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 space-like and two time-like dimensions.
- What is this deStter grouip? Let us use (x, y, z) for
the space-like dimensions, and (t, s) for the time-like directions.
This group is the Lorentz group applicable to the five-dimensional 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.
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- we can consider two Lorentz subgroups applicable to
(x, y, z, t) and
(x, y. z, s) respectively. Let us use
Li
for rotation
generators applicable to (x, y, z),
and use Ki 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 Qi
for the boost generators with respect to
time time-like variable s instead of t. In addition, there is one rotation
generator S3
which mixes up the s and t time-like 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.
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- Dirac,
in his paper of 1963, constructed the same Lie algebra of O(3,2) using the step-up and
step-down 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 Lorentz-covariant
world is constructing a representation of the inhomogeneous Lorentz group
(Lorentz transformations in the three-dimensional space with one time variable)
plus four translation generators applicable to the four-dimensional Minkowski
space of (x, y, z, t) .
- The problem is then to leave the generators Ji
and Ki intact, but
transform Qi and
S3
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 = mc2 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.
- Poincar� 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 five-by-five matrices, and
they are cumbersome. The problem is how to translate them into the language of
two-by-two matrices.
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The squeezed Gaussian function plays the pivotal role quantum optics and
entanglement problems.
- The above figure and series expansion are clearly for the two-photon coherent
state. This figure tells clearly why this two-photon 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 quantum-mechanical 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 time-separation 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 two-photon
coherent state where one of the photons is not measured.
You may
click here for a detailed explanation of this figure.
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