esoteriic.com

This is about a particular instance of intuition that I had while writing my thesis. I was going to call this post simply 'intuition' but I don't give much discussion about the intuition itself. I probably should have labelled this post with just the mathematical statement that I guessed to be true but that would be very abstruse and not likely to attract (m?)any readers.

This is what I guessed to be true (and subsequently have found to be true, I *think*): that a symplectic manifold is topologically simple.

The thought *just* came to me and seemed to be 'true' yet I didn't know how to prove it. The unconscious part of the mind is still a mystery: perhaps I had all the clues before me and then joined the dots with a logical guess. Better still is that I could see a picture in my head of why it should be true, part of which is guided by physical intuition too. Symplectic manifolds are important in physics and are also known as Hamiltonian systems.

I had a bit of difficulty finding a way to prove my intuition to be correct. A quick scour of the web revealed little that directly stated that the a symplectic manifold is topologically simple (ie, simply connected). To 'prove' this answer I made a few logical but simple connections between related ideas (there is a pun in there somewhere).

My reasoning went like this: a symplectic manifold is described by a symplectic group. It turns out that a symplectic group is an alternative name for the Abelian group. That alone should be enough to prove the statement; however, I didn't quite find the answer at that stage (at least not on wiki). The Abelian group is a sub-group of the Poincare group which IS simply connected. Hence, a symplectic manifold is simply connected. I believe that this all makes sense and that there are no flaws. The logic is simple and deductive but I grant the possibility that I could have misunderstood something.

This is a profound statement for physics. A symplectic manifold (now called phase-space) is one that applies to all Hamiltonian systems. If a symplectic manifold is simply connected then all points in phase-space can be continuously transformed from one to another. This is an inherent statement of conservation, the end points are fixed but the path between the two can be continuously deformed however there is only one path: this is much like the variational principle (essentially the principle of least action).

From Noether's theorem we know that the Poincare group, of which the Abelian (symplectic) group is a subgroup, is a fundamental statement about the symmetries and conservation laws of nature. To restate Noether: behind every conservation law is a differentiable symmetry. It is clear that the topological nature of symplectic manifolds corroborates with the definition of the manifold being smooth and differentiable. The various mathematical definitions reinforce one another and make sense with what we expect from a physical (Hamiltonian) system.