February 8, 1997

This Week's Finds in Mathematical Physics (Week 97)

John Baez

I've taken a break from writing This Week's Finds in order to finish up a paper with James Dolan in which we give a definition of "weak n-categories" for all n. This paper is now available on my website, and I'm immodestly eager to talk about it, and I will, but a lot of stuff has accumulated in the meantime which I want to discuss first.

First, I'm sure you remember a while back when atoms were first coaxed to form true Bose-Einstein condensates. The basic idea is that particles come in two basic kinds, fermions and bosons, and while the fermions have half-integer spin and obey the Pauli exclusion principle saying that no two identical fermions can be in the same state at the same time, bosons have integer spin and are gregarious: they love to be in the same state at the same time.

Why is spin related to what happens when you try to put a bunch of particles in the same state? Well, it all has to do with the relation between twisting something around:

  \  /\
    /  \
   /   /
  /  \/

and switching two things:
\     /
 \   /
  \ /
  / \
 /   \
/     \

To understand this, try

1) Spin, statistics, CPT and all that jazz, http://math.ucr.edu/home/baez/spin.stat.html

But let's consider some examples. Since photons have spin 1 they are bosons. In laser light one has a bunch of photons all in the same state. Thanks to the Heisenberg uncertainty principle, of course, we can't know both their position and momentum. In a laser we don't know the position of the photons: each photon is all over the laser beam in a spread-out sort of way. However, we do know the momentum of the photons and they all have the same momentum. This means that we have "coherent light" in which all the photons are like waves wiggling perfectly in phase. One can demonstrate this by interfering two beams of laser light and seeing beautifully perfect interference fringes, bright and dark stripes in places where the two beams are either in phase with each other and adding up, or out of phase and cancelling out.

Now, other particles are bosons as well, and they can do similar tricks. Bose and Einstein predicted that when any gas of noninteracting bosons gets sufficiently cold, all --- or at least a sizable fraction --- of them will be found in the same state: the state of least possible energy. Unfortunately, when things get cold they are usually liquids or solids rather than a gas, and the particles in a liquid or gas interact a lot, so true Bose-Einstein condensation is hard to achieve.

Some related things have been studied for decades. If you get an even number of fermions together they act approximately like a boson, at least if the density is not too high. Helium stays liquid at temperatures arbitrarily close to absolute zero, when the pressure is low enough. Since helium 4 has 2 protons, 2 neutrons, and 2 electrons, and all these particles are fermions, helium 4 acts like a boson. At really low temperatures, helium 4 becomes "superfluid" --- a substantial fraction of the atoms fall into the same state and the liquid acquires shocking properties, like zero viscosity. Similarly, in certain metals at low temperatures electrons will, by a subtle mechanism, form "Cooper pairs", and these act like bosons. When a bunch of these fall into the same state, you have a "superconductor".

But neither of these is a Bose-Einstein condensate in the technical sense of the term, because the helium atoms interact a lot in superfluid helium, and the Cooper pairs interact a lot in a superconductor. Only recently have people been able to get dilute gases of bosonic atoms cold enough to study true Bose-Einstein condensation.

The fist team to do it, the "JILA" team in Boulder, Colorado got a Bose-Einstein condensate of about 2000 rubidium atoms to form in a magnetic trap at less than 2 x 10^{-7} degrees above absolute zero. A team at Rice University did it with lithium soon after, followed by a team at MIT, who did it with sodium.

Check out:

2) Physicists create new state of matter, http://jilav1.colorado.edu/www/bose-ein.html

Atomcool home page, http://atomcool.rice.edu/

Neutral sodium ion trap at MIT, http://bink.mit.edu/dallin/nat.html

So what's the news? Well, recently the team at MIT, led by Wolfgang Ketterle, made two blobs of Bose-Einstein condensate out of sodium atoms. Ramming these into each other, they were able to see interference fringes just as in a laser! In other words, they is seeing interference of matter waves, just as quantum mechanics predicts, but involving lots of atoms in a coherent state rather than a single electron as in the famous double slit experiment. For pictures and even movies, try:

3) Matter-wave interference of two Bose condensates, http://bink.mit.edu/dallin/news.html#matterwave

In honor of this event, I hereby present the following limerick composed by the poet Lisa Raphals, with myself serving as science consultant. It may aid your appreciation if I note first that "Squantum" is an actual town in Massachusetts. With no further ado:

       A metaphysician from Squantum
       Was asked, what's the state of the quantum?
       It's all reciprocity:
       Position, velocity -
       They're never both there when you want 'em!

Now on to some more technical stuff....

