Quantization and Cohomology (Week 10)

January 17, 2007

Quantization and Cohomology (Week 10)

Posted by John Baez

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I’m back in town and eager to continue lecturing about Quantization and Cohomology!

In last Fall’s lectures, we discussed the Lagrangian and Hamiltonian approaches to the classical mechanics of point particles, and sketched how these could be generalized to strings and higher-dimensional membranes by a process that we’ll ultimately see as categorification. This quarter we’ll start by bringing the quantum aspects of the theory into the game.

We begin with a review and a quick discussion of some basic but still incompletely understood questions:

Week 10 (Jan. 16) - A quick review of classical versus quantum mechanics, in both the Lagrangian and Hamiltonian approaches. What are path integrals, really? How do we quantize a classical Hamiltonian to obtain a quantum one?
- Supplementary reading: John Baez, Geometric quantization.

The notes from last class — the final class of the Winter quarter — are here. Next week’s notes are here.

Posted at January 17, 2007 12:42 AM UTC

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16 Comments & 2 Trackbacks

Re: Quantization and Cohomology (Week 10)

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The notes say that Schrödinger

guess[ed] the quantization rule

(1)

p \mapsto \frac \hbar \i \vec \Del

But didn’t de Broglie say that first?

Posted by: Toby Bartels on January 17, 2007 2:22 AM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 10)

It’s quite possible de Broglie said it first; someone mentioned this possibility in class, and all I could say for sure is that he must have realized something like this. Maybe someone here knows the precise history?

Posted by: John Baez on January 17, 2007 5:55 AM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 10)

Could I persuade you to return to our discussion which began here? Reading some of the references I subsequently mentioned, I caught a glimpse of what Bob Coecke said:

When working with quantum mechanics a lot, one comes to the point where it feels perfectly natural and all desire to “interpret” it as anything else than what it is disappears. It then is classical physics which begins to look odd and in need of “explanation”.

How do classical particles know which way to go?

Posted by: David Corfield on January 17, 2007 1:31 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 10)

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Perhaps to tempt you with a starting point, if the wavefunction for the quantum particle is in $L^2(Q)$ , wouldn’t you expect a ‘classical wavefunction’ to be an $\mathbb{R}^{min}$ -valued function on $Q$ , whose integral (i.e, minimum) is bounded?

Posted by: David Corfield on January 17, 2007 1:58 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 10)

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if the wavefunction for the quantum particle is in $L^2(Q)$ , wouldn’t you expect a ‘classical wavefunction’ to be an $\mathbb{R}$ min-valued function on $Q$ , whose integral (i.e, minimum) is bounded?

This question ties in with the general question I tried to formulate in our recent discussion.

Namely, while I can answer your above question, I can do so only by using hindsight and additional knowledge about the setup that I have. I wouldn’t know how to answer this in a way systematic enough that I could, say, code a computer program for doing it.

Here is the unsystematic answer:

As Fuchs recalls, given a solution $\psi \in L^2(Q)$ of the Schrödinger equation, regarded as a function of $\hbar$ , we may perform a certain $\hbar \to 0$ -limit on its logarithm. $\psi$ being a solution of the Schrödinger equation will imply that the resulting limiting function is a solition of the asscociated Hamilton-Jacobi equation.

Posted by: urs on January 17, 2007 6:17 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 10)

David wrote:

Could I persuade you to return to our discussion which began here?

Yes!

Right now I’m extremely busy: having run off to that workshop in Toronto the first week of class, I’m now having to figure out exactly what to say in my courses on quantization & cohomology and computation & categorification, as well as the qualifier course I’m teaching on algebraic topology. I also have weekly meetings with five grad students, two of whom (Jeff and Derek) are finishing their theses this spring, and applying for jobs now.

Of course I want to talk to James Dolan, too — we’re starting to write stuff about categorifying quantum groups. And, to top it all off, I’m in charge of picking new grad students for the UCR math program.

So, I’ll probably be less talkative on this blog than I’d like to be.

But, yesterday I decided to discuss this ‘deformation of rigs’ approach to quantization in the course on quantization & cohomology. I realized this may be the deepest way to get a grip on how cohomology shows up when you quantize particles, strings, and higher-dimensional branes. That ‘homotopy types as categorified truth values’ business we talked about should show up somehow… I’m not quite sure yet, but I feel there must be a lot to say about this.

