Conic Sections

Click here for the source of this drawing. 
 The other lady was wearing a coneshaped Vietnamese hat. What physics
can you extract from her hat? I am not the first person to ask this question.
 Greek observed there are three different curves, namely circle (ellipse),
hyperbola, and parabola. They then believed that they
should come from one source. Indeed, a circular cone provides all
of these curves.
Greeks believed that there are three fundamental elements in one thing. For instance, they thought one ideal woman consists of three different women carrying their respective virtues. Click here for a detailed story.
 Some years later, Isaac Newton formulated the law of gravity and
his second order differential equation. In so doing,
Newton was able to unify elliptic, hyperbolic, and parabolic orbits.
This aspect of physics is well known.
Step 1, Step 2.
 Click here for a more detailed
story about how Newton defined the history of physics. Quantum mechanics
was formulated to answer the question of why energy levels are discrete
for the hydrogen atom while the Rutherford scattering formula is
applicable to the electronscattering. Here again, quantum mechanics was
formulated to unify the elliptic orbits of the electron and the hyperbolic
trajectory in Rutherford scattering. This aspect of modern physics is
also well known.
 Don't forget Einstein. Thanks to Feynman diagrams, we have a good idea of
the hyperbolic trajectories (scattering problems) in the Lorentzcovariant
world. How about
the hydrogen atom with discrete energy levels? How would this atom look
to an observer on a bicycle?
 This is John Bell's has this picture of the Lorentzboosted hydrogen orbit in his book. This picture however does not explain why the its energy levels are discrete.
 Standing waves. How to Lorentzboost?
 Evolution of the hydrogen atom the proton in the quark model.
 Quarks and Partons as two limiting cases of one Lorentzcovariant entity.
 Internal Spacetime Symmetries. The first step toward what is going on
inside the proton is to understand the internal spacetime symmetries of particles
in the Lorentzcovariant world. If the particle is at rest, then it has the
symmetry of the threedimensional rotation group. If the particle gains its
speed, we have to worry about additional degrees of freedom.
 In his 1939 paper, Wigner
formulated this problem in terms of his little groups. The Lorentz group has
six degrees of freedom. If the fourmomentum of a particle is fixed, it has
only three degrees of freedom, defining internal spacetime symmetries. For
a massive particle at rest, it has three rotational degrees of freedom.
corresponding to the spin degrees of freedom.
If you like to have a reprint of Wigner's 1939 article, send your reuest to yskim@umd.edu.
 For massless particles, the
problem was not completely solved until 1990. In his original 1939 paper, the
internal spacetime symmetry is isomorphic to the twodecisional Euclidean
group. The rotation corresponds to the helicity of the particle. There are
two translationlike degrees of freedom. It was later suggested by various
authors that they correspond to a gauge degree of freedom. However, why
the oneparameter gauge degree of freedom corresponds to two translational
degree of freedom. This question is addressed in the
1990 paper of Kim and Wigner.
 If the particle mass is imaginary, it moves faster than light, and is thus
not observable. Yet, recent trends in physics allow us to talk about
this possibility.
 Indeed, the transformation groups leaving those momenta invariant are like
O(3) (circle), like E(3) (linear or quadratic), and O(2,1) (hyperbolic)
respectively. Again the conic sections dominate even inside the particle.
 In spite of this simple picture of Wigner's view toward the internal
spacetime symmetries, his 1939 paper is regarded as one of the most
difficult papers to understand. Yes, there have been and there still are
many rising stars in physics who claimed to have solved the poblem of
internal spacetime symmetries. However, they carefully avoid Wigner's
classic paper on this problem. In so doing they are missing the main
point, and those stars fall down. Wigner's 1939 paper appears to be
the prescrition for their longevity.
 Sakura is the Japanese
word for cherry blossom. During a oneweek period of spring every year,
the city of Washington (capital city of USA) becomes brightened by
fullblown Sakura flowers. I am indeed fortunate to live near this
city to enjoy this Sakura phenomenon every year.
However, the problem is they do not last too long, they disappear one week after their peak showing. It would be nice if someone could invent a botanic process to make them last longer. Likewise, the physics community is blessed with those Sakuratype physicists. We only wish they could stay longer to give us prolonged excitements.
 In his 1939 paper, Wigner
formulated this problem in terms of his little groups. The Lorentz group has
six degrees of freedom. If the fourmomentum of a particle is fixed, it has
only three degrees of freedom, defining internal spacetime symmetries. For
a massive particle at rest, it has three rotational degrees of freedom.
corresponding to the spin degrees of freedom.
 PostWignerian Approach. Wigner's original paper divides the particle
symmetry into three distinct classes. However, we can approach the
problem as one symmetry with three branches. This viewpoint can
be derived from our experience in ray and polarization optics. Modern
optics is largely a physics of twobytwo matrices from the group SL(2,c).
The generators of this group share the same set of commutation relations
with those for the group of fourbyfour matrices performing Lorentz
transformations. For this reason, we can study Wigner's little groups,
or internal spacetime symmetries, using optical instruments and twobytwo
matrices.
 Here is a review article
on this subject. It is easy to read this paper because it contains only twobytwo
matrices. The only nontrivial aspect of this paper is that not all those
twobytwo matrices can be diagonalized. The question is how to deal with it.
 I spent some years on this subject, and learned many lessons while writing papers with my younger colleagues. We learned the most important lessons from those referees who attempted to turn down our papers. For instance, we learned how to use the traces of twobytwo matrices in our logic from one of the hostile referees. It is my pleasure to thank them for writing those valuable reports for us.
 Here is a review article
on this subject. It is easy to read this paper because it contains only twobytwo
matrices. The only nontrivial aspect of this paper is that not all those
twobytwo matrices can be diagonalized. The question is how to deal with it.
 Let us choose a real positive number. It can be smaller than two, equal to two, or greater than two. It is so simple that even God can say this. Then God created elementary particles using twobytwo matrices.
Copyright@2012 by Y. S. Kim, unless otherwise specified. The portrait of Wigner and Einstein is by Bullent Atalay.