Finite-density lattice QCD and sign problem: Current status and open problems

Finite-density lattice QCD aims for the first-principle study of QCD at finite density, which describes the system consisting of many quarks. The main targets are systems such as quark–gluon plasma, nuclei, and neutron stars. Explaining macroscopic physics from the microscopic theory is a natural path in the development of physics. To understand the strong interaction completely, we have to solve finite-density QCD. Each of the systems mentioned above has open problems which cannot easily be accessed by experiment or observation, so it is important to make progress in finite-density lattice QCD.

In this article, we summarize the past development and current status of the field of finite-density lattice QCD. The difficulty in the study of theories with the sign problem is that the numerical methods which are correct in principle do not necessarily work in practice and it is hard to know when it fails. We will introduce various approaches in this article, but all of them have pitfalls, which lead to unphysical results unless we study carefully. We will explain what kinds of studies were done in the past, to what extent they succeeded, and what kinds of obstacles they encountered, and why the approaches are correct in principle can lead to wrong answers. In this way, we would like to provide lessons from the past for ambitious researchers who plan to work on the finite-density lattice QCD.