I am now visiting the Center for Gravitational Physics and Geometry here at Penn State, which is a delightful place for people interested in the loop representation of quantum gravity (see "week77"). Right now everyone is working on using the loop representation to derive Hawking's formula which says that the entropy of a black hole is proportional to the surface area of its event horizon.

When I arrived, Jorge Pullin handed me a copy of his book:

4) Rodolfo Gambini and Jorge Pullin, "Loops, knots, gauge theories, and quantum gravity", Cambridge U. Press, Cambridge, 1996.

Here is the table of contents:

  1. Holonomies and the group of loops
  2. Loop coordinates and the extended group of loops
  3. The loop representation
  4. Maxwell theory
  5. Yang-Mills theories
  6. Lattice techniques
  7. Quantum gravity
  8. The loop representation of quantum gravity
  9. Loop representation: further developments
  10. Knot theory and physical states of quantum gravity
  11. The extended loop representation of quantum gravity
  12. Conclusions, present status and outlook

This is presently the most complete introduction available to the "loop representation" concept, as applied to electromagnetism, Yang-Mills theory, and quantum gravity. Gambini was one of the original inventors of this notion, and this book covers the whole sweep of its ramifications, with a special emphasis on a particular technical form, the "extended loop representation", which he has been developing with Pullin and other collaborators.

What the heck is the loop representation, anyway? Well, all the forces we know are described by gauge theories, and gauge theories all describe the "phase", or generalization thereof, that a particle acquires when you carry it around a loop. In the case of electromagnetism, for example, a charged quantum particle carried around a loop in space acquires a phase equal to


where q is the particle's charge, ħ is Planck's constant, and B is the magnetic flux through the loop: i.e., the integral of the magnetic field over any surface spanning the loop. Knowing these phase for all loops is the same as knowing the magnetic field. Similarly, if we knew the phase for all loops in SPACETIME instead of just space, we would know both the electric and magnetic fields throughout spacetime.

General relativity is similar except that instead of a phase one gets a rotation, or more generally a Lorentz transformation, when one parallel transports a little arrow around a loop.

The theories of the electroweak and strong forces are similar but the analog of the "phase" is a bit more abstract: an element of the group SU(2) x U(1) or SU(3), respectively.

The idea of the loop representation is to take these "phases acquired around loops" as basic variables for describing the laws of physics.

That's the idea in a nutshell. It turns out, not surprisingly, that there are many interesting relationships with such topics involving loops, such as string theory and knot theory.

Gambini and Pullin's book develops this theme in many directions. Let me say a bit about one fascinating topic that they mention, which I would like to understand better: Gerard 't Hooft's work on confinement in chromodynamics using his "order and disorder operators".

I explained some basic ideas about confinement and asymptotic freedom in "week84" and "week94", so I'll assume you've read that stuff. Remember, the basic idea of confinement is that if you take a meson and try to pull the quark and antiquark it contains apart, the force required does not decrease with distance like 1/r^2, because the chromoelectric field --- the strong force analog of the electric field --- does not spread out in all directions like an ordinary electric field does. Instead, all the field lines are confined to a "flux tube", so the force is roughly independent of the distance.

This means that the energy is roughly proportional to the distance. Since action has dimensions of energy times time, this means that if we consider the creation and subsequent annihilation of a virtual quark-antiquark pair:

                 /  \
                /    \
                \    /
                 \  /

the total action is proportional to the area of the loop traced out in spacetime. Here I am neglecting the action due to the kinetic energy of the quark and antiquark, and only worrying about the potential energy due to the flux tube joining them. This amounts to treating the quark and antiquark as "test particles" to study the behavior of the strong force.

Now, when we study quantum physics using Euclidean path integrals the basic principle is that the probability of the occurence of any process is proportional to


where S is the action of that process and ħ is Planck's constant again. So in this framework the probability of a particular virtual quark-antiquark pair tracing out a loop like the above one is proportional to


where c > 0 is some constant and A is the area of the loop. This "area law" was first proposed by Kenneth Wilson in his pioneering work on confinement; he proposed it as a way to tell, mathematically, if confinement was happening in some theory. Just compute the probability of a virtual quark-antiquark pair tracing out a particular loop and see if it decreases exponentially with the area!

Deriving confinement from chromodynamics is something that people have worked on for quite a while, and it's not easy: there is still no rigorous proof, even though there are a bunch of heuristic arguments for it, and computer simulations seem to demonstrate that it's bound to occur. One approach to studying the puzzle is due to 't Hooft and involves "order" and "disorder" operators.

