Groups and Geometry - Comments

[micromass]

micromass submitted a new PF Insights post

Groups and Geometry

Continue reading the Original PF Insights Post.

 

[fresh_42]

Very interesting article, thank you. I admit to belong to those who never walked upon that general bridge. Only occasionally on some walkways. I definitely will have a complete different view on geometries now.

 

[strangerep]

The relationship becomes even more fascinating in elementary particle theory. I.e., the insight that the state spaces of elementary (quantum) particles can be constructed by finding representations of a particular group. Also advanced classical mechanics where symmetry groups for the dynamics, and associated structure of the symplectic phase space take center stage. The notion that Minkowski spacetime is "really" just a homogeneous space for the Poincare group is also intriguing.

One might even say that these relationships are now intrinsic to most (if not all) of modern theoretical physics (after one has generalized the concept of "geometry" to "representations").

BTW, what about semigroups? Is there a well developed theory of (some alternate version of) homogeneous spaces when one is dealing with a semigroup in which some elements have no inverses? The obvious example is the heat equation for which only forward time evolution is sensible.

[Edit: Is "Erlanger" a typo? I thought it was "Erlangen".]

 

[micromass]

The relationship becomes even more fascinating in elementary particle theory. I.e., the insight that the state spaces of elementary (quantum) particles can be constructed by finding representations of a particular group. Also advanced classical mechanics where symmetry groups for the dynamics, and associated structure of the symplectic phase space take center stage. The notion that Minkowski spacetime is "really" just a homogeneous space for the Poincare group is also intriguing.

One might even say that these relationships are now intrinsic to most (if not all) of modern theoretical physics (after one has generalized the concept of "geometry" to "representations").

BTW, what about semigroups? Is there a well developed theory of (some alternate version of) homogeneous spaces when one is dealing with a semigroup in which some elements have no inverses? The obvious example is the heat equation for which only forward time evolution is sensible.

[Edit: Is "Erlanger" a typo? I thought it was "Erlangen".]

Yes, it is Erlangen. I'll fix it.

The point you bring up is very interesting, but I didn't want to go so far. But Klein had a very general method of generating geometries. Basically, he took projective space and he equipped it with a special "distance functions". Then a lot of very pathological but also natural geometries pop up. For example, of course euclidean, hyperbolic and elliptic geometry shows up this way. But also Minkowski geometry and Galilean geometry shows up in this way outlined by Klein. In this way, Klein discovered Minkowski geometry far before SR and GR, but he probably dismissed it for being not useful.

 

[martinbn]

Nice! One more example for the last section, a very important one, the upper half plane is $SL_2(\mathbb R)/SO_2(\mathbb R)$.

 

[Stephen Tashi]

There is a very deep link between group theory and geometry.

Generalizing the line of thinking in the Insight, can we say that there is a very deep link between group theory and equivalence classes (of any sort) ?

Another interesting (and probably subjective) question is "What characterizes a mathematical structure that has 'geometry'"?

( For example, at face value, elementary plane geometry has many topics besides congruence and similarity. )

 

[robphy]

But Klein had a very general method of generating geometries. Basically, he took projective space and he equipped it with a special "distance functions". Then a lot of very pathological but also natural geometries pop up. For example, of course euclidean, hyperbolic and elliptic geometry shows up this way. But also Minkowski geometry and Galilean geometry shows up in this way outlined by Klein. In this way, Klein discovered Minkowski geometry far before SR and GR, but he probably dismissed it for being not useful.

Klein built upon the idea of Cayley---hence the name Cayley-Klein Geometries ( https://en.wikipedia.org/wiki/Cayley–Klein_metric ), as mentioned at the end of the Insight.
These "distance functions" are related to the https://en.wikipedia.org/wiki/Laguerre_formula .
It might be worth noting that deSitter and anti-deSitter (spacetimes of nonzero constant curvature) and their non-relativistic limits are also in this classification of geometries.

 

[Greg Bernhardt]

Great contribution!