#symplectic stuff

[ceausescue]

mostly what inspired this is that at least at my school you start with the simple case of like one electron, maybe solve it for a well or an infinite well or a hydrogen atom. and then you think about going up to 2 and you need to bring in all this heavy duty machinery with no real prep- the fact that you could, in the right circumstances, treat the one electron case with high school calc might as well be coincidence

the actual simple version of quantum mechanics is the finite dimensional case- you get to meet your operator and work with the entire state vector, no actual physical particles in sight to give you a false sense of security. introduce symplectic stuff at the same time, treat it as part of the upgrade from regular old probability along with the complex numbers. it could be beautiful

my crank belief is that if we taught undergrads probability and quantum computing, then stat mech and actual quantum physics, they would at least have different and interesting problems. it would be good for the mathcel fake physicists anyway and they're the ones that deserve rights

[thousandmaths]

Hamiltonian Mechanics (guestpost)

Hi dedicated readers! I'm one of @thousandmaths‘s more notorious colleagues who keeps a blog about math and philosphy. In an elaborate exchange deal, he and I have written guest posts on each others' blogs. You can read his post here.

While working on a farm last summer, I started reading Roger Penrose's Road to Reality, a wonderful book about philosphy, physics, and mathematics with a beautiful set of hand-drawn diagrams. Having studied math and computer science as an undergrad, the only physics course I took was quantum mechanics and I wanted to learn more.

The book didn't disappoint and I would highly recommend it to any student studying the sciences. He always offers the intution behind the mathematics so even if you skip the symbolism, you'll see the general idea. In this post, I want to give a quick introduction to one of the two organizing frameworks of modern physics that Penrose discusses, namely, Hamiltonian mechanics. The best way to learn is by example, so I'll follow a computation from our own Peter Olver's wonderful book Applications of Lie Groups to Differential Equations.

The Lagrangian and Hamiltonian Formalisms

There are two main points of view in modern physics. Named after their alleged founders, the Lagrangian and Hamiltonian formulations are equivalent in a precise technical sense (they are dual) so they are really two different views of the same phenomena. The general idea is that both offer frameworks for deriving the equations of motions for all physical systems, from general relativity to electrodynamics.

In the Lagrangian framework, one writes down an “energy functional” the solutions seek to minimize. A standard computation leads to something called the Euler-Lagrange equations of motion.

In the Hamiltonian framework, one writes down an “energy” that the solutions conserve. A standard computation leads to something called a Hamiltonian system of equations that describe the objects motion.

Here we discuss the “conserved quantity” (Hamiltonian) point of view. This framework requires two inputs:

A Hamiltonian $H$ (a scalar quantity depending on the solution that stays constant over time).

A configuration space $G$ (a space of all possible states of the system).

Requiring only these two pieces for the setup, the Hamiltonian formalism will churn out the equations of motion.

The Hamiltonian Perspective on Rigid Body Rotation

We will extract the equations of motion for the angular velocity of a rigid body $\Omega$ by making the following two choices:

If $u = (u_1, \, u_2, \, u_3)$ is the angular velocity and $I_1, I_2, I_3$ the moments of inertia (their definitions are a story of its own) then set $$H(u) = \frac{u_1^2}{2 I_1} + \frac{u_2^2}{2 I_2} + \frac{u_3^2}{2 I_3}. $$

 The configuration space for a rigid body in three dimensional space is $SO(3)$ (the space of all rotations).

The choice of $H$ might seem a bit arbitrary, but ignoring the normalization factors $I_1, I_2, I_3$, it looks like the usual kinetic energy $1/2 v^2$ in basic physics. Roughly speaking, the moments of inertia track the resistance of a body to rotation (increase in angular momentum via torque).

The choice of the configuration space is clear. Any rigid body floating in space will rotate around a fixed point (its center of mass) and thus at any time can be described via a rotation from an initial orientation. Hence, the motion of of a cube rotating about its center can be described as a series of rotation matrices $R(t) \in SO(3)$.

