For a given Hamiltonian, is the space of histories of a classical system the same as the symplectic manifold?

Do I have to take care of gauge equivalences and if so, is this only an issue for fields (not for trajectories)?

  • $\begingroup$ Could you clarify you second question? Which gauge equivalences are you referring to? Also, is this question about classical mechanics with finite-dimensional phase spaces (as with systems of particles) or is it about classical field theory? $\endgroup$ – joshphysics Dec 15 '13 at 21:42
  • $\begingroup$ @joshphysics: The picture in my mind is a classical phase space for trajectories, but the question for a bijection between the configuration space and the histories of somehthing extends automatically for any quantity whos time evolution is uniquely given by a full set of initial conditions. The second question regards any degrees of freedom in the kernel of the time evolution (gauge fields or "co-moving" morphisms of the quantity of interest), which descibe the same physical system. If these need their own dynamics, then I figure comparing the phase space alone to the history might fail. $\endgroup$ – Nikolaj-K Dec 15 '13 at 21:52

Basically, the answer to both questions is yes. In the field theory case, however, this is true (optimistically speaking according to Witten-1) - in the case when the field equations are hyperbolic wave equations. In this case one needs the initial conditions to be defined on global Cauchy hypersurface. In this case there is a one to one correspondence between the space of solutions and the space of initial data.

Actually, we always construct the symplectic structure from the Lagrangian in the course of the procedure of canonical quantization, when we construct the conjugate momenta and declare canonical Poisson brackets between the coordinates and their momenta. As a matter of fact, when the coordinates define an affine space (or superspace) like in the theory of spinless point particles or the case of scalar, spinor or Yang-Mills fields, then the phase space is an affine symplectic vector (super)space.

Now, your second assertion is also true (please see Witten-1 and: Witten-2 ). In the presence of gauge freedom, (which is a choice or a declaration that we make); the space of solutions modulo the gauge freedom is symplectic and is identified with the physical phase space of the theory. This identification is natural because we measure only gauge invariant quantities. The first assertion that this space is symplectic stems from a deep theorem in symplectic geometry called the Marsden-Weinstein reduction. (This theorem is true in finite dimensions but was generalized to many infinite dimensional cases), which asserts that the reduction of a symplectic manifold by a gauge freedom is also symplectic.

Another way to look at the gauge redundancy is to observe that there are certain functions of the initial data that can be arbitrarily evolved in time and still the field equations are satisfied, please see Belot.

Now, in contrast to the unreduced symplectic manifolds which are in our cases nice symplectic vector spaces, the reduced manifolds are actually orbifolds. These spaces contain singular submanifold which may be classically harmless as long as one stays away from them, but have profound effect after quantization, which is not very well understood until now.

There are cases when the unreduced phase spaces of certain gauge theories are infinite dimensional while the reduced ones are finite dimensional. These cases led Witten to one of his greatest achievements, namely in the Chern-Simons theory or in the 2+1 dimensional gravity theory in Witten-2.

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  • $\begingroup$ Thanks for your answer! You say "the space of solutions modulo the gauge freedom is symplectic and is identified with the physical phase space of the theory. This identification is natural because we measure only gauge invariant quantities.", do you also mean that formally in the sense of quotient spaces. I've heard a voice saying this might be a dangerous thing to do, something about global behavior and anomalies. Might be related to the profound effect line of yours. But in any case, you say the space of physical histories is bigger than the data space, that's what I take away here. $\endgroup$ – Nikolaj-K Dec 16 '13 at 22:43
  • $\begingroup$ @Nick Kidman 1) Yes, it is a quotient space often denoted as $ \mathcal{M}//G$ (with two lines) called the Marsden-Weinstein quotient or the symplectic quotient. $ \mathcal{M}$ is the unreduced phase space and $G$ is the gauge group.In a finite number of dimensions (where we can count), the dimension of the reduced space is $ \mathrm{dim] \mathcal{M} - 2\mathrm{dim] G$. The factor 2 comes from the fact that we need to remove one dimension per symmetry and another dimension for gauge fixing. 2) We can reduce only by the anomaly free subgroup of the gauge group. $\endgroup$ – David Bar Moshe Dec 17 '13 at 10:19

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