Quantum field theory in strong background fields

The processes we are interested in only happen in extremely strong fields, or at extremely high intensity, where “high” means more than 10²⁰ times the intensity of a laser pointer. It was for a long time impossible to reach such intensities, but it is now becoming possible thanks to the development of new high-intensity laser facilities. This development is partly driven by practical applications, e.g. using such lasers as compact particle accelerators. This will also allow us to explore uncharted regions of fundamental physics. But we need to develop better theoretical methods to improve our abilities to design, predict and analyze future experiments.

The Standard Model of particle physics gives a list of all known elementary particles and a description of how they interact. The theory is written in the language of quantum field theory, which is the combination of quantum mechanics and special relativity. In principle, solid-state physics, chemistry, biology etc. follow from the Standard Model, but, of course, knowing the ingredients and rules of the Standard Model does not mean that one knows how to actually derive chemistry etc. As P. W. Anderson wrote, “more is different”. As one increases the number of particles, new phenomena emerge, and to study them one needs new methods.

The standard textbook method in quantum field theory is to make a perturbative (i.e. a Taylor) expansion in the coupling constant. Basically, at order N, a particle can interact with N other particles. To study the interaction of many particles, one would need to calculate many orders in the perturbative expansion. But that is not practical or even possible. This explains why one cannot use quantum field theory to study biology. But the perturbative approach breaks down at a much lower level, e.g. inside high-intensity EM fields, where a large number of photons participate in the interactions. This can be seen as “just” one level above the most fundamental level. But new phenomena emerge already at this level; phenomena which cannot happen at any fixed order in a perturbative expansion.

To explore such non-perturbative phenomena, we are working on two types of non-perturbative methods:

1. Quantum field theory in the Furry picture, where the total Hamiltonian is split into a “free” and interacting part such that the “free” part takes the interaction with the background field into account exactly. To do so one needs the exact solution to the Dirac equation in the background field. Finding such solutions is difficult, which forces one to consider fields which only depend on one space-time coordinate (e.g. t+z). This approach still uses the language of quantum fields.
2. The worldline formalism, where quantum fields are replaced by Feynman path integrals over particle worldlines. A worldline is a trajectory in space time [t(τ), x(τ), y(τ), z(τ)] parametrized by proper time τ. This is similar to worldlines in classical electrodynamics, except that the path integral is a sum over all trajectories, which include both complex trajectories and ones that go backwards in time. One might naively think that some information must have been thrown away when going from fields to worldlines, but these two formulations are in fact equivalent. However, the worldline formalism allows us to study realistic fields which depend on all 4 space-time coordinates, which one has not been able to do with any field-based approach.