Recent progress in the formulation of relativistic hydrodynamics for particles with spin one-half is reviewed. We start with general arguments advising introduction of a tensor spin chemical potential that plays a role of the Lagrange multiplier coupled to the spin angular momentum. Then, we turn to a discussion of spin-dependent distribution functions that have been recently proposed to construct a hydrodynamic framework including spin and serve as a tool in phenomenological studies of hadron polarization. Distribution functions of this type are subsequently used to construct the equilibrium Wigner functions that are employed in the semi-classical kinetic equation. The semi-classical expansion elucidates several aspects of the hydrodynamic approach, in particular, shows the ways in which different possible versions of hydrodynamics with spin can be connected by pseudo-gauge transformations. These results point out at using the de Groot - van Leeuwen - van Weert versions of the energy-momentum and spin tensors as the most natural and complete physical variables. Finally, a totally new method is proposed to design hydrodynamics with spin, which is based on the classical treatment of spin degrees of freedom. Interestingly, for small values of the spin chemical potential the new scheme brings the results that coincide with those obtained before. The classical approach also helps us to resolve problems connected with the normalization of the spin polarization three-vector. In addition, it clarifies the role of the Pauli-Lubanski vector and the entropy current conservation.

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自旋极化流体的相对论流体动力学

综述了自旋二分之一粒子的相对论流体动力学公式化的最新进展。我们从建议引入张量自旋化学势的一般论点开始，该化学势在与自旋角动量耦合的拉格朗日乘数中发挥作用。然后，我们转向讨论自旋相关分布函数，这些函数最近被提出用来构建包括自旋在内的流体动力学框架，并作为强子极化现象学研究的工具。这种类型的分布函数随后用于构建在半经典动力学方程中使用的平衡 Wigner 函数。半经典扩展阐明了流体动力学方法的几个方面，特别是，显示了可以通过伪规范变换连接具有自旋的流体动力学的不同可能版本的方式。这些结果指出使用 de Groot - van Leeuwen - van Weert 版本的能量动量和自旋张量作为最自然和最完整的物理变量。最后，提出了一种全新的方法来设计带自旋的流体动力学，该方法基于自旋自由度的经典处理。有趣的是，对于小的自旋化学势值，新方案带来的结果与之前获得的结果一致。经典方法还帮助我们解决与自旋极化三矢量归一化相关的问题。此外，它还阐明了 Pauli-Lubanski 向量和熵流守恒的作用。