Symplectic Geometry

A 1993 licenciatura thesis from Instituto Superior Técnico, transcribed and annotated. It asks what survives of symplectic geometry when the form is allowed to degenerate — when some directions become invisible to it — and shows that what remains, once those directions are collapsed away, is again a phase space on which mechanics works.

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This is a short version. Read the complete edition →

Abstract. We examine what survives of symplectic geometry when the non-degeneracy of ω is given up, and how far the algebra of observables can still be built. Non-degeneracy is what makes ω♭ : TM → T*M an isomorphism, and therefore what assigns to each observable g a single field X_g with ι_Xgω = dg; once it fails, X_g is determined only up to the kernel of ω♭, and the Poisson bracket loses its meaning.

We show that if ω has constant rank the kernel is a subbundle of TM which, because ω is closed, is involutive and hence arises from a regular foliation. Taking the quotient of TM by the kernel directions, and of M by the leaves, yields a base which is again a symplectic manifold — and on it the Poisson brackets, the canonical transformations and the remaining structure are recovered.

Non-degeneracy, rank and kernel at a single point: what the form sees, and what it does not.

A word on where this started. In the longer account I put it like this:

I did not set out to weaken anything. I set out to write down the algebra of observables, and found that I could not.

There was no decision to generalise anything. There was an ordinary task — write down the bracket that turns a list of measurable quantities into a mechanics — and the discovery that it fails the moment the form has a direction it cannot see.

That task is less parochial than it sounds, and it is worth saying why. The bracket is what quantum mechanics inherits: quantisation replaces observables by operators and the Poisson bracket by the commutator, so a classical theory with no well-defined bracket has nowhere to be quantised to. And general relativity is precisely a theory whose form degenerates — its Hamiltonian is itself a sum of constraints, and the directions the form cannot see are the relabellings of points that leave the geometry alone. So the question of what a bracket can still mean once non-degeneracy fails is not a corner of the formalism. It is the first obstacle between general relativity and a quantum theory of it. That was the horizon this small piece of work was pointed at — even though, as the quotation says, it began from something a good deal more modest.

The second part takes up Abraham’s intrinsic definition of a second-order equation — a vector field X on TM with Tτ_M ∘ X = Id — exhibiting the rhombic diagram it obeys, the two distinct projections τ_TM and Tτ_M of TTM onto TM, and how the condition on the integral curves reduces, in local coordinates, to the classical second-order equation of motion.

The complete edition carries a commentary that follows the argument in nine steps, an afterword on gauge theories and general relativity, two glossaries and a set of illustrated ones, and the seven manuscript pages in facsimile. This, for instance, is where the phase space of a pendulum lives:

The phase space of the pendulum. On the left, the plane of the angle and the momentum, with the level sets of the energy: closed orbits about the centre, where the pendulum swings; open ones above and below, where it goes over the top; and between them the separatrix, through the saddle where the pendulum stands inverted. On the right, the same curves once the two edges are recognised as one state, so that the plane rolls into a cylinder.

Read it in full

The complete edition, the Portuguese transcription, the transcription set beside the original hand-drawn diagrams, a facsimile of the manuscript, an outside assessment, and the commentary.

Open the thesis  →

ferrelm.github.io/tese