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.

有限密度晶格 QCD 的目标是在有限密度下对 QCD 进行第一性原理研究，它描述了由许多夸克组成的系统。主要目标是夸克-胶子等离子体、原子核和中子星等系统。从微观理论解释宏观物理是物理学发展的一条自然路径。要完全理解强相互作用，我们必须求解有限密度 QCD。上述每个系统都存在无法通过实验或观察轻易解决的开放性问题，因此在有限密度晶格 QCD 方面取得进展非常重要。

在本文中，我们总结了有限密度晶格QCD领域的过去发展和现状。符号问题理论研究的难点在于，原则上正确的数值方法在实践中不一定奏效，何时失效也很难知道。我们将在本文中介绍各种方法，但它们都有缺陷，除非我们仔细研究，否则会导致不实际的结果。我们将解释过去进行了哪些研究，他们成功的程度，遇到了哪些障碍，以及为什么这些方法原则上是正确的会导致错误的答案。通过这种方式，我们想为计划研究有限密度晶格 QCD 的雄心勃勃的研究人员提供过去的经验教训。