FQXi Essay Contest: The Axiom of Measurement

I’ve submitted an entry for the Foundational Questions Institute Essay Contest: “Trick or Truth: the Mysterious Connection Between Physics and Mathematics”. You can check out my entry here.

The backstory to this is that I was participating in a weekly mathematical physics seminar back at Florida State (although I use the word “seminar” pretty loosely – it was regularly attended by only myself and one other individual!), and in the process of working on presenting on some NCG topic, I came across “The Bost-Connes System”. This is a particular C^*-algbera, on which you can define some dynamics. What makes it special is that if you calculate the partition function for this dynamical system, you get the Riemann Zeta function! Since the partition function can be used to generate predictions for a statistical mechanical system, I wondered how possible it was to construct a real physical system with the same symmetry as the Bost-Connes system. Then you would have experimental access to (at least some features of) the Riemann Zeta. There is a great deal of mathematical important to Zeta, including a $1 million dollar prize for finding the zeros!

I wasn’t exactly thinking about which Benz to buy with my prize money yet, but I thought it was an interesting idea – experimental verification of a mathematical theorem. I wasn’t aware that anything like this had been done before. Normally the “flow of ideas” works the other way – constructions in mathematics find usefulness in physics, or theoretical models become interesting mathematical systems. It stuck in my head for a while, I did a few calculations to determine what the zeros of a partition function might look like, but nothing really came of it.

When this essay contest came around, I thought it might be an opportunity to share this idea. I figured that if it was going to be taken seriously, you needed to raise experimental verification to the level of mathematics – after all, if I prove a conjecture is true outside of the field in which the conjecture is stated, we should not take the proof very seriously! I needed to make experimental physics a subfield of mathematics. It turns out that this is pretty easy, and so that’s what the essay is about. If you take your physical model as a set of formal axioms, and add in an additional axiom which can be used to experimentally verify a theorem (I call this “an axiom of measurement”), you can formulate physics as a complete formal system. As a bonus, the axiom can be used to add a little more structure to the Platonist viewpoint on universal versus physical forms.

Now, the FQXi Essay Contests are Contests; the community and the public can vote on the quality of the essays, and the quality of the essays vary widely, since nearly anyone is allowed to submit an entry. I actually think  my essay represents a pretty mainstream viewpoint about physics – that we are not really studying “nature” or “the universe” when we do physics, we are really studying a “model for the universe”, which is confirmed by our everyday observations as well as carefully constructed experiments. Since it’s not a new, dramatic viewpoint on any particular aspect of the relationship between the two fields, I don’t expect to be winning any awards. But, I had an opinion with an interesting idea behind it, and an essay seemed like the ideal place to explore it.

Anyway, if you’re so inclined go over and check out my entry as well as all the others.

Scattering Amplitude Workshop at SCGP – the Amplituhedron!

I’ve just attended a workshop on Scattering Amplitudes at the Simons center (posting here: http://scgp.stonybrook.edu/archives/7136, and check out the video here), so I thought I would put a few thoughts about the most interesting recent result in this direction – the Amplituhedron. Naturally, since this is the Simons center you need to translate “Scattering Amplitudes” to mean “scattering of strings and super-things”, but the techniques come from good old QFT.

Also, since this is not my direct area of expertise, the content of this post will be from the “interested outsider”…anyway…

The point of all this focus on scattering amplitudes is that although Feynman diagrams provide us a nice way of organizing diagrams for complicated QFT calculations, in the end we always end up calculating things which look like

\int d^4l \frac{\mathcal{N}}{l^2(l+p_1)^2(l+p_1+p_2)^2(l-p_1)^2}

(specifically, this is the scattering of 4 particles with momentum p_1, p_2, p_3, and p_4=-(p_1+p_2+p_3) at 1-loop with momentum l). The numerator \mathcal{N} might be some complicated thing, but when these particles are all massless, all the interesting information is contained at the poles (where l=p_1, for example). There is nothing too new in wanting to understand these kinds of integrals – this was the point of “The S-matrix Program”, which lead to all kinds of interesting work in the 1960s and 70s, but has died out a little since then. The revival has occured because apparently when one restricts to N=4 super Yang-Mills, enough simplifications occur so that further progress can be made.

This further progress has lead to “the amptliduhedron”, which has been propagandized as “the end of locality and unitarity in physics!” by Quanta magazine (link) – although is it worth it to note that Quanta is an “editorially independent division of the Simons Foundation”? Anyway, it’s really discussed much better at Sean Carrol’s blog (link). Nearly everything is discussed much better there. This was topic of much debate during this workshop, but last week I got the chance to see one of the original workers on the subject (Nima Arkani-Hamed) give a review of it. Having seen him talk, it’s easy to see why everyone is so excited – he is a fantastic speaker, very passionate and energetic. He also has some “colorful metaphors” to describe how he thinks and works, so it’s not surprising that the blogosphere has attached themselves to him to deliver us from these 19th century notations that the universe should make sense.

So I will try and relay some of his talk to describe this beast (so this next part is not mine). Essentially, look at the integral above. If you were dumb, you might say “the integrand is a product of d(log l^2)s”. If you were a little smarter, you would notice that not only does the numerator screw up that nice description, but also you can’t take the log of a dimensional number. However, if you were *a lot* smarter, you would see there is a change of coordinates (which is very similar to the duality transformations that Feynman came up with back in the day) which takes the integrand to the form

\frac{d\alpha}{\alpha}\frac{d\beta}{\beta}\frac{d\gamma}{\gamma}\frac{d\rho}{\rho}

(This is the part that only seems to work for N=4 super-YM). The amplituhedron is the geometric shape which describes a form with log-singularities on its boundaries, and thus completely encodes all the information about this scattering amplitude. The sweeping claims made by Arkani-Hamed and others is that *this describes everything*, so the universe can be reduced to a bunch of amplitudhedrons, which do not require locality and unitarity because these transformations are independent of them.

(just a note that unitary is still encoded when you actually *do* the integral, but my understanding is that if they can make this integrand into an honest volume form, unitary will not be needed either).

So the work is very interesting and sounds totally reasonable – but what about the claim that “this describes everything”? Well, if you believe that QFT describes everything (seems reasonable – I guess you have to assume we have no souls), that scattering amplitudes in QFT describe everything (which they don’t – various topological properties are not detected by them), and further that supersymmetry is real (not my bag, but I think 80% of HEP physicists would agree), then the amplitudhedron should “describe everything”. At least, it provides a method to construct any interaction by referring only to a specific geometric object. A rather fascinating idea, but don’t we already have such an object (at least for the standard model), called a fibre bundle?

Anyway, more questions remain, but my impression is that this is the most productive area of S-matrix work – since they have actually been able to analyze a scattering amplitude in terms of some very concrete geometry.  I expect more interesting work and grand claims in the near future!