Heisenberg's Gift to Einstein
Among the webpages I built on the history of physics, the page entitled
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.

Einstein's house in Princeton.

 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 the Poisson brackets lead to Einstein's relativity.
He could not tell this because he was not aware of this in 1954.

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}.
 Wigner got his prize in 1963, but not for
his 1939 paper on internal spacetime symmetries
 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.
Contents of Einstein's E = mc^{2}

Particle 
Massive/Slow 
between 
Massless/Fast 

Einstein 
Energy Momentum 
E = p^{2}/2m 
E = [m^{2}c^{4} + (cp)^{2}]^{1/2 } 
E = cp 

Wigner 
Helicity spin, Gauge 
S_{3} S_{1}
S_{2}

Wigner's Little Groups 
Helicity Gauge Trans.


This table is from
one of my papers published in 1986.
 Let us go back to the Heisenberg issue. Heisenberg could have
made Einstein by telling his E = m^{2} is a consequence
of the symmetries derivable from his uncertainty commutation relations.
He did not know this in 1954, but he could have said this based on a
paper I published in 2019,
with Sibel Baskal and Marilyn Noz.

Here comes the key question. Did you know the Poisson brackets
lead to the Lorentz group? Don't worry. I am not the first person
who observed this. It was Paul A. M. Dirac who used two harmonic
oscillators to derive the group of Lorentz transformations. Dirac
in fact derived two coupled Lorentz groups or O(3,2) group starting
from two oscillators.

Both Dirac and Heisenberg were interested in the Poisson brackets.
Einstein did not like Heisenberg's interpretation of those brackets.
On the other hand, Dirac was interested in extending those brackets to
Einstein's Lorentzcovariant world.
It is not clear whether they knew the Poisson bracket for a single pair
of position and momentum variables has the symmetry of the Sp(2) group
isomorphic to the Lorentz group applicable to two spacelike and one
timelike directions, as specified in this figure.



In his 1963 paper, Dirac considered two harmonic oscillators and constructed
the following ten operators.


Dirac then noted that these operators satisfy the algebra for the generators for
the O(3,2) deSitter group, with O(3,1) as a subgroup with three rotation and
three boost generators applicable to the Monkowski space of (x, y, z, t).
The O(3,2) group requires an extra time variable. There are thus three additional
boost generators with respect to this new time variable. In addition, there is
a rotation generator for these two time variables. There are thus four
additional generators.


The issue is then to transform these four extra generators into the four spacetime
translation generators the Minkowski space.



Dirac published this result in 1963 in the
Journal of Mathematical Physics. This paper
is largely unknown to the present generation of physicists, in spite of the
fact that it provides the mathematical base for many branches of physics,
including squeezed states, entanglements, entropy, highenergy physics,
Bogoliubov transformation in superconductivity, O(3,2) supersymmetry,
and presumably many future theories..
 Then, while you did not know, how do I know about this paper? In
the fall of 1962, spent many hours with Dirac. How did this happen?
Click here for a story. At
that time, I did not like what was going on the physics world.
Click here to see how much I disliked
the physics environment at that time. I had to be born again, like
Nicodemus after seeing Jesus (Bible story from the Gospel of John).
Dirac's 1963 paper is difficult to read, because it consists of a
mathematical poem consisting of ten generators and thus sixty commutation
relations. It is fun to provide interpretations to this poem using
physical examples, and this has been my research line for many years.
 Then, what does this have to do with Heisenberg? If Dirac got the
Lorentz group from the commutation relations for harmonic oscillators,
there must be the basic element of Lorentzian symmetry in Heisenberg's
Poisson brackets. Indeed, this symmetry is well known, and it is called
the group of canonical transformations.
The Poisson bracket consists of two conjugate variable x and p.
This bracket is invariant under rotations in the phase space of those two
variables. It is also invariant when x increases while p decreases while
the product xp remains constant. This is a squeeze in phase space. These
operations are enough to construct the Lorentz group applicable to the two
spacelike and one timelike directions.
This symmetry is not rich enough address the symmetries in the
Minkowski space of three spacelike and one timelike dimensions.
Dirac in 1963 was able to construct his richer symmetry using
two Poisson brackets. Yes, Einstein could have shown his interest
in this symmetry of the Poisson brackets, but he did not.
 The rotation in the phase space of x and p leads to the rotation around
y axis in the threedimensional space. The squeeze along the x~p
directions leads to the Lorentz boost along the z direction. The squeeze
along the 45degree direction corresponds to the Lorentz boost along the
x direction as shown in this figure.
 While the Poisson bracket for a single pair of positionmomentum variable
leads to the Lorentz group applicable to two spacelike and one timelike
directions, Paul A. M. Dirac considered two harmonic oscillators. Each
oscillator corresponds to one Poisson bracket. He then ended up with
the O(3,2) symmetry which corresponds to the Lorentz group applicable to
three spacelike dimensions and two timelike directions.
 We are familiar with the procedure of contracting the Lorentz group O(3,1)
to the Galilean group, which includes three rotations and three translations.
Likewise, we can contract O(3,2) into the Lorentz group O(3,1) and four
translation in fourdimensional Minkowski space (x, y, z, t). This group
is known as the inhomogeneous Lorentz group which serves the basic
symmetry group of quantum mechanics in the Lorentz covariant world.
You may click here for a comprehensive
discussion of this matter.

