Poincaré, Einstein, and Poincaré with Decoherence

Arthur Wightman in 2002. I was in his class on the Lorentz group in 1960,
and I learned this two-by-two representation from him.

Two-by-two matrices enable us to do to many wonderful things. On this webpage, I would like to use them to write a history of physics.

People do not like to do Lorentz transformations because their four-by-four transformation matrixes are two clumsy for them. Things could become quite different when they can be represented by two-by-two matrices. Then then become parts of sentences.
I had my first exposure to the two-by-two representation while I was in Wightman's class on the Lorentz group in 1960 when I was a second-year graduate student at Princeton. Here is the note I took on the two-by-two representation of the Minkowski four-vector
Later, I had to study this representation more thoroughly from Naimark's book on Linear Representations of the Lorentz Group, when I was writing a book on the Theory and Applications of the Poincaré Group, with Marilyn Noz. since I cannot find earlier references than Naimark's work, I choose to call this two-by-two formalism Naimark representation.
This Naimark representation is not only useful in special relativity, but also in other areas of physics, including Stokes parameters, Poincaré sphere, density Matrix, Feynman's rest of the universe. I am still looking for other places where this representation can play its role.
In this Naimark, the space-time four vector takes the form, and its Lorentz transformation can be written as
Here is the explicit expression for this transformation.
This G matrix is unimodular with six independent parameters. It is possible to translate this two-by-two matrix into a corresponding four-by-four transformation matrix applicable to the Minkowskian four-vector.
Click here to see how.
However, for all practical purposes, it is enough to use four transformation matrices and their four-by-four counterparts.
With this Naimark representation, we can clearly see what Poincaré did, and what Einstein did for the formulation of special relativity. Click here for the story.
In 1939, Wigner considered subsets of the G matrix which leaves a given momentum invariant. They are different for massive, massless, and superluminal particles. These subgroups are called Wigner's little groups. As you probably know, this is my favorite subject. Wigner used to become very happy whenever I talked about this subject.
Not many particle physicists seem to know that Wigner's little groups dictate internal space-time symmetries of elementary particles. This is particularly true for those who claim to be studying the internal space-time symmetries.
It may be easier to explain this aspect of Wigner's little group if I use the Naimark representation. Click here for an explanation.
Perhaps, it could be even easier if we use this formalism for another subject of physics. In polarization optics, we use Jones vectors and Stokes parameters and their transformation matrices.

Let us study Polarization optics with the Naimark representation of the Lorentz group.

The Jones vector consists of the x and y components of electric field for a light wave propagating along the z direction. These electric field component can have different phases and they can be changed. The electric fields can be rotated around the direction of the wave propagation. In traditional textbooks, the polarizer matrix is discussed. However, it is an extreme case of the attenuation matrix with two different attenuation rates. Thus, this polarizer matrix can be replaced with a squeeze matrix. Click here for the formulas.

Click on the figure to expand it.

Then the set of matrices applicable to the Jones vector is identical to the set used for special relativity. Click here for the comparison. Indeed, the role of the Lorentz group between optics and special relativity and like the second order differential equation connecting forced mechanical oscillator and LCR circuit in electronics. Click here for an illustration.
Stokes parameters. The Jones vector with its two components cannot address the issue of whether two orthogonal electric fields are coherent with each other. This is the reason why we need the two-by-two coherency matrix. Its off-diagonal elements measure the degree of coherence. Since the parameter for this is necessarily smaller than one and greater than zero, we choose this number to be cos\chi, and call the angle \chi the coherence angle. The degree of decoherence is zero when \chi = 0, and is maximum when \chi = 90^o .

The coherency matrix is transformed by optical operations according to the transformation matrices defined for the four-dimensional space-time. Both rotations and boosts in space-time physics can be translated into corresponding operations in optics, according to this table.
However, the determinant of the coherency matrix remains invariant under these transformations, and the value of the determinant is
This quantity is invariant under all available transformations in the Lorentz group, and is therefore a Lorentz-invariant quantity. As this table indicates, the Lorentz-invariant quantity in the mass of the particle in space-time physics. Thus, this decoherence parameter occupies a very important place from the mathematical point of view.
We can now place the four Stokes parameters into the coherency matrix, as we did for the space-time coordinate into the Naimark representation of the Lorentz group. These parameters can be computed from the elements of the coherency matrix.
The radius of the Poincaré sphere takes a value between its maximum and it minimum, depending on the decoherence parameter.
It is now clear that the parameters
constitute the three-dimensional space with are quite familiar with. The Poincaré sphere is the sphere defined in this space. The radius of this sphere can be computed as the
This radius takes its maximum value of S₀ when the decoherence angle is zero, and takes its minimum value when the angle is 90^o. This is what the Poincaré sphere is all about.
However, if we include Lorentz boosts applicable to the two-by-two coherency matrix, all these variables are shaken up. However, its determinant
remains invariant, as noted above.
The two-by-two coherency matrix can also serve as the density matrix, if it is sp normalised that its trace is one. I also learned this while I was in Wightman's class in 1960. See the 1960 Les Houches Lectures. Since the coherency matrix is Hermitian, it can be diagonalized. It is then straight-forward to compute the entropy. The entropy depends only on the decoherence angle, and therefore invariant under Lorentz transformations on the coherency matrix. Here are the formulas.
Since this density matrix is determined by (sin\chi)² , we can consider another density matrix with (cos\chi)² and the corresponding Poincaré sphere. It is clear that the entropy increases in the first Poincaré sphere, the entropy for the second Poincaré sphere should decrease. Thus, these two Poincaré spheres could lead to another example of Feynman's rest of the universe.
Click here for the formulas.
With Sibel Baskal, I published a paper on possible enlargement of space-time group to accommodate this aspect of Feynman's rest of the universe. You may click here for the paper. I like to study more. It is an interesting subject.
If you are not accustomed to read physics from webpages, you can go to a written article on this subject. Click here. This paper is based on my talk presented at the Fedorov Memorial Symposium: International Conference "Spins and Photonic Beams at Interface," dedicated to the 100th anniversary of F.I.Fedorov (Minsk, Belarus, 2011). I would like to thank Professor Sergei Kilin for inviting me to this conference. My way of thinking him is to make a webpage for him and his city.
It is also my pleasure to acknowledge what I learned from Arthur Wightman while I was a student at Princeton. He was not my thesis advisor, but he has been kind and helpful to me throughout my professional life. I mentioned Wigner and Wheeler many times before, but my thesis advisor was Sam Treiman. Here is my photo with him taken in 1987. Since I gained so much from Princeton, I maintain a webpage for the campus of the University. This page seems to be popular among those high school students who want to go to Princeton for their undergraduate education.
Y. S, Kim (March 2012)

copyright@2010 by Y. S. Kim, unless otherwise specified. The image of Feynman is from the public domain.
Click here for his home page.