3} to the translation generators.
For thais purpose, we use another concept developed in quantum optics, namely
Squeeze.
In the language of group contractions,
These generators lead to the following translations in the (x,y,z,t) coordinates:
In this way, Dirac's O(3,2) become contracted to IO(3,1). Diriac's O(3,2) is derivable from the symmetry of the Poisson brackets.
 copyright@2018 by Y. S. Kim, unless otherwise specified.
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