Symplectic Geometry in Physics: String Theory & Beyond

In summary, symplectic geometry plays a crucial role in fundamental physics, particularly in Hamiltonian mechanics and optics. Despite being promising, it has not received much attention in the past 15 years. However, it has been the foundation of mechanics for over a century and has influenced areas such as quantum gravity. Learning symplectic geometry is essential in understanding string theory.
  • #1
pivoxa15
2,255
1
How crucial is symplectic geometry to fundamental physics?

Any examples? I know it is related to string theory.
 
Physics news on Phys.org
  • #2
Symplectic Geometry underlies Hamiltonian Mechanics and Optics.

http://www.math.hawaii.edu/~gotay/Symplectization.pdf
 
Last edited by a moderator:
  • #3
Just read the article. They sound very promising but 15 years, sympletic geometry still hasn't got much attention.
 
  • #4
pivoxa15 said:
Just read the article. They sound very promising but 15 years, sympletic geometry still hasn't got much attention.

What time scale is more appealing to you?
...Gotten the attention from whom?Here's a more recent article:
http://www.math.princeton.edu/~acannas/04_SG/
 
Last edited by a moderator:
  • #5
The first article was making huge claims about symplectic geometry but I don't see them happening. The title of the article 'Symplectization of Science' was too catchy.
 
  • #6
Every known (or maybe even conceivable) system of classical mechanics can be cast in the form of symplectic geometry.
Poisson after whom the bracket that introduces the symplectic structure on phase space is named, died in 1840. Through Hamiltons recasting of the equations of motion as the flow of a vectorfield induced by an observable and a symplectic structure, through the Noether theorems, Heisenbergs quantization, Diracs cannonical analysis of gauge systems to todays work on the interpretation of Quantum Gravity, symplectic structures have provided the foundation (sometimes implicitly) of almost the entire apparatus of mechanics for the last century and a half at least.
 
  • #7
How crucial is it to learn symplectic geometry if one wants to understand string theory?
 

1. What is symplectic geometry?

Symplectic geometry is a branch of mathematics that studies the geometric properties of spaces that preserve a specific structure known as a symplectic form. This structure is important in physics as it describes the dynamics of systems with multiple degrees of freedom, such as particles in motion or fields in space.

2. How is symplectic geometry related to physics?

Symplectic geometry has been used extensively in theoretical physics, particularly in the study of dynamical systems and classical mechanics. It has also played a crucial role in modern physics, specifically in the development of string theory and its applications to quantum gravity and other areas of fundamental physics.

3. What is the role of symplectic geometry in string theory?

In string theory, symplectic geometry is used to study the dynamics of strings and their interactions in a multi-dimensional space. The symplectic structure of this space determines how strings move and interact with each other, and it is crucial for understanding the underlying principles of string theory.

4. How does symplectic geometry extend beyond string theory?

While symplectic geometry has been primarily applied in string theory, it has also been used in other areas of physics, such as quantum field theory, statistical mechanics, and cosmology. It has also found applications in mathematics, including algebraic geometry and topology.

5. What are some current research topics in symplectic geometry and its applications in physics?

Some current research topics include the application of symplectic geometry to topological phases of matter, the study of chaotic systems using symplectic methods, and the use of symplectic techniques in understanding the dynamics of black holes in general relativity. Other areas of interest include the role of symplectic geometry in non-commutative and higher-dimensional theories, as well as its connections to other branches of mathematics, such as algebraic topology and mirror symmetry.

Similar threads

A Can Membrane Field Theory Elucidate M-Theory?
    • Beyond the Standard Models
Replies
9
Views
2K
A Symplectic Majorana Spinors in 5 Dimension
    • Beyond the Standard Models
Replies
5
Views
2K
I All possible QFTs from geometry?
    • Beyond the Standard Models
Replies
3
Views
1K
A Finding the Hermitian generator of a Symplectic transformation
    • Quantum Physics
Replies
17
Views
1K
A Symplectic geometry of phase space
    • Classical Physics
Replies
3
Views
1K
I String theory vs Quantum field theory
    • Beyond the Standard Models
Replies
1
Views
172
A Lorentz Invariant Phase Space: Symplectic Geometry
    • Special and General Relativity
Replies
1
Views
1K
A The no boundary proposal and quantum gravity
    • Beyond the Standard Models
Replies
7
Views
520
I MOND from MacDowell-Mansouri geometrical formulation
    • Beyond the Standard Models
Replies
19
Views
2K
I Noncommutative geometry and asymptotic safe gravity and MOND
    • Beyond the Standard Models
Replies
3
Views
2K

Hot Threads

Recent Insights

Back
Top