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So I've studied a bit about symplectic geometry, and I know that phase space is a symplectic manifold, and the symplectic form induces a poisson bracket. However, what is the physical meaning of the symplectic form? Perhaps it's simply the same as asking what the meaning is of the poisson bracket. What does the poisson bracket say about a system? And what is the physical meaning of observables?

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    $\begingroup$ Have you looked at the first chapter of McDuff and Salamon's book on symplectic topology? I don't know if that will answer your question but it should be useful. $\endgroup$
    – user98602
    Feb 1 '19 at 23:20
  • $\begingroup$ a little bit, just scanned through it $\endgroup$ Feb 1 '19 at 23:54
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    $\begingroup$ You might find "What is symplectic geometry?" by McDuff helpful for getting intuition. Abstract: "In this talk we explain the elements of symplectic geometry, and sketch the proof of one of its foundational results — Gromov’s nonsqueezing theorem — using J-holomorphic curves." $\endgroup$ Feb 2 '19 at 3:55
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This topic is incredibly deep so my answer barely skims the surface, however I believe you're looking for an intuitive meaning of the symplectic form. Consider the standard symplectic form

$$ \omega = \sum_i d x_i \wedge d \lambda_i $$

This can also be written as

$$ \omega = \begin{bmatrix} \mathbf{0} & \mathbf{I} \\ -\mathbf{I} & \mathbf{0} \end{bmatrix} $$

Now consider a rotation matrix

$$ R_\theta = \begin{bmatrix} \cos(\theta) & \sin(\theta) \\ -\sin(\theta) & \cos(\theta) \end{bmatrix} $$

By setting $\theta = 90^o$, you find the data of these tensors are equivalent. Ergo the standard symplectic form is very similar to a rotation matrix that rotates vectors by 90 degrees. Consider the symplectic form of Hamiltonian mechanics

$$ dH = \iota_{X_H} \omega $$

where $dH$ is the exterior derivative of the Hamiltonian and $\iota$ is the interior derivative. This exterior derivative generates a covector field whose covectors point in the direction of steepest descent. Rotate these covectors by 90 degrees and they point tangential to the level sets of $H$. In Hamiltonian mechanics, motion almost always lies on level sets of $H$.

Ultimately, I like to think about the symplectic form as a rotation matrix that properly encodes the structure of mechanics, but be aware that due to its differential geometric nature, it's much deeper than I lead on.

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Before I start, I would like to readily admit that the math behind this topic is beyond my skill. Question what I write. And please, if you see anything wrong, comment so that I can correct it.

I came across symplectic geometry as part of studying groups. Group theory can be summarized as the study of symmetry. Symmetry is fascinating to me because it permits you to make statements about a system that you do not otherwise understand. For example, if I create a pair of entangled electrons, I may not know whether one is spin up or down, but I can create them in a symmetric way such that if one is up the other is down, and vice versa. I can make experiments which take advantage of this, as we do with quantum encryption. At a more macroscopic level, if I am sparring in martial arts, I may not know where my opponent is punching, but I do know that for wherever he punches, there's a corresponding position he must contort his body into to avoid tearing tendons. This can be used in fascinating ways. If I can find a place that his body must be if he threw a punch, but where his body wont be if he doesn't throw a punch, and I can get there first, I can make sure that I punish him for throwing a punch if he throws it, but not hurt him if he didn't throw it. (This is often overlooked in combat, but in more civilized venues, being able to do something that only hurts them if they are doing bad things is highly prized).

So what is the Poisson bracket? The philosopher in me would say either that its a curiosity about how objects change state over time (time evolution), or that we may have developed our sense of time to take advantage of an apparent symmetry in the world around us. Physical systems seem to evolve this way if you think of them in terms of phase space. Phrased in the other way, if we can't sense the time evolution of the system captured by the Poisson bracket, we typically don't think of it as physical. We apply metaphysical terms to it, such as the "agency" of individuals.

What is important about the symplectic form of the Hamiltonian equations is that it points out that any symmetry in the time evolution of the physical world must be a symplectomorphism (in particular, a Hamiltonian symplectomorphism). This ties into observables. If you want to state anything about the state of a system, you either need to know enough about a state in the past, or you need to state something that can be proven with symmetry. If I want to state something about the state of an electron, I either need to have measured it in that state, or I need to state something I can know by symmetry, such as the result of interacting it with its entangled pair.

One of the key aspects of Hamiltonian mechanics is the ability to define canonical transformations. These are transforms of how we model a system which preserve Hamiltonian mechanics (read: don't break physics). They let us start with one system which is governed by physical laws (perhaps the states of electrons in cartesian coordinates) and transform it into another system which is governed by physical laws (perhaps the same electrons, but viewed in spherical coordinates).

The symplectic form of the Hamiltonian equations gives rise to Liouville's theorem, which states that these canonical transformations preserve volumes in state space. In other words, if we choose to model the world using symplectic forms, we find this volume preserving behavior occurs. If we model it with other tools, it doesn't necessarily occur.

This is a big deal for observables. In a quantum system, there is a phase space describing the system. When we observe it, we configure our test apparatus to test in a way which is easy to understand in one coordinate system (such as "spin up" "spin down"). We show that the symmetries in the system will ensure a time evolution that leads to one answer or the other. Now since we knew the phase space from the setup of the system, we can apply a canonical transformation to go from the coordinate system we used to describe the system at first to the coordinate system we want to use for test measurement.

And when we do this, we instantly see uncertainty. Since the volume of the phase space must be preserved, and we're effectively squeezing the phase space in one dimension, another dimension must expand in order to preserve the volume. You can demonstrate this in any number of ways, not just with symplectic forms, but symplectic forms make it obvious to see.

But in the end, I must wax philosophic. What is important about these forms is not necessarily that they're fundamental to reality. They do indeed appear to be fundamental to physics. If you believe that reality only consists of the physical world, then you will find that they must be fundamental to reality too. But if you consider that its possible there might be more, the importance of these forms gets bit more peculiar. It may be that they are important because humans have found them to be a useful lens with which to view reality around them. We may thus ascribe importance to these forms simply because they are important to what we saw in the past, and we see them because we ascribed importance to them in the past.

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