vanhees71 said:It just uses that the determinant of a matrix is given by the product of its eigenvalues. The eigenvalues of ##D(0)## come in pairs, ##\gamma_i## and ##\gamma_i^*##. The Mass matrix is proportional to the unit matrix and thus the eigenvalues are ##\gamma_i+m_q## and ##\gamma_i^*+m_q##. Do you get
$$\det (D(0)+m_q)=\prod_i (\gamma_i+m_q)(\gamma_i^*+m_q)$$
which is Eq. (20) in the paper.
Thanks!vanhees71 said:The point is to show that without baryo-chemical potential you have always pairs of conjugate omplex eigenvalues and that's why in this case the fermion determinant is positive. For finite ##\mu_{\text{B}}## it's not longer real (except for imaginary chemical potential). That's why you cannot use Lattice QCD so easily to evaluate the QCD phase diagram at finite ##\mu_{\text{B}}##. Ways out of this trouble is subject of vigorous ungoing research in the nuclear-physics/finite-temperature-lattice community.
The determinant problem in the QCD phase diagram refers to the difficulty in accurately calculating the phase structure of Quantum Chromodynamics (QCD) at finite baryon density. This is due to the presence of a complex phase in the QCD partition function, which makes it challenging to use traditional numerical methods to study the phase diagram.
The determinant problem hinders our ability to accurately map out the phase diagram of QCD at finite baryon density, making it difficult to understand the behavior of nuclear matter under extreme conditions such as those found in neutron stars. It also limits our understanding of the nature of the transition between the confined and deconfined phases of QCD.
Several approaches have been taken to address the determinant problem, including the use of analytical approximations, reweighting techniques, and various numerical methods such as lattice QCD simulations. However, none of these methods have been able to fully solve the problem and provide a complete understanding of the QCD phase diagram at finite baryon density.
The determinant problem remains an active area of research in theoretical and computational physics. Scientists continue to explore new techniques and approaches to solve this problem and gain a better understanding of the QCD phase diagram. Some promising developments include the use of machine learning algorithms and new theoretical frameworks.
Solving the determinant problem is crucial for our understanding of the universe at a fundamental level. The QCD phase diagram plays a significant role in the behavior of matter in extreme conditions, such as those found in the early universe or in the cores of neutron stars. A complete understanding of the phase diagram will help us unravel the mysteries of the universe and shed light on the fundamental laws of nature.