Attacking quantum field theories from first principles

What are quantum field theories?

Quantum field theories (QFT) are used in many areas of physics. The formalism was originally developed in the 1930s to combine the principles of quantum physics with those of the theory of special relativity. This enables us to describe, for example, what happens in particle accelerators, such as the one at CERN. As the name suggests, fields are the main concept of this formalism. A field describes a quantity that varies in time and space. In particle physics, a field is associated with each type of particle (photons, electrons, protons...) and the particles themselves represent 'only' particular states of these fields. Beyond that, other applications include understanding changes of state in various materials (so-called 2nd order phase transitions). QFT gives us a way to calculate critical exponents, which are key parameters describing these phenomena. It is also a tool in string theory that seeks to reconcile gravitation with quantum physics. In summary, this formalism has been widely used in physics. But there are still limits.

What are these limits?

Despite these achievements, we still have only a rudimentary understanding of what a quantum field theory really is. What are the underlying mathematical structures? We have ideas about some special cases but not in general. In addition to this theoretical problem, which is one of the motivations for my research, practical consequences also exist. To make predictions, which are used to compare theory and experiments, calculations are based on sums over all possible configurations of the fields. Exact calculations are complicated and often impossible. One of the standard numerical methods involves starting with a simpler problem and approaching the real problem by successive small "perturbations". This is the principle of Feynman diagrams invented by the physicist of the same name. However, this method reaches its limits when the interactions between different fields are strong, for example, in the case of proton collisions that occur in the CERN accelerator. Other approaches have been developed to overcome these difficulties, and I am trying to push them further at the Center for Theoretical Physics (CPHT*).

What are these approaches?

These approaches start by asking what are the possible values of the parameter we are trying to find out. Starting from the first principles of the theory and from very general conditions such as the symmetries of the problem, the aim is to constrain the possible values as much as possible, rather than engaging in the calculations mentioned earlier. These so-called "bootstrap" approaches were developed from the 1960s onwards. In some very symmetrical cases, they allow us to completely determine the value we are looking for. In other cases, rigorous constraints that provide an interval of possible values can be obtained. Over the last 15 years, there have been many advances in this field, particularly in numerical methods that make it possible to extract such constraints, especially for the calculation of critical exponents.

Why is your project called 'QFTinAdS'?

QFT stands for quantum field theories. AdS is the acronym for 'Anti de-Sitter', which describes a particular type of space-time geometry where the curvature is negative (Willem de Sitter was a Dutch physicist). The particularity of this space-time is that it has an edge. A bit like a circle is the edge of a disc. I mentioned string theory and quantum gravitation: one of the interests is that there is an equivalence between a theory of gravitation valid in AdS space and another theory valid only at the edge of this space. The edge is one dimension less than the space itself, in the same way that a circle (one-dimensional space) is the edge of a disc (two-dimensional space). Therefore, it is easier to study the theory on the edge to deduce the properties of the theory valid in the AdS space. The idea of QFTinAdS is not to study a theory of quantum gravitation, but a quantum field theory in AdS space, such as the quantum field theory used to describe proton collisions. It can be shown that at the edge of this space, this theory is similar to another quantum field theory that describes, for example, critical exponents. Even if there is no total equivalence, a connection exists between the two. By looking at these theories at the edge, for which there are good numerical methods, one could deduce information about the theory valid in the whole space. This would allow a better estimation of the key values of proton collision phenomena. Of course, an AdS space is not a space-time like that of our Universe, which has almost zero curvature. But it could provide useful information when the curvature approaches zero. So it could be used to make predictions for CERN experiments. I think this can lead to fruitful results.

*CPHT: a joint research unit CNRS, École Polytechnique, Institut Polytechnique de Paris, 91120 Palaiseau, France