Atiyah
We shall now recall
the data of a classical theory as understood by physicists and then
reinterpret them in geometrical form. Geometrically or mechanically we
can interpret this data as follows. Imagine a structured particle, that
is a particle which has a location at a point x of R4 and an internal structure, or set of states, labeled by elements g of G.
In
classical, or non-quantum-mechanical, general relativity a particle
moves along the world line that minimizes the so-called action of the
particle: its energy as it moves through time. The action is
proportional to the length of the world line, and so a path of least
action is a geodesic, or the shortest distance between two points in
spacetime. The motion of a string is treated in an analogous way. In a
non-quantum mechanical approximation the string also moves in a way
that minimizes its action. The action is proportional to the area swept
out by the string, and so the world sheet must be a surface of minimum
area. If time is regarded as a spatial dimension, the world sheet swept
out by a closed string can be thought of as a kind of soap film that
joins the string at its starting point and at the end of its path in
spacetime. There is an enormous symmetry embodied in the condition that
the motion of the string is determined by minimizing the area of its
world sheet.
Green
Green
Schwarz
The thing that impressed other physicists most about the general theory of relativity is that it is based on very general physical principles—the equivalence principle and general coordinate invariance—and very beautiful mathematical concepts. The relevant mathematics is called differential geometry (specifically, Riemannian geometry). The idea is that gravity is a manifestation of the curvature of space-time. Also, the geometry of space-time is determined by the distribution of energy and momentum. The basic equation of motion is
It is a most beautiful and awe-inspiring fact that all the fundamental laws of Classical Physics can be understood in terms of one mathematical construct called the Action. It yields the classical equations of motion, and analysis of its invariances leads to quantities conserved in the course of the classical motion. In addition, as Dirac and Feynman have shown, the Action acquires its full importance in Quantum Physics.
Ramond
Weinberg
It is increasingly clear that the symmetry group of nature is the deepest thing that we understand about nature today.
Furthermore,
and now this is the point, this is the punch line, the symmetries
determine the action. This action, this form of the dynamics, is the
only one consistent with these symmetries ... This, I think, is the
first time that this has happened in a dynamical theory: that the
symmetries of the theory have completely determined the structure of
the dynamics, i.e., have completely determined the quantity that
produces the rate of change of the state vector with time.
state vector
symmetry
[All] chemical binding is electromagnetic in origin, and so are all phenomena of nerve impulses.
Salam
Greene
Groleau, Elegant Universe
Epistemologically
it is not without interest that in addition to ordinary space there
exists quite another domain of intuitively given entities, namely the
colors, which forms a continuum capable of geometric treatment.
Weyl
Kaluza
Klein
Simple Calabi-Yau space
From the above brief review, we find there are three versions of geometrization of non- gravitational gauge interactions:
1. Fibre-bundle version, in which the gauge interactions are correlated with the geometrical structures of internal space. [...] the essence of the internal space is still a vexing problem: Is it a physical reality as real as space-time, or just a mathematical structure?
2. Kaluza-Klein version, in which extra space dimensions which compactify in low-energy experiments are introduced and the gauge symmetries by which the forms of gauge interactions are fixed are just the manifestation of the geometrical symmetries of the compactified space. [...] The assumption of the reality of the compactified space is substantial and is in principle testable [...]
3. Superstring version, in which the introduction of extra compactified space dimensions is due to different considerations from just reproducing the gauge symmetry.
Cao
Calabi
compactify
Yau