The Hamiltonian Formalism in Action

Okay, so we have made our two necessary choices. Running through the Hamiltonian formalism will produce the final equations of motion. One can phrase Hamiltonian dynamics in terms of symplectic geometry, but for our purposes, we can take the less jargon-intensive approach of the so-called "Poisson bracket". If $H$ is our conserved quantity chosen above, the Hamiltonian formalism tells us that the angular velocity $u$ evolves as

$$ \frac{du}{dt} = \{ H, \, u \}. $$

The Poisson bracket is thus the key part of the theory. Where does it come from? The Poisson bracket actually accepts two functions so the notation $\{ H, \, u \}$ means that we apply the Poisson bracket to each component $(u_1, \, u_2, \, u_3)$ individually and assemble back into a vector.

A choice of Poisson bracket introduces tons of structure to a manifold. But we can go the other way, and use the structure of a manifold to generate a Poisson bracket.

The space $SO(3)$ of all rotations in three dimension is a in fact a group under matrix multiplication. Moreover, it is a highly symmetric group: a so-called Lie group. That is, it is both an algebraic group and a manifold. These groups are fantastic because they are so shockingly symmetric that restricting attention to infinitesimal perturbations of the identity (tangent space to the 'do nothing' rotation) is enough to (almost) completely describe the group. That is, we can "linearize" the group and lose (almost) no information.

The “linearization” of a Lie group is called its corresponding Lie algebra. A Lie algebra is a vector space with a special operation called the Lie bracket that it inherits from the Lie group structure. In the case of $SO(3)$, the Lie algebra $\mathfrak{so}(3)$ is a three-dimensional space of matrices. This space has enough structure to produce a Poisson bracket. However, we must again pivot to one other space.

The “magical” feature of the Hamiltonian framework is that the correct setting is the dual space $\mathfrak{so}^*(3)$ to $\mathfrak{so}(3)$. Intuitively, we operate in the dual since Hamiltonian flows are described by $\nabla H$ (gradient of $H$). As one learns in differential geometry, the gradient is best viewed as a linear functional (element of the dual space) to ensure coordinate invariance.

Following definitions and such, one computes that the Poisson bracket on $\mathfrak{so}^*(3)$ is given by

$$ \{ H, u \} = -u \times \nabla H $$

where $\times$ denotes the cross product (the geometric reason for this is the subject of another post).

Since $u$ roughly represents the velocity of rotation (it is not precisely the time derivative $R'(t)$ however), we can view the evolution

$$ \frac{du}{dt} = \{ H, u \} = -u \times \nabla H $$

as describing a flow in $\mathfrak{so}(3)$. Expanding our definition for $H$ and the cross product finally gives the following equations of motion

$$ \begin{align*} \frac{du_1}{dt} &= \frac{u_2 u_3}{I_2} - \frac{u_2 u_3}{I_3}, \\\ \frac{du_2}{dt} &= \frac{u_1 u_3}{I_3} - \frac{u_1 u_3}{I_1}, \\\ \frac{du_3}{dt} &= \frac{u_1 u_2}{I_1} - \frac{u_1 u_2}{I_2}. \end{align*}$$

One can then reconstruct the actual trajectory $R(t)$ of the rigid body by standard formulas.

Further thoughts

In addition to simple rigid body dynamics, all of the field theories, from quantum mechanics to fluid dynamics can be extracted from a Lagrangian or Hamiltonian point of view. On the surface, this level of grand unification is almost magical. Does that mean physics is “solved”? After all, for any new theory, one must simply generate an appropriate Lagrangian or Hamiltonian and then “turn the crank" to derive the equations of motion. Nothing could be easier.

Of course, physics is far from “complete” (in many senses of the word) and even in the Lagrangian / Hamiltonian frameworks there is always the problem of choosing the “right” Lagrangian or Hamiltonian to plug into the framework.

More broadly, these frameworks are not truths but merely formalisms and the value of a formalism is precisely how much it guides and organizes our fallible intuition. Penrose hints at some problems with these approaches:

However, I must confess my unease with this as a fundamental approach. I have difficulties in formulating my unease, but it has something to do with the generality of the Lagrangian approach, so that little guidance may be provided towards finding the correct theories. (p 491)

Both frameworks are merely tools for organizing good ideas. As always, there is a “conservation of creativity” in the sense that humans must guess at good models in the first place. No amount of symbol pushes will remove this requirement. At best, abstraction simply focuses our attention on the important features.

