Counting Microstates with Lateral Thinking

I’ve spent some time this morning stuck on the following undergraduate problem

Consider a quantum system with N distinguishable particles, each of which can have energy E=n\epsilon. Show that the number of microstates consistent with a macrostate of energy E is given by the binomial coefficient

\displaystyle \left(\frac{E/\epsilon + N - 1}{N - 1}\right)

This is a classic example of a problem which needs some lateral thinking! It’s pretty trivial when you get the right idea, but I think it’s not entirely obvious at first. I’ll share with you my (embarrassingly slow) reasoning – let me know whether you agree with my philosophy in the comments.

To start with, I tried some examples – attempting to arrange 3 particles in 5 energy levels for instance. My approach was to work out how I could partition 5 as a sum of smaller numbers, then evaluate the number of possible configurations consistent with this.

More concretely, I had a configuration where the particles had energies \{0,1,4\}. There are 3 \times 2=6 such arrangements, because the particles can be distinguished (think of them as different coloured balls, if you like).

That’s all very well, but generalizing this idea is tricky. The problem is that the total energy constraint \sum E_i = E makes it hard to enumerate all possible configurations. So I sat, stumped, for a good few minutes.

But thankfully, my failure contained a vital clue. My difficulties lay with that irritating total energy constraint. What if I could remove it from the problem altogether?

To do this requires a bit of lateral thinking. We’ve been trying to fit particles into energy levels. But you can turn the problem around. Equivalently we can try to distribute the E/\epsilon units of energy among the particles. This effectively trivializes the troublesome constraint.

We’re not quite out of the woods yet. We need to work out how to distribute the energy blocks into the particle buckets. Here a second piece of lateral thinking helps us out. Rather than throwing the energy into buckets, we can think of partitioning it into sections. It’s just like being at the supermarket till – different customers (particles) separate their shopping (energy) with plastic dividers.

So how many ways can we divvy up the shopping on the conveyor belt? Well, there are N customers so we’ll need N-1 dividers. We’ve also got E/\epsilon items ready to be bought. This means that you have

\displaystyle (E/\epsilon + N - 1)!

possible arrangements of the items and dividers. But hang on, every unit of energy looks exactly the same. It’s as if every customer has bought exactly the same product! And clearly it doesn’t matter if you exchange the dividers – the overall partition is unchanged. Taking this into account, the correct number of microstates is

\displaystyle \frac{(E/\epsilon + N - 1)!}{(E/\epsilon)!(N - 1)!}

This is exactly the binomial coefficient in the question above!

Although this problem was pretty simple, there are two important morals. First, always examine a problem from every angle. Second, never completely discard your failed attempts. Chances are they hold vital clues which will point you in the right direction!

Why Should Undergraduates Attend Classes?

I’ve just finished teaching classes for the Quantum Mechanics B course here at Queen Mary. It’s been an enjoyable few weeks, watching the students grapple with bra-ket notation, spherical harmonics and the Stern-Gerlach experiment. All in all, I’ve found it rather rewarding.

A perennial bugbear is that many undergraduates don’t turn up for the classes. Although nominally compulsory, the university rarely imposes any sanctions for lack of attendance. It’s natural to wonder whether there’s any benefit in running the sessions!

I decided to do some elementary analysis to determine any correlation between class attendance and performance. Fortunately, the results were favourable – students who come to class tend to do better than those who don’t. And what’s more, the gap widens as the term goes on!

Effect of Attending Classes

I’d like to conclude that this effect is due to the usefulness of my teaching, of course. But my scientific brain doesn’t allow such an easy deduction. After all, correlation does not imply causation! To say anything more we’d need a control study, which is unlikely to happen any time soon!

Still, I can now tell my undergraduate students that they’re more likely to succeed if they come to class. Flawed logic aside, surely that means I’m doing something right?

Financial Times Christmas Carol!

Away from my physics life I spend a lot of time singing. About 6 months ago I co-founded a choir. The development of the group has been phenomenally rapid, culminating in recording a new Christmas carol for the Financial Times, composed by the acclaimed baritone Roderick Williams. Do have a listen and let me know what you think!

I see mathematics and music as natural intellectual cousins. Both involve artistic creativity within certain constraints. As a researcher, I must learn the subject and find new concepts. As a musician I’ll certainly study the notes, but it’s that spark of original interpretation which brings the music alive.

If you liked the FT carol, have a listen to our other recent recording below!

Asking the Obvious Question

I’m now about to finish my first year as a PhD student. Along the way I’ve done a lot of physics! Some of the concepts are very hard. I’ve sure spent my fair share of hours battling with abstract maths! But I’ve learnt something much tougher and infinitely more valuable in the past 12 months – how to do research.

