Homepage of Matthijs Hogervorst

On this web page, I plan to put some images, links and scripts pertaining to my interests. Currently, I'm preparing my "thèse de doctorat" (PhD thesis) at the LPTENS at the École normale supérieure (ENS) in Paris. I'm mainly based at the theory division (PH-TH) at CERN in Geneva, Switzerland. Finally, my supervisor is Vyacheslav Rychkov.


My work concerns conformal field theory (CFT). CFT is a framework that describes a lot of interesting models: it controls what happens at the surfaces of strings in string theory, but at the same time it tells you how water can evaporate and how magnets suddenly fail when you heat them. CFT has also inspired many models of particle physics. In any case, conformal symmetry is very special, and we try to exploit it maximally to learn about all of these models. The field is very active at the moment, and the dream of finding and understanding all CFT's seems slowly to turn into reality.

September 2014

Today finally marks the completion of a long project, done together with Slava Rychkov and Balt van Rees (CERN). We have been exploring the Truncated Conformal Space Approach for around a year - it's a Hamiltonian method that allows you to do controlled calculations of RG flows that start at a conformal fixed point (e.g. Lagrangian theories). Different people have explored this method over the last 20 years, but only in two dimensions. We have been able to pass to D dimensions, focusing (in this paper) on the massive boson and the so-called Landau-Ginzburg theory. Details are on the arXiv. The manuscript is not a complete premiere -- I've talked about the TCSA at EPFL, CERN and the Back to the Bootstrap 4 workshop in Porto (with Balt joining in Porto).

July 2013

I went to EPSHEP 2013 (a large particle physics summer conference) to give a talk about the projects described below and to explain how they relate to the conformal bootstrap. [slides]

May 2013

Together with Hugh Osborn (Cambridge) and Slava Rychkov, I worked on a paper describing special features of conformal blocks (see below for what they mean). Conformal blocks depend on two parameters, say x and z. We consider exclusively the region where x equals z. These ''kinematics'' are sufficient for all applications considered so far, and the setup works in any number of dimensions. In the paper, we sketch how to turn our results into an efficient algorithm, which will hopefully help people tackling more practical issues in CFT.

March 2013

Earlier this year, I wrote a paper with Slava Rychkov on the structure of conformal blocks. These functions tell you what happens when an operator is exchanged in a 2-2 scattering (schematically, at least). If you know these conformal blocks, then measuring four-point functions teaches you a lot about your theory!
So far, only explicit (and intruiging!) expressions were found by Dolan and Osborn in two, four, six, ... dimensions. We advocate that it's not necessary to look for these complete expressions in three, five, ... dimensions, but that given a smart choice of coordinates, all conformal blocks can be calculated in a series expansions up to a fixed order, in a systematic way.
In pictures, this choice of coordinates corresponds to placing four fields symmetrically around the origin. The state-operator correspondence tells you that you can calculate everything in using ordinary quantum mechanics, when you place the theory on a cylinder (or a sphere, in > 2 dimensions). Time runs vertically, in the image on the right.

Before 2013

Last year, I reviewed some CFT basics. The image below shows a geometric way of thinking about conformal symmetries (discovered by Dirac, in 1936). Basically, the conformal group acts linearly in the lightcone of an embedding space. The red line on the cone, called the Poincaré section, corresponds to the real world, and the thin blue rays correspond to points in the physical space. Mathematically, it's a bijection between d-dimensional real or Minkowski space and a compactification on a cone in d+2-dimensional projective space.

In 2011, I worked on phi^4 theory at the Humboldt Universität in Berlin. In particular, I looked at a way to check whether phi^4 is really an interacting theory in the continuum limit using a worm algorithm, conceived by Ulli Wolff. We wrote an article with more details.
Although only the case of four dimensions is interesting, below you can get an idea of how things work in two dimensions. Instead of simulating the field phi directly, or even the spins on the lattice, the simulations use objects called 'worms' which live on the lattice. Their heads and tails - the fat dots in the image - crawl around the (anti-)periodic lattice, and the renormalized coupling constant of the theory depends on the percolation between the two worms. The image below is a measurement close to the continuum limit; the different worms are displayed in red and blue. You can see that they form large clusters, wrapping around the 16x16 lattice.

Finally, an accessible article I wrote on the Casimir effect (in Dutch!) still floats around the Internet.

Other stuff

Outside of physics, I like contemporary and classical music, competitive cycling, and I try to keep up with what's happening in Europe. I'm also interested in free culture, technology, and programming. I try to contribute to Wikipedia, especially the Dutch version.

This site is kindly hosted by Pablo Rauzy, a computer scientist at the ENS, who also helped me out with the stylesheet.