Do you think that symplectic reduction (Marsden Weinstein reduction) is interesting from a physical point of view? If so, why? Does it give you some new physical insights?
There are some possible answers I often heard of, but I don't really understand it. Perhaps you could comment these points. Explain and illustrate why they are good reasons or if not, explain why they are nonsense:
1) Symplectic reduction is interesting because "it simplifies" the system under consideration because you exploit symmetries to eliminate some redundant degrees of freedom. I do not really understand what's the point here because in general reduction leads to a more complicated geometry. (Or even to singular spaces if you consider more general reduction settings).
2) It's interesting because it is a toy model for gauge theories.
3) It's interesting because if you want to "quantize" a system, from a conceptual point of view, one should start from the reduced system, from the "real" phase space. I don't see why one should do this for nonrelativistic quantum systems. Even for gauge theories I don't get the point, because the usual procedure is quantize the unreduced system (via gauge fixing), isn't it?
If there are points which make symplectic reduction interesting from a physical point of view, are there physical reasons why one should study reduction by stages?
Added; after reading José Figueroa-O'Farrill's answer I had some thoughts I should add:
I am still by far not an expert in gauge theories. But I think, that in gauge theories, one typically has redundant variables, which have no or at least not a direct physical interpretation. So I would agree that the "physical" dynamics takes place at the quotient in the case that the gauge theory itself has a physical meaning (in particular experimental evidence). Concerning quantization, however, if I am not mistaken, the only known quantum gauge theory which corresponding classical gauge theory which experimental support is quantum electrodynamics. For the other physical relevant quantum gauge theories, I think, the classical counterparts play just the role of auxiliary theories in some sense. In this case I would agree, that on the quantum side only the reduced space has physical meaning, but I think for the corresponding classical one, this seems to be a rather pointless question. So the question remains, why it is physically interesting to study on the classical side the reduced phase space in case of gauge theories. Moreover as José Figueroa-O'Farrill points out, the classical reduced space in most cases too complicated to quantize it directly, one would use some kind of extrinsic quantization as BRST instead. I don't know exactly how the situation for gravitation is. I think one can formulate ART as classical gauge theory. But makes it sense in this case to study the reduced classical phase space for quantization purposes? I guess not.
Best Answer
Symplectic reduction arises naturally in constrained hamiltonian systems, e.g., gauge theories. So it is not just a question of it being "interesting" as much as a fact of life.
The way to deal with coisotropic constraints -- those whose zero locus is a coisotropic submanifold -- is via symplectic reduction. The real (read, physically meaningful) dynamics are taking place in the symplectic quotient, which is the standard quotient of the zero locus of the constraints by the (integrable) distribution defined by the hamiltonian vector fields associated to the constraints.
Now, as you point out, the symplectic quotient is usually much more complicated geometrically than the original symplectic manifold and this makes working there cumbersome. For instance, quantising the symplectic quotient is usually difficult. Luckily, one can go the other way: instead of performing the symplectic quotient and then quantising, one can first quantise the constrained system and then do a quantum version of the symplectic quotient. One such procedure, which works in many gauge theories, is BRST quantisation. This is a homological approach to the quantisation of constrained hamiltonian systems. It has the virtue that it preserves the symmetries of the original system which "gauge fixing" typically destroys.