The blessing and curse of research is it’s very hard to teach. You need just the right combination of perserverance, creativity and inspiration. Unlike most forms of employment, science is wonderfully, frustratingly unpredictable!

There’s one principle that stands out through every success and failure this year. Ask the obvious question! Whether in a seminar, a conversation with colleagues or in front of your desk, never be afraid to say something stupid. Often it’s the most basic idea which leads to the richest consequences.

At the end of the day, research is something of a confidence game. It’s a bit similar to my limited experience on a snooker table. If I think I’m going to win, I usually do. But when those doubts creep in, it’s much harder to keep the break going!

That’s why it’s so important that scientists communicate. Sadly the human brain doesn’t seem to be wired up to think deeply and laterally simultaneously. Regular breaks for discussion, evaluation and presentation of your work are vital!

I’ve had my clearest thoughts on walks to the tube, after chatting over coffee or writing a blog post. Although the life of a scientist might appear relaxed, ours is not a job where you can just clock in and out!

Asking the obvious question is not just important for researchers. Students, journalists, politicians, civil servants, lawyers, managers, even executives pose simple questions every day. In fact, it’s when public figures disguise their questions and answers with complex language that we struggle to relate.

A stupid, obvious question can do no harm. And more often than not, it’s exactly what you need to say.

Calabi-Yau Manifolds and Moduli Stabilization

For better or worse, string theory dominates modern research in theoretical physics. Naively, you might expect a theory consisting of tiny strings to be pretty simple. But the subject has grown into a vast and exciting playground for new ideas.

String theory is popular not just because it might unify physics or quantize gravity. In fact, many unexpected offshoots have proved more successful than the original idea! From particle physics to superconductors, string theory is having a surprising indirect impact. It’s certainly useful, even if it doesn’t prove to be the ultimate description of reality.

But what of the original plan – to describe nature using strings? A key sticking point is the existence of extra dimensions. String theory needs 6 of these to work consistently. Another problem is supersymmetry. 10 dimensional string theory must have lots of this to work correctly. But 4 dimensional physics only has a little bit of supersymmetry at most!

It turns out that these problems can be solved in one step. By coiling up the extra dimensions into a Calabi-Yau manifold we can make the extra dimensions effectively invisible, while reducing the supersymmetry we end up with in our 4D world. So what is this Calabi-Yau manifold, I hear you ask!

Well, Calabi-Yau is just a technical term for the shape of the compact extra dimensions. Different shapes break different amounts of symmetries, leaving us with different theories. Calabi-Yau’s are just symmetrical enough to break the right amount of supersymmetry, giving us a sensible theory in the end!

Technically Calabi-Yau manifolds must have a metric which is Kaehler and Ricci flat. These properties provide the correct information about the shape of the curled up dimensions. So we must look for 6-real dimensional manifolds with these properties.

Generically, you don’t have to put a notion of distance on a space. When I go for a walk, I don’t always carry around a yardstick so I can measure how far I’ve gone! You can have a perfectly good manifold without giving it a metric, but you get extra information once you have defined what distance means.

As it happens, finding a metric which is Calabi-Yau is quite difficult. But due to the genius of Shing-Tung Yau, we know that you don’t need to do this! There’s an equivalent definition of a Calabi-Yau manifold which doesn’t depend on metrics at all. All you need to know is the topological information about the manifold – roughly speaking, how “holey” it is.

If you know something about differential geometry, this kind of equivalence might sound familiar. Yau’s theorem relating geometry and topology is like a (much) more complicated version of the classic Gauss-Bonnet theorem!

It’s a darn sight easier to discover Calabi-Yau’s when you know it’s only the topological data that matters. At first people thought there might only be a few, but now we know there’s a huge number of potential candidates! The problem then becomes choosing one which produces the physics of our universe.

While people have made progress on this, the going is tough. One reason is that nobody knows the metric on a compact Calabi-Yau. This isn’t so important for string calculations, but it makes a big difference when you need to consider branes. So people have come up with various workarounds, which give promising physical results. One such success story is provided by my colleague Zac Kenton, who recently wrote a paper on brane inflation with his PhD supervisor, Steve Thomas.

There’s one final complication that I should mention. If string theory is to be a fundamental theory, then the Calabi-Yau shape should be dynamic. More specifically it will squeeze and stretch over time, unless there’s some mechanism to keep it stable. From the perspective of the 4 large dimensions, this freedom is seen as free scalar fields. These so-called “moduli” fields are bad, because we don’t observe anything like them in nature!

To solve this problem, we must find a way of constraining the fluctuations of the Calabi-Yau. Put another way we have to stabilize the moduli fields, by giving them potential terms, so that their fluctuations are small and essentially negligible at low energies. Hence this is known as the problem of moduli stabilization.

