I'm asking this question because of an article in New Scientist about a recent preprint by a group including Lee Smolin. I haven't taken the time to comprehend the paper completely. My knowledge of differential and pseudo-Riemannian geometry is out of practice, so it'd take me a while to do so. Even so, the paper put some ideas in my head about phase space and symplectic manifolds that can be put as simple questions but I couldn't answer with some basic Googling. Also, this is resting on the (mis)understanding that phase space is a symplectic manifold. If this is wrong, these questions might be meaningless.

  1. Is there a canonical symplectic manifold that is associated with a given pseudo-Riemannian manifold? I'm guessing yes, because a pseudo-Riemannian manifold is itself a differential manifold, and I think the tangent bundle is a symplectic manifold.

  2. If the space we started with has a metric, does this imbue the associated symplectic manifold with any structure? e.g. is the symplectic form limited by or related to the metric or something similar?

  3. Could any ideas from SR/GR be extended to this symplectic space such that we learn something new? e.g. could there be something like the Einstein Field Equations (EFEs), written for the 8-dimensional phase space, that have new solutions that are realistic but wouldn't be found using the EFEs? Or could there be non-canonical symplectic manifolds associated with a given pseudo-Riemannian space? (I suspect Darboux's theorem says "no".)

This is mostly crazy speculation based on knowledge I haven't exercised in a few years now. For all I know, there's some underlying concept that makes this pointless or I've just totally misunderstood the paper in the first place. Also, I've posted here because of the theoretical physics nature of the problem, but I accept it may be better served on another SE.

  • $\begingroup$ Phase space is a symplectic manifold but tangent bundle isn't. Only cotangent bundle carries the canonical symplectic form. But except for these details, the question is very nice, +1 $\endgroup$ – Marek Aug 16 '11 at 8:33
  • $\begingroup$ As a quick comment, I'd like to point out that taking cotangent bundle of a space-time manifold (while possible mathematically) doesn't give you what you want. Classically, you only take the space and model dynamics by flows on that space. If you want to generalize this to GR you need to introduce folliations -- slicing of the space-time to space-like hypersurfaces -- and you can proceed from this. Obviously, this is not canonical. The classical case is a special case where the slicing is given by the same space at each time and this lets you avoid all technicalities involved. $\endgroup$ – Marek Aug 16 '11 at 8:38
  • $\begingroup$ To add two points to Marek: 1. the canonical/Hamiltonian formulation of GR is called the ADM formulation; 2. if you have a singularity/black hole, to maintain a Hamiltonian formulation in the presence of a horizon, one should use the concept of a "dynamical horizon" and eventually an "isolated horizon", rather than the event horizon. $\endgroup$ – genneth Aug 16 '11 at 9:58

This question addresses many aspects of the physics of indefinite signature spaces. I'll try to answer you by examples.

  1. As already mentioned in the comments, the cotangent bundle of a pseudo-Riemannian manifold is a symplectic manifold. The basic example of a physical system having this phase space is the relativistic free particle.

It is very interesting that the quantization of the relativistic particle produces another type of indefinte signature on the cotangent bundle: The indefinite metric in the free particle Lagrangians produces a "wrong" sign in the time component of the symplectic structure of the cotangent bundle. The canonical quantization of this symplectic structure produces negative norm states which are eliminated by the imposition of a physical constraint. One possible way to restrict to the physical subspace is the Gupta-Bleuler method, originated in the covariant quantization of electrodynamics, but can be used in the case of the free relativistic particle.

  1. The pseudo-Riemannian metric on the base manifold results the association of the cotangent bundle with an $SO(M,N)$ principal bundle. ($(M,N)$ is the metric's signature) via the contragradient fundamental representation. This allows to define tensor fields on the base space as sections of associated vector bundles. If in addition the base space has a spin structure, it is possible to lift this bundle to a $Spin(M,N)$ bundle and define spinor fields as sections of aspinor bundle (The associated bundle corresponding to the spinor representation).

  2. There are many Hamiltonian approaches, in which the Einstein equations are obtained as the equations of motion. I'll describe to you the the method that I think is the most interesting: MacDowell–Mansouri Gravity, see for example the following article by Derek Wise. In this approach one construct a gauge theory based on the de-Sitter or anti de-Sitter group. One identifies the spin connection with the components corresponding to the Lorentz subgroup generators, and the Vielbeins with the components of the generators dS/Lorentz, or AdS/Lorentz. In this setting it is possible to construct a hodge star operation not depending on a metric. It turns out that the Yang Mills Lagrangian, when written in terms of the spin connection and the Vielbeins turns out to be equal to the Einstein-Cartan Lagrangian (with a cosmological constant) which gives the Einstein equations of motion + a topological term.

Now, the symplectic geometry of this theory is similar to the well known Yang-Mills with a compact gauge group, for example the vanishing momenta of the time components of the connection give rise to the Gauss law constraints.


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