Using the Complex Langevin Eqn. to solve the Sign Problem

In this project we study (among other things) the interactions of quarks and gluons. These particles build up the nuclei of the atoms that we and everything else around us are built of. In the material around us, the quarks are confined into protons and neutrons, and the gluons are the particles that hold them together. The protons and neutrons then build up the nuclei of atoms. However, if we increase the temperature, the quarks start to "vibrate" more, and at some very high temperature the properties of the nuclei change: they break up, and quarks, gluons move more freely in a very hot plasma state. This is comparable to how water behaves: at low temperature it is solid ice, when heated it melts and eventually also evaporates. To study this phase change, we solve the equations governing quarks and gluons on a (super)computer. To achieve this, we must overcome a very important technical problem: In certain situations (when the net number of quarks and antiquarks in the system is zero), we can use a probabilistic description of the system, such that we can collect so called "configurations" which describe a probable state of the system. At nonzero net quark number, however, the theory dictates that the probabilities of configurations can become negative, therefore this descripition is unusable. (This is called the sign problem). This means that it is very hard to predict theoretically how the inside of neutron stars, where a large quark density is present, behaves. To overcome this problem we use and test a proposed solution called Complex Lanegvin equation (CLE). In this proposal we the a mathematical structure called complex numbers to enlarge the number of variables we use to describe the system. This extension allows to turn the equations back into a probabilistic process which is easy to solve on computers. The theory of the CLE tells us that in some cases this allows to recover the results of the system on the original domain, however in some cases this leads to the extended theory giving results different from the original. In this project we use and test the CLE further for the theory of quarks and gluons, quantify how much discrepancy is to be expected (in the cases there is one). This method turns out to be useful in other cases too: if we want to calculate the time evolution of quantum systems (as opposed to thermal equilibrium), we have again to deal with the sign problem. In this project we also study scalar fields to test the applicability of an alteration of the CLE for this problem, and we search for the right alteration using machine learning, which should deliver correct results and give a stable process in the extended field manifold.