#015: More in depth exploration of coordinate invariant measures
Category: Measure theory - Tags: General theory, Classical mechanics, Complex analysis
Find if there are more mathematically interesting and precise ways to express the idea of invariant measures.
Mathematical problem. In our derivation of classical mechanics, we recover symplectic spaces by requiring that the measure needed to quantify the number of states in a region must be coordinate invariant. That is, we need to be able to change coordinates while keeping the measure invariant. This leads to the need for a conjugate quantity such that \(dq \wedge dp = dq' \wedge dp'\). The question is whether we can formalize this requirement in a more abstract way, since other theories (e.g. quantum mechanics) must have a similar requirement (e.g. for differentiable manifolds, there is the idea of generalized complex structure).
One way to approach the problem is to see what properties measures possess on the complex plane. This may be related to understanding why integration on the complex plane is more powerful than on the real line. Trying to justify those properties from the physics may lead to new ideas.
Physical significance. The requirement of a coordinate invariant measure stems from the principle of relativity and the idea that nature must account for preservation of number of states during deterministic and reversible evolution without specific reference to reference frames.
Here we want to find the most minimal and general mathematical argument to that effect.