Quantum mechanics and geometry

Here’s a nice little story about quantum mechanics, which surprisingly few mathematicians seem to know about. The essential idea is “quantum mechanics on the projective space looks remarkably like classical mechanics”! Everything I say here comes from two papers Geometrical Formulation of Quantum Mechanics (gr-qc/9706069), Ashtekar and Schilling, and Geometry of stochastic state vector reduction (#), Hughston. If you’re interested in more details, I’d encourage you to read these papers — they’re well written and contain many further references.

As you’ll recall, quantum mechanics says that systems are described by Hilbert spaces, with states given by vectors. I’ll stick with finite-dimensional systems (e.g. particles with spin) for simplicity, but this isn’t essential for what follows. A particular self-adjoint operator H, called the Hamiltonian, governs the dynamics of the system via the Schrodinger equation \frac{d}{dt} \psi = i H \psi. Quantum mechanics also says something about measurement, which we’ll come to in a moment.

Now the Schrodinger equation defines a one parameter flow via U(t) = exp(i H t). This preserves the unit sphere in our Hilbert space, and descends to a flow on the projective space. The projective space is naturally a Kahler manifold, and in particular a symplectic manifold, so we immediately ask if this flow is Hamiltonian. The answer is unsurprising but underappreciated: yes, the flow is Hamiltonian, and the Hamiltonian function is just the expectation value of the Hamiltonian operator \langle \psi, H \psi \rangle.

The example you should have in mind at this point is a simple spin 1/2 system in a magnetic field, whose Hilbert space is \mathbb{C}^2, Hamiltonian \begin{pmatrix}1 &0 \\ 0 & -1\end{pmatrix}. The projective space is \mathbb{CP}^1 and the Hamiltonian function we get as the expectation value is just the usual z coordinate of the standard embedding of \mathbb{CP}^1 = S^2 in R^3. The Hamiltonian flow rotates points along lines of latitude, completing each orbit in \pi units of time (go calculate the unitary).

Eigenvectors for the Hamiltonian operator correspond to critical points for the Hamiltonian function, and in particular fixed points of the flow. (That’s the north and south poles in the example above.) The flow described above is just a rigid rotation of the sphere, and in fact this is generally true: the flow on projective space coming from a self-adjoint operator is Killing, that is, it preserves the metric. This is the first appearance of the metric, but it’s really essential, because the converse of this statement is true — Hamiltonian functions whose corresponding flows preserve the metric are precisely those which arise as expectation values of self-adjoint operators on the Hilbert space.

That’s not all the metric is good for! Quantum mechanics also tells us something about what happens during “measurement”. This is that when a “measurement” (yes, I’m going to keep using scare quotes, so you’re not allowed to argue with me about what measurement means) occurs, the system jumps discontinuously to one of the eigenvector of the Hamiltonian, and the probabilities of reaching the the various different eigenvectors  are given by the absolute value squared of the inner product of the current state and the eigenvector. This probability is exactly \cos^2 \theta, where \theta is the metric distance between the current state and the corresponding fixed point. (In the spin 1/2 example, let’s normalise this metric so it just measures angles between points on S^2.)

It gets even better, but at this point I’m going to stop talking about the conventional description of quantum mechanics, and begin describing a proposed modification of quantum mechanics. Physicists have already thought a lot about whether modifications like this are reasonable, but I’ll postpone that for now. At this point, if you’re reading the actual articles, we’re switching from the Ashtekar/Schilling paper to the Hughston one.

