is for the hyperbola. In terms of the (t +z) and
(t - z) variables, this equation can be written as

(t + z) (t = z) = 1 .

The equation for the circle takes the form

t^{2} + z^{2} = 1 .

This equation can be also be written as

(t + z) ^{2} + (t - z)^{2} = 2.

This circle can then be squeezed to

e^{-x}(t + z) ^{2} +
e^{x}(t - z)^{2} = 2 .

When x = 0 , this equation is for the circle.
As x increases from zero to a positive number, the circle
becomes squeezed to the ellipse.

Exercise: The circle is tangent to the hyperbola at x = 0.
The tangential point moves along the hyperbola when this parameter
increases. Find the exact location of the tangential point as a
function of x.

What does this algebraic geometry mean to you?

How did I get this idea?
Click here. I met Paul A. M. Dirac in 1962. It was like
Nicodemus meeting Jesus.

Moses talked to God by writing books about God. They are of course the
Five Books of Moses in the Old Testament. I wrote books and papers to approach
Eugene Wigner. I am talking to Einstein by constructing this webpage.

If you are interested in symmetry problems, you should be aware that
Wigner's 1939 paper deals with
the subgroups of the Lorentz group whose transformations leave the momentum
of a given particle invariant. Thus, these subgroups dictate the
internal
space-time symmetry of the particle.

It is generally agreed that Wigner deserved a Nobel prize for this paper alone,
but he did not. He got the prize for other issues.
Click here for my explanation of where
the confusion was. Wigner liked my story. This is the reason why he had
photos with me, and I am regarded as Wigner's youngest student, even though
my thesis advisor at Princeton was Sam Treiman.

You have been wondering why the proton appears as a bound state of three
quarks to Gell-Mann, while it appears like a collection of an infinitie number
of partons to Feynman.

A massive particle at rest has three rotation degrees of freedom. However,
a massless particle has only one degree of freedom, namely around the
direction of its momentum. What happens to rotations around the two
transverse directions when the particle is Lorentz-boosted?

You also have been wondering why massless neutrinos are polarized, while
massless photons are not.

If you are interested in further contents of Einstein's E = mc^{2},
click here.

The Gaussian wave function leads to the minimum uncertainty product. How
would this product appear to a moving observer. Einstein must have been
interested in this question.
Click here for the resolution of this
issue.

If you like modern optics, including coherent and squeezed states, beam transfer
matrices, polarization optics, as well as periodic systems,
click here.

If you are interested in entanglement problems, particularly
Gaussian entanglements,
click here.

If you are interested in entropy problems and Feynman's rest of the
universe, click here, and
here.

If you are interested in history of physics,
click here.

If you are interested where Einstein stands among the philosophers,
click here.

Click here for the cartoons
serving as a powerful language in physics. Example: Feynman diagrams.

If you like know how powerful two-by-two matrices are in physics,
click here, and
here.

Acknowledgments. This page is based on the papers I published since 1973.
I wrote many of those papers in collaboration with a number of co-authors,
especially, Sibel Baskal, Elena Georgieva, Daesoo Han, and Marilyn Noz. I
would like to thank them. I am also grateful to Professor Eugene Wigner for
clarifying some critical issues concerning his 1939 paper on the internal
space-time symmetries of particles in the Lorentz-covariant world.