Dirac then ended up with ten operators satisfying the Lie algebra for the
O(3,2) deSitter group,
which is the Lorentz group applicable
to three spacelike and two timelike dimensions.
 What is this deStter grouip? Let us use (x, y, z) for
the spacelike dimensions, and (t, s) for the timelike directions.
This group is the Lorentz group applicable to the fivedimensional space of
(x, y, z, t, ).
This O(3,2) group has its own merits in general relativity and other areas of physics.
Yet, we can ask whether whether
 this group can be constructed from the Heisenberg brackets for his uncertainty
principle,
 this group can be transformed into other symmetry problems in physics.
With these questions in mind,
Lie algebra for the O(3,2) group.


 we can consider two Lorentz subgroups applicable to
(x, y, z, t) and
(x, y. z, s) respectively. Let us use
L_{i}
for rotation
generators applicable to (x, y, z),
and use K_{i} for
the three boost generators with the time variable t. They are the six
generators for the Lorentz group familiar to all of us.
 There are four additional generators involving the extra time variable s.
We can use Q_{i}
for the boost generators with respect to
time timelike variable s instead of t. In addition, there is one rotation
generator S_{3}
which mixes up the s and t timelike variables.
 These ten generators constitute a closed set of commutation relations (Lie algebra)
for the O(3,2) group.
These operators satisfy the Lie algebra of the O(3,2) group.
Clearly this algebra is derivable from the Heisenberg brackets.


 Dirac,
in his paper of 1963, constructed the same Lie algebra of O(3,2) using the stepup and
stepdown operators for two harmonic oscillators, and these generators are given
in this table. The question still is whether this
group can be transformed into other symmetry problems in physics.
To address this question, let us go to Dirac's earlier paper entitled
Forms of Relativistic Dynamics Published in 1949. In this paper,
he sates that the task of constructing quantum mechanics in the Lorentzcovariant
world is constructing a representation of the inhomogeneous Lorentz group
(Lorentz transformations in the threedimensional space with one time variable)
plus four translation generators applicable to the fourdimensional Minkowski
space of (x, y, z, t) .
 The problem is then to leave the generators J_{i}
and K_{i} intact, but
transform Q_{i} and
S_{3}
into four translation operators. This is not a difficult
problem. We can use the group contraction procedure to achieve this purpose.
In 2019, I was fortunate enough to write the following three papers on this subject.
 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.
 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.
 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.
In dealing with O(3,2) problems, we have to use many fivebyfive matrices, and
they are cumbersome. The problem is how to translate them into the language of
twobytwo matrices.


The squeezed Gaussian function plays the pivotal role quantum optics and
entanglement problems.
 The above figure and series expansion are clearly for the twophoton coherent
state. This figure tells clearly why this twophoton state is called
"squeezed state." This mathematics is a product of the Lorentz group,
as illustrated in this figure.
 This mathematics is in the current literature called the Gaussian entanglement.
 The concept of the entanglement was first formulated by Richard Feynman.
In his book on statistical
mechanics (1972), Feynman says
When we solve a quantummechanical problem, what we really do is
divide the universe into two parts  the system in which we are interested
and the rest of the universe. We then usually act as if the system in
which we are interested comprised the entire universe. To motivate the use
of density matrices, let us see what happens when we include the part of
the universe outside the system.
Feynman then used the density matrices and Wigner functions to illustrate
his rest of the universe. However, he used only one oscillator to
illustrate what he said about the rest of the universe. Yes! The harmonic
oscillator is the basic tool to illustrate the Wigner function. But how
could he explain two different worlds with one oscillator? Go to
this paper for clarification.
 The following figure explains that timeseparation variable is not measurable,
and thus it belongs to the rest of the universe. In the physics of two photons,
they are entangled in the squeezed state, one of them is in the rest of the
universe if we do not measure it.
This figure serves also the case of twophoton
coherent state where one of the photons is not measured.
You may
click here for a detailed explanation of this figure.


 copyright@2019 by Y. S. Kim, unless otherwise specified.
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I did this mathematical exercise in 1953 during my high school years.
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