So what is this proposed modification? Well, let’s imagine the symplectic flow as some differential equations describing the trajectory of our state. We now want to add in a stochastic term, in particular an isotropic Brownian motion term with an amplitude that depends on the position in the projective space. This amplitude will be (some simple function of?) the energy uncertainty, namely the quantity \langle \psi, H^2 \psi \rangle - \langle \psi, H \psi \rangle^2. In fact, this energy uncertainty is exactly the squared velocity of the symplectic flow with respect to the metric. In our spin 1/2 example this velocity is \sqrt{1-z^2} (remember we have rigid rotation) and since H^2=1, \langle \psi, H^2 \psi \rangle - \langle \psi, H \psi \rangle^2 = 1-z^2. What happens? Well, at the fixed points it’s easy to see that the energy uncertainty is zero, so we might expect that the Brownian motion term drives the state away from areas with high energy uncertainty, towards the eigenstates — just like what is supposed to happen during “measurement”. This is precisely what happens: Hughston does a lot of financial mathematics, and he knows his stochastic calculus. His Proposition 5 says the energy uncertainty in this model is a supermartingale, that is, an on average decreasing function. As time passes, you expect to end up at one of the fixed points, each with various probabilities. Note that these are honest, stochastic probabilities, not just numbers we’ve declared to be interpreted as probabilities as in the naive set up. (ED: see below for Greg’s comment on this.) His next result, of course, is that these probabilities match up with what we want, namely that they are given simply by metric distances on the projective space.

I think this is a beautiful picture. The measurement process is now something more concrete, a stochastic term in the governing equation, and we can resume thinking probabilistically about quantum mechanical probabilities.Very roughly, you’re meant to think that in an “isolated quantum system” the stochastic term is extremely small, and symplectic flow dominates. On the other hand, during a “measurement”, presumably when the system is coupled with the macroscopic world, the scale of energy uncertainties becomes extremely large and the stochastic terms dominates, and the system is quickly driven to a fixed point of the symplectic flow.

You have to think hard, however, about where this stochastic terms comes from, and what it means. Hughston has some ideas about quantum gravity, but I’m not so sure I like them! There are also lots of no-go theorems ruling out stochastic variations on quantum mechanics, and I have to admit to not being clear about whether these results affect Hughston’s model.

A final idea for further thought, from the Ashtekar/Schilling paper: we can fully describe quantum mechanics solely in terms of the Kahler manifold structure of the projective space, so why not drop the requirement that it’s a projective space? That is, can we imagine systems on other Kahler manifolds? It seems that all we lose is the fact that on \mathbb{CP}^n any two points have a canonical \mathbb{CP}^1 through them — i.e. that we’re allowed to form linear superpositions of states. Is this really essential? Where might we look for finite dimensional systems described by “exotic” Kahler manifolds? And all you quantum topologist gallium-arsenide engineers out there — how might we try to make one?

20 questions

Two Berkeley grad students, Pablo Solis and Andrew Critch, just organised a “20 questions seminar”. The premise is pretty simple — everyone gets 2 minutes to ask a question they’d like an answer to, and we spend any remaining time (and tea afterwards) talking about them.

I wasn’t sure beforehand whether or not it was going to work out, but ended up pretty pleased I came. We didn’t quite make it to 20 questions (13 that I counted); they appear below. There’s also apparently going to be a wiki page for the seminar. The questions range from easy to weird to awesome. I left out one or two for various reasons, and my apologies for the lame TeX. Decide for yourself which you like, and feel free to give answers — if there’s good discussion here I’ll advertise that at next week’s “20 questions”.
1 Scott:
in R^2, you can tile the plane with hexagons. However any closed trivalent graph has a face that’s smaller than a hexagon. You can tile R^3 with vertex-truncated octahedrons. Say we have a “generic” closed finite cell-complex (every edge has 3 incident faces, every vertex has 4 incident edges). Is there something “smaller” than a vertex-truncated octahedron (or the other polytopes that give generic tilings)?

2 Critch:
Is there a space with trivial homology, non trivial homotopy?
(Anton: isn’t there a result that say that first nontrivial homology and homotopy agree?)

4 Yael:
Out(G) = Aut(G)/Inn(G). Is there a nice description of cosets, beyond that they’re cosets?

5 Mike:
X a banach space, f:X->R convex.
If X is infinite dimenionsal, what extra conditions guarantee that f is continuous?

6 Darsh:
Take a triangle in R^2 with coordinates at rational points. Can we find the smallest denominator point in the interior? (Take the lcm of the denominators of the coordinates.)
(Hint: you can do the 1-d version using continued fractions.)