So, I’ll try to tackle a bunch of your questions, or at least similar questions, in this course!

Posted by: John Baez on January 17, 2007 10:13 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 10)

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Here’s another tack. You can just about convince yourself that the Lagrangian story works as a rig change, but what about the Hamiltonian? How do we get from the cotangent space to $L^2(Q)$ ?

Is there anything to the observation that:

(1)

\epsilon ^m + \epsilon ^n \approx \epsilon^{min(m, n)}?

Posted by: David Corfield on January 17, 2007 8:48 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 10)

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After attending this week’s lecture, where I compared four approaches to physics:


Lagrangian classical           Lagrangian quantum

Hamiltonian classical          Hamiltonian quantum

someone inquired:

BTW, what would be wrong with labelling the two columns in the table today “particle mechanics” and “wave mechanics” (where the latter is not restricted to the deBroglie usage) as in the old distinction between ray optics and wave optics? The connotation of that distinction is slightly different since it positions the quantization as a consequence of the wave-ization of mechanics.

I replied:

It would be sort of confusing, since the distinction between particles and waves is often treated as orthogonal to the distinction between classical and quantum.

It’s fun to consider this square:
    classical particles        quantum particles
(e.g. Newtonian mechanics)   (e.g. Schrödinger's 
                                 equation)

    classical waves               quantum waves
(e.g. the part of Maxwell's   (e.g. the part of QED
 equations describing light)     describing light)
The part of quantum electrodynamics describing light can be obtained by quantizing Maxwell’s equations in perfect parallel with how Schrödinger’s equation is obtained by quantizing F = ma.

One can also take the viewpoint that quantization is the same as the passage from particles to waves. I find this a bit less fruitful… but there’s something to it, as evidenced by the fact that people often use ‘second quantization’ to mean the quantization of a theory of waves!

There’s a lot more to be said about all this… it took me decades to feel I understand it, and I’m just scratching the surface here!

If we think of the particle/wave distinction as orthogonal to the classical/quantum and Lagrangian/Hamiltonian distinction, we get $2^3 = 8$ kinds of physical theory — and a good physicist needs to learn all eight!

Posted by: John Baez on January 20, 2007 3:44 AM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 10)

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How should we fit your $2^2 = 4$ table:

classical statics         classical dynamics
thermal statics           quantum dynamics

to this $2^3 = 8$ picture?

Posted by: David Corfield on January 20, 2007 3:03 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 10)

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Good question, David! I leave it as an exercise for the ambitious reader.

Some hints: the process of reinterpreting ‘ dynamics for $p$ -branes’ as ‘statics for $(p+1)$ -branes’ is a process of categorification, together with Wick rotation. For clarity, we should probably separate this categorification process from the Wick rotation. The Wick rotation amounts to changing $\hbar$ in the expression $exp(iS/\hbar)$ from real to imaginary. It’s part of the overall ‘change of rig’ business that relates thermal physics, quantum physics and classical physics.

Posted by: John Baez on January 21, 2007 2:18 AM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 10)

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What confuses me a little about this is that 3 of the entries in your $2^2$ table are classical, and I don’t see where quantum thermal theories fit in?

Posted by: David Corfield on January 22, 2007 9:25 AM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 10)

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Before worrying about that, maybe you should think about quantum statics and thermal dynamics?

Posted by: Tim Silverman on January 23, 2007 10:25 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 10)

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David asked how the wavefunction of a quantum particle reduces to the ‘wavefunction of a classical particle’ — whatever that is — in the $\hbar \to 0$ limit.

I think I finally figured it out. The answer is pretty simple: a ‘classical wavefunction’ is a real-valued function $\psi$ on spacetime. We can think of the number $\psi(t,x)$ as the least possible action for a particle to reach the point $x$ at time $t$ . We then evolve this in time following the principle of least action. It’s sort of obvious how — but before I describe that, let me bring in a handy analogy.

We globe-trotting academics are all familiar with the problem of minimizing the cost of travelling from here to there. So, the principle of least action becomes more intuitive if we call it the principle of ‘least cost’.