One popular way to solve the conundrum is to turn on some supergravity fields at high energy. These so-called fluxes generate potential terms for the moduli, solving the stabilization problem. Initially this idea was unpopular because of a famous no-go theorem by Witten. But since the advent of the D-brane revolution, the concept is back in vogue!

So there you have it – a 5 minute snapshot of “real” string theory. Now it’s time to get back to my calculations, where string theory is more the background Muse, and certainly not the main protagonist!

Why are D-Branes Non-Perturbative?

It’s frequently said that D-branes are non-pertubative objects. In other words, you can’t learn about them by doing a series expansion in the string coupling g. That’s because the DBI action which encodes the dynamics of D-branes couples to the dilaton field via a term e^{-\phi}. Now recall that the dilaton VEV yields the string coupling and Bob’s your uncle!

But there’s a more subtle point at work here. What determines the coupling term e^{-\phi}? For this, we must remember that D-brane dynamics may equivalently be viewed from an open string viewpoint. To get a feel for the DBI action, we can look at the low energy effective action of open strings. Lo and behold we find our promised factor of e^{-\phi}.

Yet erudite readers will know that the story doesn’t end there. Recall that in theories of gravity, we can change the metric by altering our frame of reference. In particular we can effect a Weyl rescaling to eliminate the pesky e^{-\phi}. From our new perspective, D-branes aren’t non-perturbative any more!

There’s a price to pay for this, and it’s a steep one. It turns out that this dual description turns strings into non-perturbative objects. This is quite unhelpful, since we know a fair amount about the perturbative behaviour of fundamental strings. So most people stick with the “string frame” in which D-branes are immune to the charms of perturbation theory.

Thanks to Felix Rudolph for an enlightening discussion and for bringing to my attention Tomas Ortin’s excellent volume on Gravity and Strings.

Thanks to David Berman, for pointing out my ridiculous and erroneous claim that D-branes are conformally invariant. My mistake is utterly obvious when you recall the Polyakov action for D-branes, namely

\displaystyle S[X,\gamma]= \int d^{p+1}\xi \sqrt{|\gamma|}\left(\gamma^{ij}g_{ij}+(1-p)\right)

where \xi are worldsheet coordinates, \gamma is the worldsheet metric, and g is the pull-back of the spacetime metric. Clearly conformal invariance is violated unless p=1.

From this perspective it’s even more remarkable that one gets conformally invariant \mathcal{N}=4 SYM in 4D from the low energy action of a stack of D-branes. This dimensional conspiracy is lucidly unravelled on page 192 of David Tong’s notes. But even for a 3-brane, the higher derivative operators in the \alpha' expansion ruin conformal invariance.

Incidentally, the lack of conformal invariance is a key reason why D-branes remain so mysterious. When we quantize strings, conformal invariance is enormously helpful. Without this crutch, quantization of D-branes becomes unpleasantly technical, hence our lack of knowledge!

Renormalization and Super Yang Mills Theory

It’s well known that \mathcal{N}=4 super Yang-Mills theory is perturbatively finite. This means that there’s no need to introduce a regulating cutoff to get sensible answers for scattering amplitude computations. In particular the \beta and \gamma functions for the theory vanish.

Recall that the \gamma function tells us about the anomalous dimensions of elementary fields. More specifically, if \phi is some field appearing in the Lagrangian, it must be rescaled to Z \phi during renormalization. The \gamma function then satisfies

\displaystyle \gamma(g)=\frac{1}{2}\mu\frac{d}{d\mu}\log Z(g,\mu)

where g is the coupling and \mu the renormalization scale. At a fixed point g_* of the renormalization group flow, it can be shown that \gamma(g_*) exactly encodes the difference between the classical dimension of \phi and it’s quantum scaling dimension.

Thankfully we can replace all that dense technical detail with the simple picture of a river above. This represents the space of all possible theories, and the mass scale \mu takes the place of usual time evolution. An elementary field operator travelling downstream will experience a change in scaling dimension. If it happens to get drawn into the fixed point in the middle of the whirlpool(!) the anomaly will exactly be encoded by the \gamma function.

For our beloved \mathcal{N}=4 though the river doesn’t flow at all. The theory just lives in one spot all the time, so the elementary field operators just keep their simple, classical dimensions forever!

But there’s a subtle twist in the tale, when you start considering composite operators. These are built up as products of known objects. Naively you might expect that these don’t get renormalized either, but there you would be wrong!

So what’s the problem? Well, we know that propagators have short distance singularities when their separation becomes small. To get sensible answers for the expectation value of composite operators we must regulate these. And that brings back the pesky problem of renormalization with a vengeance.

The punchline is the although \mathcal{N}=4 is finite, the full spectrum of primary operators does contain some with non-trivial scaling dimensions. And that’s just as well really, because otherwise the AdS/CFT correspondence wouldn’t be quite as interesting!

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