High School Geometry

High School Algebra

  • The equation

      t2 - z2 = 1

    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

      t2 + z2 = 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 + ex(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.

    Einstein's World

    		Massive/Slow between Massless/Fast
    Energy
    Momentum     		E=p2/2m Einstein's
    E=(m2 + p2)1/2
    		E=p
    Spin, Gauge,
    Helicity     		S3
    S1 S2     		Wigner's
    Little Group     		S3
    Gauge Trans.
    Hadrons,
    Bound States     		Gell-Mann's
    Quark Model    		 One
    Lorentz-Covariant
    Entity     		Feynman's
    Parton Picture

  • If you are a high-energy physicist,

    1. 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.

    2. 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?

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

    4. Click here for explanastions.

  • If you are interested in further contents of Einstein's E = mc2, 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.

  • Poincaré and Einstein? 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.