In these terms, you should think of $\psi(t_0,x_0)$ as the price you have to pay to start a trip at the point $x_0$ at time $t_0$ . The ‘startup cost’, as it were.

Given this, we define $\psi(t_1,x_1)$ to be the least cost of getting to the point $x_1$ by the time $t_1$ .

It’s easy to compute this function if we know the price of any path that takes us from the point $x_0$ at time $t_0$ to the point $x_1$ at time $t_1$ .

To compute $\psi(t_1,x_1)$ , we first take the startup cost $\psi(t_0,x_0)$ for each point $x_0$ . Then we add the cheapest price of a path from $x_0$ at time $t_0$ to $x_1$ at time $t_1$ . Then we minimize this over all such paths and all points $x_0$ .

Note this is exactly like the path integral at the top right of this week’s notes, but with the ‘cost’ or action $S$ replacing the exponentiated action $exp(iS/\hbar)$ , addition replacing multiplication, and minimization replacing integration. This is just what we’d expect: classical mechanics is just like quantum mechanics, but with the rig $\mathbb{R}^{min}$ replacing the ring of complex numbers!

Starting from what I’ve said, we can write down a differential equation describing the time derivative of the wavefunction $\psi(t,x)$ . In the quantum case this is called Schrödinger’s equation; in the classical case it’s called the Hamilton–Jacobi equation.

If you try to work out how the quantum wavefunction reduces to the classical one in the $\hbar \to 0$ limit, you need to see the ring $\mathbb{C}$ , or at least $\mathbb{R}$ , as a one-parameter deformation of the rig $\mathbb{R}^{min}$ . The formulas for doing this involve exponentials, which is why Urs mentioned logarithms in his answer to your question.

It’s all very pretty — and it’s a bit shocking that more people don’t discuss this stuff. So, maybe it’s my duty to figure it out and explain it thoroughly in this course.

Posted by: John Baez on January 21, 2007 1:56 AM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 10)

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So can you stitch together a classical trajectory from its pieces, or does it kinda need to know that it’s a limit as $\hbar \to 0$ of a quantum sum?

Obviously a particle can’t find a global minimum cost if the initial part of the trajectory would be expensive.

Posted by: David Corfield on January 22, 2007 9:14 AM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 10)

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a ‘classical wavefunction’ is a real-valued function $\psi$ on spacetime. We can think of the number $\psi (t, x)$ as the least possible action for a particle to reach the point $x$ at time $t$ .

Shouldn’t that be an $\mathbb{R}^{min}$ -valued function to allow for it being impossible to get to $x$ at time $t$ , i.e., infinite action?

Posted by: David Corfield on January 25, 2007 9:46 AM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 10)

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David just corrected my discussion of the wavefunction in classical mechanics. I sloppily said it was a real-valued function on configuration space, and he said:

Shouldn’t it be an $\mathbb{R}^{min}$ -valued function to allow for it being impossible to get to $x$ at time $t$ , i.e., infinite action?

He’s right!

Yesterday my student John Huerta revealed that he’s already learned some stuff about this from James Dolan. He pointed out that the classical analogue of a ‘position eigenstate’ is a wavefunction

$\psi: Q \to \mathbb{R}^{min}$

which equals $\infty$ everywhere except at one point. This means it’s impossible for the particle to be anywhere except that point.

This sort of wavefunction has an amusing upside-down similarity to the ‘delta function’ used to describe a position eigenstate in quantum mechanics. This comes from the upside-down nature of the Boltzmann homomorphism

$E \mapsto exp(-E/k T)$

sending energies to probabilities (though in this discussion we’re actually talking about actions and amplitudes — confusing, eh?).

Posted by: John Baez on January 26, 2007 12:26 AM | Permalink | Reply to this

Read the post Quantization and Cohomology (Week 11)
Weblog: The n-Category Café
Excerpt: What's really going on with quantization?
Tracked: January 24, 2007 1:51 AM

Read the post Quantization and Cohomology (Week 9)
Weblog: The n-Category Café
Excerpt: A glimpse of geometric quantization.
Tracked: January 31, 2007 2:39 AM

The n-Category Café

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January 17, 2007