Remarks on Poisson Brackets and Symplectic Structure
String Theory without the Baggage, Episode 28
In Classical Dynamics, the Poisson bracket packages the dynamics of the theory into a that Algebraists can be pleased with. In terms of the basic, nonrelativistic Hamiltonian:
Here q is a vector of coordinates on the configuration space Q and p is its canonical momentum. Phase space is then the cotangent bundle of Q,
Physicists call this construction of phase space canonical, and for once that aligns with a mathematicians’ use of the word: that is, there is a definitive way to do this.
Let’s work specifically with the components of p and q.
For a physicist that involve the Lagrangian:
This Hamiltonian structure comes with a Poisson Bracket, defined by
These are devised so that
The classic example would be a point particle moving in three-dimensions.
The full phase space,
carries1 a somewhat distinguished two-form ω,
Careful consideration of ω reveals that is essentially defines the Poisson Bracket posted above.
This is sort of a trivial exercise for R^6. As with other things we’ve discussed, this is heavy technology which seems trivial for familiar systems. The point is that ω defines how the dynamics of a system works. In this simple case, you still need to specify the potential V(q), but once you do that, ω helps you compute the trajectories of the given particle. ω, in some coarse sense, captures the laws of motion.
Aside from its utility much later on, we bring it out to highlight one point in particular: the existence of symplectomorphisms of P.
An Aside on Symplectomorphisms
You are probably aware that when solving problems of dynamics, we are free to pick whatever coordinate system on Q suits our needs best. Or even one that doesn’t. Spherical coordinates and central potentials come to mind. The physics doesn’t matter on what coordinates you choose.
General Relativity of course takes this observation to the extreme and demands local invariance under these coordinate changes. This local invariance gives a locally invariant object: curvature, which we interpret as the force of Gravitation.
For almost all physical systems of interest, dynamics takes place on the very phase space we constructed above: the cotangent bundle of the configuration space. Of course, what the precise nature of Q isn’t always obvious from the physical problem. We’ve already discussed how constrained systems like some descriptions of special relativity imposes some ambiguity around the definition of that phase space.
Nevertheless, the cotangent bundle construction guarantees us a two-form ω, the symplectic two-form, which characterizes the dynamics in terms of the Poisson Brackets.
We mention this because there is a broader class of coordinate transformations we may consider, the symplectomorphisms of P. Any change of coordinates that preserves ω represents the same dynamical system.
For example, if we replace q with p and vice versa2, ω will remain the same and the dynamics will be unchanged3. Historically, physicists have called this generalized coordinate transformations.
An even-dimensional manifold equipped4 with such an ω can be equipped with a physical, dynamical system. Such symplectic manifolds - these phase spaces - afford coordinate transformations that preserve ω. Unlike the relationship between smooth manifolds and curvature - which is a local invariant of coordinate symmetries - symplectic manifolds carry no local invariants. They are strictly global. Hence they are harder to work with.
In particular, we cannot zoom in on a patch of phase space and divine anything about the dynamical rules - or their consistency - from only this coordinate patch. We cannot use some kind of normal coordinate linear approximation to simplify our calculations, as we can with studying smooth manifolds.
As is hopefully clear to you now, these global concerns and ideas are easy to state. But once we start adding constraints into our system - and infinite dimensional gauge symmetries such as diffeomorphism invariance - the details aren’t easy to work out.
TL;DR
Given a dynamical system, there is typically no such thing as the unique, correct definition momentum. We’ve already seen this in studying the relativistic point particle:
Ensuring your dynamics are correct amounts to demanding that phase space be a symplectic manifold. Ensuring this is a global problem that is particularly tricky in the case of constrained systems.
The symplectic structure is encapsulated in the Poisson Brackets of the classical system. As the experts are surely aware, Poisson Brackets immediately5 generalize to commutators in the quantum theory.
Next time, we’ll apply these ideas to the relativistic string.
Here we are talking about differential forms and the exterior algebra over R^6. Roughly, these are equivalent to integral measures. For a familiar example, Q = R^3, we can perform line integrals, surface integrals or volume integrals. So we have three independent line integrals dq_i, three independent surface integral measures dq_i ^ dq_j (antisymmetric in ij, or otherwise vectors normal to the surface) , and one independent volume measure, dxdydz, which is totally antisymmetric. For R^6 the combinatorics are a little more complicated, but no by much. For ω here, we are interested in specifically in linear subspace of these “surface integral measures” which has enjoys one momentum and one position direction.
Up to a possible minus sign. Careful with those minus signs!
Note that these transformations can also be continuous, as with rotations between coordinates. However, some transformations are clearly not allowed. For example, we cannot take x to be the momentum for y. That would not preserve ω.
There are other technical requirements that define dynamics here, such as the need for ω to be closed, that we don’t concern ourselves with at this point.
Again, assuming you’ve got your constraints consistently worked out. Dirac and others have spent a lot of effort on this problem.