7 Jakob
Take a “sparse” (every vertex has reasonably small degree) graph. Consider a maximal independent set for the graph (a maximal set of disconnected vertices). Can we make a new graph, with vertices the set, and whatever edges we like, that is as topologically similar to the original graph as possible?

8 Andrew:
What’s the deal with algebraic geometry? Just kidding!
Consider the sequence x0=0, x1=1, x_{n+2} = a x_{n+1} + b x_n, generalizing the Fibonacci sequence. Fix p a prime. If k is minimal so p|x_k and p|x_l implies k|l, then v_p(x_nk) = v_p(x_k) + v_p(n). (Here v_p(z) is the power of p dividing z.) Is there some framework that makes this sort of result obvious? Andrew only knows strange proofs.

9 Anton:
Take I=[0,1), the half open interval. Do there exist topological spaces X and Y, with X and Y not homeomorphic, but XxI and YxI are homeomorphic?
E.g., if instead I=[0,1], the closed interval, you can take X=mickey mouse=disc with two discs removed, Y=cross-eyed frog=disk with two linked bands glued on the boundary.

10 Pablo:
x^x^x^x … converges if x \in [e^{-1}, e^{1/e}], e.g. with x=\sqrt{2}, this converges to 2. Given a sequence (a_i), when does the “power tower sequence” converge?

13 Andrew again:
Can you define the set of all primes with a first-order theory?

Given a first-order theory, let S(T)={|M| | M is a finite model of T}.
You can get all prime powers with the field axioms. Is there some T so S(T)={primes}.

14 Yuhao:
Let A be an abelian category, that might not have enough injectives? Can you embed into another abelian category with enough injectives? Is there a universal way?
e.g. finite abelian groups embeds into Z-modules
e.g. coherent sheaves embeds into quasi-coherent sheaves
(Anton: the Freyd embedding theorem says every abelian category embeds in Z-mod. But this doesn’t help universality.)

Man and machine thinking about SPC4

I’ve just uploaded a paper to the arXiv, Man and machine thinking about the smooth 4-dimensional Poincaré conjecture, joint with Michael Freedman, Robert Gompf, and Kevin Walker.

The smooth 4-dimensional Poincaré conjecture (SPC4) is the “last man standing in geometric topology”: the last open problem immediately recognizable to a topologist from the 1950s. It says, of course:

A smooth four dimensional manifold \Sigma homeomorphic to the 4-sphere S^4 is actually diffeomorphic to it, \Sigma = S^4.

We try to have it both ways in this paper, hoping to both prove and disprove the conjecture! Unsuprisingly we’re not particularly successful in either direction, but we think there are some interesting things to say regardless. When I say we “hope to prove the conjecture”, really I mean that we suggest a conjecture equivalent to SPC4, but perhaps friendlier looking to 3-manifold topologists. When I say we “hope to disprove the conjecture”, really I mean that we explain an potential computable obstruction, which might suffice to establish a counterexample. We also get to draw some amazingly complicated links:

SPC4 link

Continue reading

Writing a math paper on a wiki

Combining a couple of previous topics, I was wondering: is there a good platform for writing a math paper on a wiki? This seems like a desirable goal, both for small groups of collaborators and for any MMORPG’s (massively multiplayer online research project groups), and I’ve never seen such a thing, but I’ll hold off on crankily complaining about its absence until the blog readership has had a chance to tell me whether it’s out there.

Here’s what such a thing would have to include:

  • The ability to take in proper TeX code, including packages, bibtex, anything else people use in arXiv papers, and produce some kind of reasonable preview. Obviously, it wouldn’t have to be precisely what LaTeX would produce, but it would have to be readable. Clearly this is somewhat possible, since WordPress and Wikipedia do a decent job with it.
  • The ability to sync with a local copy quickly and easily (hopefully with something roughly approximating svn).
  • All the usual wiki features (user control, full history, etc.)

I feel like this is not a lot to ask, since all aspects of it seem to be in wide use in different programs, but I’ve never seen the whole package brought together. Am I just missing out?

Is massively collaborative mathematics scalable?