I'll explain what these are, and what they have to do with knot theory, but not how 't Hooft actually uses them in his argument for confinement. For the actual argument, try Gambini and Pullin's book, or else 't Hooft's paper:

5) Gerard 't Hooft, Nucl. Phys. B138, (1978) 1.

Let us work in space at a given time, rather than in the Euclidean path integral approach. We'll do "canonical quantization", meaning that now observables will be operators on some Hilbert space.

If we have any loop g in space, there is an observable called the "Wilson loop" W(g), which is the trace of the holonomy of the connection around g. The precise way of stating Wilson's area law for confinement in this context is that

<W(g)> ~ exp(-cA)

where <W(g)> is the vacuum expectation value of the Wilson loop, and A is the area spanned by the loop g. The point is that <W(g)> is the same as what I was (a bit sloppily) calling the probability of the quark-antiquark pair tracing out the loop g.

't Hooft calls the Wilson loops "order operators". We don't really need to worry why he calls them this, but if you know how physicists think, you may know that the Wilson loops are keeping track of a kind of "order parameter" of the vacuum state. Anyway, his idea was to study the Wilson loops by introducing some other operators he called "disorder operators".

Chromodynamics is an SU(3) gauge theory but it's a little clearer if we work with any SU(N) gauge theory. Notice that the center of the group SU(N) consists of the matrices of the form

exp(2 π i n/N)

where n is an integer. So if we have a loop h, we can imagine an operator that does the following thing: it modifies the connection, or vector potential, in such a way that if you do parallel transport around a tiny loop linking h once, the holonomy changes to exp(2 π i/N) times what it had been. Note: this is a gauge-invariant thing to do, because that exp(2 π i/N) is in the center of SU(N)! So just as the Wilson loop observables are gauge-invariant, we can hope for some some "disorder operators" V(h) that modify the connection in this way.

If you think about it, what this means is that the following commutation relations hold:

W(g) V(h) = V(h) W(g) exp(2 π i L(g,h)/N)

where L(g,h) is the linking number of the loops g and h, which counts how many times g wraps around h.

There is an obvious symmetry or "duality" between the V's and the W's going on here. In fact, just as W's satisfy an area law where there is confinement of chromoelectric field lines into flux tubes, I believe the V's satisfy an area law when there is confinement of chromomagnetic field lines into flux tubes. The simplest case of this kind of thing occurs in plain old electromagnetism, where plain old magnetic field lines are confined into flux tubes in type II superconductors. For this reason confinement of electric field lines is sometimes called "dual superconductivity".

Perhaps the simplest way of beginning to understand this stuff more deeply is to understand the wonderful formula proved by Ashtekar and Corichi in the following paper:

6) Abhay Ashtekar and Alejandro Corichi, Gauss linking number and electro-magnetic uncertainty principle, preprint available as hep-th/9701136.

This formula applies to plain old electromagnetism, or more precisely, quantum electrodynamics. If we work in units where ħ = 1, and consider a particle of charge 1, the Wilson loop operator W(g) in electromagnetism is just

W(g) = exp(-iB(g))

where B is the magnetic flux flowing through the loop g. But instead we can just work with B(g) directly. Similarly, instead of V(h)'s we can work with the operator E(h) corresponding to the electric flux through the loop h. Then we have

B(g) E(h) - E(h) B(g) = i L(g,h)

In other words, the electric and magnetic fields don't commute in quantum electrodynamics, and the Heisenberg uncertainty of the electric field flowing through a loop g and the magnetic field flowing through a loop h is proportional to the linking number of g and h!

Quantum mechanics, electromagnetism, and knot theory are clearly quite tangled up here. Since the linking number was first discovered by Gauss in his work on magnetism, it's all quite fitting.

And that leads me to the last paper I want to mention this week. It should be of great interest to Vassiliev invariant fans; see "week3" if you don't know what a Vassiliev invariant is.

7) Dror Bar-Natan and Alexander Stoimenow, The fundamental theorem of Vassiliev invariants, preprint available as q-alg/9702009.

Let me just quote the abstract here:

The "fundamental theorem of Vassiliev invariants" says that every weight system can be integrated to a knot invariant. We discuss four different approaches to the proof of this theorem: a topological/combinatorial approach following M. Hutchings, a geometrical approach following Kontsevich, an algebraic approach following Drinfel'd's theory of associators, and a physical approach coming from the Chern-Simons quantum field theory. Each of these approaches is unsatisfactory in one way or another, and hence we argue that we still don't really understand the fundamental theorem of Vassiliev invariants.

© 1997 John Baez