Topological Examples in the Real World
In this section, a list is displayed of a number of physical effects which - to
first order - do not depend explicity on size or shape. Therefore, these
phenomena must have a topological basis for their explanation. Key topological
properties are the number of parts (connectivity), the dimension, and
orientability. Concepts of size and shape, metric, or connection can not enter
into the explanation of such phenomena, except in an auxiliary way..
- Planck's Radiation Formula.
The distribution law is independent from the size and shape of the hot
- Coaxial Wave-Guide Propagation.
A hollow wave guide is a high-pass filter, but a non-simply connected co-axial
cable can pass DC.
- Chaos does not occur in Dimension 2.
A system of ODE's has a PDE equivalent. If the PDE satisfies the Frobenius
integrability theorem, according to Darboux there is always a representation in
terms of 2 functions.
- Bohm-Aharanov effect.
Flux quanta in superconductors come in integer multiples of h/2e.
- Gauss Law.
The integral of the D over a bounding surface only depends upon the
number, e, of
electrons in the interior of the closed surface, no matter what the size or
shape of the surface may be.
- Topological Torsion and Spin.
The Bohm-Aharonov effect is a 1-D period integral. Gauss Law is a 2-D period
integral. There are two 3-D period integrals that have been little studied in
science, but appear to have significance in hydrodynamics and plasmas as robust
coherent vortex-like helical structures - insensitive to deformations in space
- Quantum Transition Probability.
Fermi's Golden Rule demonstrates that the
transition probability is a cross-ratio projective invariant,
independent from scales.
- Thermodynamic Irreversibility.
The idea that thermodynamic irreversibility is defined
by the failure of the Frobenius Theorem for the 1 form of work implies
that the topological dimension of the Action 1-form must be 4. In other words,
thermodynamic irreversibility is an artifact of four dimensions.
Can you help us add to this list?