Evolutionary chart of Theories of Physics & some related mathematical areas
---------------Warren D. Smith April 2001----------------------------------

(A)              Euclidean        Arithmetic, rational &
                 geometry \       Irrational #s
                    /      \      /
                   /        Real #s,
                  /         Calculus
                 /          |       \
(B) Newton's    |       Maxwell      Newton's                
    Law of      |    equations of     laws of ----      
    Gravity     |   electromagnetism   Motion     \
       |        |          \   (E&M)    /        "Wave" -----
       |   NonEuclidean     \          /         notions     \
       |   geometry by       \        /                DeBroglie-Bohr "old"
       |   Bolyai, Gauss,     \      /    Complex      quantum mechanics
       |    Lobachevsky      Einstein's    Analysis          |
       |        |            "Special           \            |         Operator
       |    Riemannian       Relativity"        Schrodinger equation   analysis
        \    geometry       /       \            & operator-based     / 
         \      |          /         \           quantum mechanics---
          \     |         /           \              /
(C)    Einstein's "GR" -     Quantum Electrodynamics ("QED"):        
      "General Relativity"     Dirac (& related) equations,      Yang-Mills
      - & Einstein-Maxwell     Feynman diagrams and rules      gauge fields
      equations" unifying     /           |                     /
      E&M with gravity       /            |                    /
                  |  \      /            Quantum chromodynamics,
Linear   Group    |   QFT in            Glashow-Salam-Weinberg
algebra  theory   |   curved space,     Electroweak theory,
       \  |       |   Hawking           "standard model"
        Lie       |   radiation         quantum field theories (QFT)
       groups      \                     /
             \      \                   /
(D)          5 kinds of "String theories",
             "supersymmetry", "supergravity"
                             |
                             |
                      "M-brane" theory

Leval A: Due to the Ancient Greeks, especially Eudoxus, Euclid,
  Archimedes, Pythagoras; arithmetic is prehistoric; and then
  the calculus arose from European renaissance workers especially Newton 
  and Leibniz.

Level B: Well understood, mathematically rigorous, 
  existence and uniqueness results known, beginnings of understanding
  of algorithmicity and computational complexity, superb
  agreement with experiment, fairly easy to use.

Level C: mathematically non-rigorous, neither existence nor uniqueness
  results known, may be non-algorithmic. Hard to use. GR probably is
  incomplete theory in that it simply cannot address certain metric situations.
  QED (& descendants) are known to involve infinite self-energies
  for point particles, which are dodged for now, and hopefully are not infinite
  in some future deeper theory. Thus as QED-like theories presently stand, they
  are manifestly incomplete. QED and GR are known to contradict each other...
  However, still, considerable agreement with experiments...
  (The major ingredient of the standard model still not observed experimentally
  is the "Higgs Boson," but many suspect it will be found in the next 10 years;
  the recent experimental discovery of neutrino mass requires revision of most
  expositions of the standard model and the addition of new "right handed"
  forms of all the neutrinos, but the standard model framework
  seems able to handle it; QED is superbly well confirmed experimentally;
  GR, QCD, and electroweak all have good, but not great, confirmation.)

Level D: Inaccessible to experiments. (Also true of QFT in curved spaces
   and the Maxwell-Einstein unification... There is some hope a
   supersymmetric-partner particle will be found in the next 10-20 years,
   and the fact that most of the mass of the universe is known to consist
   of "dark matter" which does not interact by electromagnetic forces,
   and perhaps not by the weak and strong forces either - but it
   does interact gravitationally - can be taken as evidence for stable
   supersymmetric partner particles or anyhow as decisive evidence that
   there is physics, including at least one common stable particle, not known
   to the "standard model.") Therefore highly speculative.
   Incomprehensible without years of training; almost impossible to
   use & understand.

Some major still unsatisfactory problems:

1. Interpretation of quantum mechanics (and all its descendants)
not understood: "measurement", "decoherence".

2. Why is the cosmical constant zero - or small and nonzero?

3. Is there any sense in which QED can be done WITHOUT making every
computation involve a diverging asymptotic series, whose physical meaning
(since alpha=1/137.0... is not actually infinitesimal, so 
"asymptoticity" does not actually hold) is entirely unclear?
Ditto for QCD, etc.

4. Where is the "entropy of a black hole" hiding? How can
the unitarity of quantum mechanics and QFT conservation laws
be reconciled with their apparent forced violation during
black hole formation during stellar collapse and later
Hawking evaporation? What is the ultimate fate of a black hole?

5. What are the right "boundary conditions" in GR?

6. Why is the universe made of matter with antimatter extremely rare?

7. How can more of physics be made mathematically rigorous, existence and
uniqueness results proven, algorithmicity established, and its
computational complexity understood?  (Example: the Navier-Stokes equation
of fluid flow?)