I’ve been watching, though not particularly intently, Tim Gowers’s attempt massively collaborative mathematics. I’m not sure if I’ve looked hard enough to judge, but it certainly looks as though it were quite successful. This of course, answers Tim’s original question “is massively collaborative mathematics possible?” positively, but I still have to wonder if it’s sustainable in the long term. Of course, it never seems smart to bet against the possibilities of the Internet combining disperate contributions into valuable knowledge. Certainly, I would say people have tended to underestimate the possibilities of real advances coming from the technology of wikis and blogs. At the same time, it seems hard to imagine that people will really have the energy and time, not to mention mental organization, to follow several such projects at all closely. One of Tim’s take away lessons from the project seemed to be that it shrank in number of participants faster than he expected. And this was in a collaboration prominently featuring two Fields medalists and promoted on what is probably the world’s most prestigious math blog! It seems more likely that as the number of such projects expands the average number of participants will shrink until most are functionally equivalent of the collaborations we are used to today, just with more efficient coauthor location. By which I mean, the important advance will not be the number of people involved, but rather the identity of them.

Not that the value of efficient coauthor location should be minimized! The broader array of people we can stay in contact with due to the Internet is a huge boon to mathematics. It’s just that I suspect any concern over how we will deal with the allocating credit in a 20 person collaboration is a bit premature, at least outside of exceptional cases.

On the other hand, I’m kind of excited about the possibility of proving myself wrong, but haven’t been able to come up with any good projects. Does anyone wanna do that massively collaboratively?

A suggestion: Good refereeing certificates

Some one tell me what’s wrong with this idea: Journal editors should publicly acknowledge their best referees. Obviously, they can’t say which papers the referee worked on, but they could write a note saying

To whom it may concern: Jane Doe has refereed more than twenty papers for the Journal of Isotropic Widgets, and she has always done a through and careful job. Our journal is greatly improved by her efforts.

Professor Doe could then list this on her webpage and CV, and hopefully it would help her tenure case and her professional reputation. As things currently stand, referees can list the journals we have worked for on our CV’s, but there is no way to indicate the quality of that work.

The only argument I see against this is that the process of writing these letters would be very subjective. But that’s also true of the writing of recommendation letters and of the acceptance of papers to journals. Am I missing something?

More F_un

Incidentally, I hope you’ve all been reading F_un mathematics. Even if you aren’t all that interested in the field with one element, it’s a beautifully designed site and might give you some ideas about pushing Web 2.0 in mathematics a bit further than just blogs. While I like our blog, with all its messy diversity (as my collaborators can tell you, messy diversity is a core component of my mathematical style), F_un mathematics has a much more organized focused feel, which I think maybe more promising for getting actual mathematics done. I also think the division of the posts into “outreach,” “undergraduate,” “graduate,” and “research” has some interesting potential and sort of makes me feel like we should be doing a better job of indicating the background level for our post.

L’affaire El Naschie

So I know I’m a little late to the party on this, but I couldn’t resist commenting on the strange case of M. el Naschie (I assume that this is just the German transliteration of the name English speakers would be more likely to spell al Nashi). Zoran Škoda brought him up in the comments to a post at the n-Category Cafe, and John Baez did an excellent job exposing the level of intellectual bankruptcy at the journal Chaos, Solitons and Fractals. The details are better recounted elsewhere, (unfortunately the posts above have been removed. Those interested in following the case can try Richard Poynder’s blog Open and Shut) but in a nutshell, El Naschie published dozens of papers in his own journal (he’s the editor-in-chief) which appear to be of no scientific or mathematical merit (this is my judgment based on excerpts and titles, and also seems to be the consensus of commenters at nCC), which make rather grandiose claims based on rather incoherent numerology. John Baez characterized him as “worse than the Bogdanov brothers,” which is pretty high up in the food chain of physics hoaxes.

But my intent here is not to beat up on El Naschie. He’s already set to retire in shame. The people who really have egg on their face here are those who enabled the man who is for all intents and purposes a crank to run a superficially prestigious-seeming journal. Continue reading