I recently read the following passage on page 137 in volume I of 'Quantum Fields and Strings: A course for Mathematicians' by Pierre Deligne and others (note that I am no mathematician and have not gotten too far into reading the book, so bear with me):

*A physical system is usually described in terms of states and observable. In the Hamiltonian framework of classical mechanics, the states form a symplectic manifold $(M,\omega)$ and the observables are functions on $M$. The dynamics of a (time invariant) system is a one parameter group of symplectic diffeomorphisms; the generating function is the energy or Hamiltonian. The system is said to be free if $(M,\omega)$ is an affine symplectic space and the motion is by a one-parameter group of symplectic transformations. This general descriptions applies to any system that includes classical particles, fields, strings and other types of objects.*

The last sentence, in particular, has really intrigued me. It implies a most general procedure for quantizing all systems encountered in physics. I haven't understood the part on symplectic diffeomorphisms or free systems. Here are my questions:

Given a constraint-free phase-space, equipped with the symplectic 2-form, we can construct a Hilbert space of states and a set of observables and start calculating expectation values and probability amplitudes. Since the passage says that this applies to point particles, fields and strings, I assume this is all there is to quantization of any system. Is this true?

What is the general procedure for such a construction, given $M$ and $\omega$?

For classical fields and strings what does this symplectic 2-form look like? (isn't it of infinite dimension?)

Also I assume for constrained systems like in loop quantum gravity, one needs to solve for the constraints and cast the system as a constraint-free before constructing the phase, am I correct?

I don't know what 'the one-parameter group of symplectic diffeomorphisms' are. How are the different from ordinary diffeomorphisms on a manifold? Since diffeomorphisms may be looked at as a tiny co-ordinate changes, are these diffeomorphisms canonical transformations? (is time or its equivalent the parameter mentioned above?)

What is meant by a 'free' system as given above?

By 'affine' I assume they mean that the connection on $M$ is flat and torsion free, what would this physically mean in the case of a one dimensional-oscillator or in the case of systems with strings and fields?

In systems that do not permit a Lagrangian description, how exactly do we define the cotangent bundle necessary for the conjugate momenta? If we can't, then how do we construct the symplectic 2-form? If we can't construct the symplectic 2-form, then how do we quantize the system?

I have asked a lot of long questions, so please answer as many as you can and link relevant articles.

This post imported from StackExchange Physics at 2014-04-12 19:04 (UCT), posted by SE-user dj_mummy