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Register a new userSpin-dependent distribution functions for relativistic hydrodynamics of spin-1/2 particles
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Recently advocated expressions for the phase-space dependent spin-1/2 density matrices of particles and antiparticles are analyzed in detail and reduced to the forms linear in the Dirac spin operator. This allows for a natural determination of the spin polarization vectors of particles and antiparticles by the trace of products of the spin density matrices and the Pauli matrices. We demonstrate that the total spin polarization vector obtained in this way agrees with the Pauli-Lubanski four-vector, constructed from an appropriately chosen spin tensor and boosted to the particle rest frame. We further show that several forms of the spin tensor used in the literature give the same Pauli-Lubanski four-vector.
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The formulation of relativistic hydrodynamics for massive particles with spin 1/2 is shortly reviewed. The proposed framework is based on the Wigner function treated in a semi-classical approximation or, alternatively, on a classical treatment of spin 1/2. Several theoretical issues regarding the choice of the energy-momentum and spin tensors used to construct the hydrodynamic framework with spin are discussed in parallel.
We analyze algebraic structure of a relativistic semi-classical Wigner function of particles with spin 1/2 and show that it consistently includes information about the spin density matrix both in two-dimensional spin and four-dimensional spinor spaces. This result is subsequently used to explore various forms of equilibrium functions that differ by specific incorporation of spin chemical potential. We argue that a scalar spin chemical potential should be momentum dependent, while its tensor form may be a function of space-time coordinates only. This allows for the use of the tensor form in local thermodynamic relations. We furthermore show how scalar and tensor forms can be linked to each other.
A newly proposed framework of perfect-fluid relativistic hydrodynamics for particles with spin 1/2 is briefly reviewed. The hydrodynamic equations follow entirely from the conservation laws for energy, momentum, and angular momentum. The incorporation of the angular-momentum conservation requires that the spin polarization tensor is introduced. It plays a role of a Lagrange multiplier conjugated to the spin tensor. The space-time evolution of the spin polarization tensor depends on the specific form chosen for the spin tensor.
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 possib
We have studied analytically the longitudinally boost-invariant motion of a relativistic dissipative fluid with spin. We have derived the analytic solutions of spin density and spin chemical potential as a function of proper time $tau$ in the presence of viscous tensor and the second order relaxation time corrections for spin. Interestingly, analogous to the ordinary particle number density and chemical potential, we find that the spin density and spin chemical potential decay as $simtau^{-1}$ and $simtau^{-1/3}$, respectively. It implies that the initial spin density may not survive at the freezeout hyper-surface. These solutions can serve both to gain insight on the dynamics of spin polarization in relativistic heavy-ion collisions and as testbeds for further numerical codes.
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