Geometry and Physics have a long joint history, dating at least all the way back to the Greekphilosophers and geometers.
Modern examples of this deep link include Einstein’s geometrization of gravity and more recently the unificationof forces and gravity in the context of string theory. My aim in this talk is to review some deep links we have witnessed between geometry and physics in the context of string theory.
Newton
Apple: Where it all starts!
Classical Mechanics
FLAT SPACE
Add Time + Space Not Flat!
Geometry of Space and Time is Curved
Particles move on geodesics
curvature + geodesic looks like force
Matter Curvature
Even light bends as it moves on geodesics
The planetary orbits can also be explained geometrically: Sun curves spacetime and planets follow geodesics
Theory is relevant also for cosmological questions
Particles are ``fuzzy’’ at smaller scales
Quantum mechanics was born
Bohr
This was so radical, even Einstein was intrigued!
Principle of quantum mechanics seems valideven at nuclear and subnuclear scales
These ideas were checked by scattering experiments of high energy particles
Feynman
Interaction takes place by exchange of particles
Gravity+Quantum Mechanics?
Particles+quantum mechanics+gravity
Difficult to make sense of !
Strings come to the rescue!
At yet smaller scales ``elementary particles’’ look like strings
String Interaction
Joining of strings
Joining and splitting of strings
String interactions are described by the beautiful geometry of surfaces
Everything seems to be in place with strings at a very tiny (at present unobservable) scale
Smooth geometry of stringsseems to explain all known interactions (at least in principle)
String’s connection between geometry and physics seems to start on the wrong foot:
String theory demands that spacetime not be four dimensional! d=10 (or 11 in M-theory)
This seems to raise a dilemma:
How can we hope to use string theory toanswer questions relevant to 4d physics?!
First attempt at the answer: We can demand that the extra dimensions be curled up into a tiny compact space, thus rendering it unobservable and avoiding a direct clash with reality!Not totally satisfactory: What are the extra dimensions good for?!
But this answer is not totally satisfactory:
What are the extra dimensions
good for?!
Charged matter
U(3)
U(1)
U(2)
SO(10)
1/g2
Naïve continuation does not make geometric sense
Instead a new geometry opens up!
Smooth Transition
On the other hand there are a number of ons for 4d physics which require atter answer: Black Hole Entropy: Where are the microstates of the black hole hidden?
Bekenstein-Hawking formula:S=A/4A= area of the horizon
Black holeas wrapped branes in the internal dimensions
Wheeler: Quantum mechanics wildfluctuations of space at Planck Scale -33 (10 cm)
Not only the space is fluctuating in sizeat Planck scale but even its topology shouldfluctuate. Very difficult to describe.
Space and time should look likeA ``quantum foam’’ at Planck scale
The geometryof the corners is exactly the same as for strings movingin a complex 3D space!
Macroscopic crystal geometry Stringy geometryAtomic structure of crystal ?
A simple modelfor melting crystal as removing atoms starting from the atoms near the corners
Typical configuration of molten crystal
We are witnessing anongoing revolution linkingGeometry and Physics
We are witnessing anongoing revolution linkingGeometry and Physics
We have learned a lot, buta lot remains to be learned.It is a `work in progress’!