[transgenderer]

i was talking with one of the upper year grad students at my uni and apparently the only prof who does riem geo stuff is a category theorist....theres a bunch of people who like symplectic stuff tho. so maybe ill just do that.

[macla539ac]

"Only the intelletual intuition can save you, but only those, who are predestined by mysterious might."

the intuition is not something fixed like an extra toe or even ingrained through a lifetime like a sexual fetish. The core of what humans do is acquire new abstractions in such a way that what was previously convoluted is now intuitive. "Intuitive" is a word for "stuff I already know". Symplectic geometry is unintuitive until you learn it.

[onions-and-oranges-blog]

Emily Does Summer School (also some stuff about Riemannian surfaces)

So, considering that I knew nothing about Kahler Geometry before starting this program (or algebraic geometry) I think things are going pretty well!! I have been following the lectures pretty closely (except algebraic geometry.. that will definitely take further study later). I got a pretty hard look at Truth today because I was completely lost during the algebraic geometry (ag) lecture and then tried to tackle the problem set to see if I could figure stuff out... and was not successful at all. Fortunately the TAs and professors who were with us helped me out, but I needed a lot of help and I can’t do math with an audience unless I am really comfortable/confident. Which... I wasn’t. Tomorrow should be better though. I am doing well at talking to students; everyone is very nice and, guess what, math people in general are not super outgoing and so I am fitting in very well socially. It is interesting to talk to the other students and learn about their schools, there are some students from really good math programs here! After today I should be better about talking to (not being scared by) the professors.

tl;dr, things are going better than expected!!

On to the math! According to the wikipedia page, a Kahler manifold is one with three compatible structures: (1) a complex structure, (2) a Riemannian structure, and (3) a symplectic structure. We skipped talking about (1) because most people should be pretty familiar with that by now (lol I haven’t taken complex analysis but w/e I’m managing). We also aren’t going to talk about (3) because apparently it is a fairly new branch of research and isn’t as accessible as (2).

SO! What is a Riemannian surface? We are working with Gabor Szekelyhidi slowly toward this concept from an analysis perspective; so far, (from what I can tell), a Riemannian surface is one you can create by stitching together copies of $\IC$ and adding points at infinity to create a compact manifold. You can then map this back to the complex numbers using holomorphic fuctions, and the identifications constructed through these functions allows you to do calculus on this manifold where you can circumvent certain problems, such as not having an injective square root function in the complex numbers (this part I am still unclear, so apologies if I made a mistake), as opposed to sitting and crying (idk, if you’re me you might do this anyway). The professor lecturing on analysis is explaining mainly through pictures, and I am hoping to be able to post something later with more detailed notes. 

We are working much faster toward this concept from an algebraic perspective, because the dude teaching ag, Claudiu Raicu, an associate professor at Notre Dame, is really whipping us along. This is an interesting way to go about doing things, because as far as I know, only a few students have enough of a background in ag to keep up, or even have really any idea what is going on. According to ag, a Riemann surface is a non-singular (affine or projective) curve over the field of complex numbers. A Riemann surface is compact if and only if it is projective. 

There are a *lot* of concepts to unpack here, and each is pretty heavy. A lot of what we are doing involves quotienting the ring of polynomials in 2 or 3 variables by polynomials and determining the size of the result to give you information about the multiplicities of intersections of the polynomials. It’s nice that I am using my algebraic knowledge (oh man, am I digging out ideas that I honestly thought I wouldn’t use again. Turns out local rings are *super* important, and sou is Bezout’s Theorem), and I will need to dredge through this very carefully this week and in the future to make sure I am understanding everything.

The final lecturer is Andrei Jorza, who is lecturing on applications of Riemannian surfaces to number theory. We constructed a fundamental domain for the upper half plane in the complex numbers, and are using the properties of arithmetic on that domain to help prove an equivalence discovered by Ramanujan... That I am not going to get in to right now. To be honest I have been paying the least amount of attention to these problems because they appear to be straight forward; showing that summations are equivalent, and etc. They are definitely good practice and I will do my best to type up those notes and problems later as well.

I will try to update when I can!