Is God a Geometer
Lorenzo Sadun
February 20, 2008
A few years ago the Clay Mathematics Institute announced the Millennium
problems. These were meant to serve the same purpose as Hilbert’s Problems
at the turn of the twentieth century. Dr. Sadun was asked by the institute to
give a talk on Yang-Mills expansion and the mass gap.
Many of the fundamental theories of physics are expressed in geometric form.
There is a tension between intuitive ideas (even when written formally) and
making them rigorous.
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150 years of Physics in 15 minutes
Coulomb figured out the laws of electrostatics Like charges repel. Opposite
charges attract. Force obeys r12 law. What is special about r12 . Since the force
decays at r12 and the area of a sphere is 4πr2 , the flux is constant. Therefore
the force doesn’t die out with distance it just spreads out.
The original laws of electrodynamics and electrostatics were inconsistent
without adding missing a term. Adding this term showed the existence of very
high speed waves. This lead Maxwell to discover that light obeyed the laws of
electrostatics and electrodynamics.
Quantum mechanics was invented to deal with the fact that light radiation
from a heated object diverged. Max Plank showed you could deal with this if
you dealt with light as discrete chunks. This lead to the idea of particle wave
duality.
In non-relativistic math energy grows with the square of the momentum.
Einstein showed that this was only a low energy approximation. The correct
formula is E 2 = p2 c2 + m2 c4 . If you are going really fast the old formula falls
apart.
de Broglie invented matter waves by asking ”If light behaves as both a particle and wave why doesn’t matter do the same.”
Shrodinger’s equations describe the evolution of the wave equation. You can
use these to predict that the heavier a particle is the shorter it’s wavelength.
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100 years of Math in 15 min
If you are given two vectors in the plane can you give me the angle between
them. If you are in Euclidean space this is easy, because of the concept of
parallelism. On most other spaces there is no notion of parallelism so you can’t.
You can talk about relative angles but not absolute angles. On a sphere you
could try moving the vectors together on a path. Unfortunately vectors on a
sphere twist as the move. If you move the vectors in a triangle you find that the
twist is area
r 2 . If you define curvature as twist per unit area you find that the
curvature of a sphere is r12 . For more complicated surfaces with nonconstant
curvature you need a tensor to keep track of curvature.
The notion of a covariant derivative was introduced to keep track of the rate
of twisting. Einstein had the idea of making space time a 4 − d manifold with
curvature defined by mass and energy. Weyl invented gauge theory to keep
track of how vectors stretch as they are moved. Einstein showed there were
problems with this idea as moving an object in a loop does not change its size.
Eventually quantum mechanics resurrected Weyl’s idea to deal with the fact
that you cannot measure the phase of an object.
Key features in Quantum Mechanics
(1) Equations are linear
(2) Electric/Magnetic fields act on matter
(3) Matter produces EM fields
(4) But EM fields do not act on EM fields
To understand how particles emit particles it was necessary to change the
wave equations from scalar to vector valued functions. The idea was to treat
protons and neutrons as different states of a nucleon (isospin) and make emitted
particles linear transformations that change the state. Murray Gelman showed
that nucleons are made up of three quarks.
Weinberg showed that quarks were not scalar values but were instead vectors (quarks have colors). You can use these to predict the weak, strong, and
electromagnetic forces.
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Mysteries
Physics
1) Why are the nuclear forces short range? Why are there no massless gauge
particles.
2) Why do we never see a bare quark. We only see composite particles with no
net color.
Math
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1) Quantum mechanics works on a fixed number of quantities
2) How do you have a quantum field theory
Problem: Naive calculations give infinite answers. Solution: Renormalization Problem: Only works for toy models.
For 1, 000, 000 construct a fully renormalized quantum version of Yang Mills
theory on R4 and show it has a mass gap.
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