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arXiv:2505.02657

Local equilibrium Wigner function for spin-1/2 particles

Samapan Bhadury, Zbigniew Drogosz, Wojciech Florkowski, Kar, Sudip Kumar, Valeriya Mykhaylova

Published May 5, 2025• 0 citations
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Abstract

Formal connections between the spin density matrix and the Wigner function for spin-1/2 particles forming a relativistic gas are explored to determine their general structures. They suggest that the commonly used form of the local equilibrium Wigner function should be replaced by a new expression. The latter fulfills the necessary condition for the normalization of the mean spin polarization, which the former fails to reproduce. The new definition of the Wigner function leads to generalized thermodynamic relations for perfect spin hydrodynamics, identical to those obtained earlier using the classical concept of spin. Moreover, one can prove that the perfect spin hydrodynamics based on the new equilibrium Wigner function is nonlinearly causal and stable. Finally, the selection rule for the Lagrange multipliers, which is satisfied by real systems, is discussed.

DOI
10.48550/arxiv.2505.02657

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Formal connections between the spin density matrix and the Wigner function for spin-1/2 particles forming a relativistic gas are explored to determine their general structures. They suggest that the commonly used form of the local equilibrium Wigner function should be replaced by a new expression. The latter fulfills the necessary condition for the normalization of the mean spin polarization, which the former fails to reproduce. The new definition of the Wigner function leads to generalized thermodynamic relations for perfect spin hydrodynamics, identical to those obtained earlier using the classical concept of spin. Moreover, one can prove that the perfect spin hydrodynamics based on the new equilibrium Wigner function is nonlinearly causal and stable. Finally, the selection rule for the Lagrange multipliers, which is satisfied by real systems, is discussed.
A recently proposed Boltzmann local equilibrium Wigner function for massive spin-1/2 particles is generalized to the case of Fermi-Dirac statistics. The resulting formula ensures the correct normalization of the mean polarization vector and reproduces the generalized thermodynamic relations with spin that were obtained in earlier studies. Moreover, we show that the macroscopic currents constructed from the Fermi-Dirac Wigner function can be obtained as derivatives of a suitably defined generating function with respect to the Lagrange multipliers (temperature, hydrodynamic flow, and chemical potentials). The identified generating function also indicates that the underlying framework can be classified as a divergence-type theory.
Using the recently developed relativistic Wigner formulation for the density matrix of spin-1/2 particles, we study the $^{\mathrm{\ensuremath{\pi}}}$=1/2+{2}^{+}2+ Coulomb-like and cavity relativistic states. One of our objectives is to understand the sharing of the total angular momentum of a quantum state between the spin and rotational degrees of freedom, arising due to the spin-orbit coupling. Another is to demonstrate that the 4\ifmmode\times\else\texttimes\fi{}4-matrix Wigner function is the appropriate generalization from the 2\ifmmode\times\else\texttimes\fi{}2 form of the nonrelativistic theory.
This thesis provides a detailed derivation of dissipative spin hydrodynamics from quantum field theory for systems composed of spin-0, spin-1/2, or spin-1 particles. The Wigner function formalism is introduced for quantum fields in the respective representations of the Poincaré group, and the conserved currents, i.e., the energy-momentum tensor and the total angular momentum tensor, in various so-called pseudogauges are derived. An expansion around the semiclassical limit in powers of the Planck constant is performed. Subsequently, kinetic equations are obtained for binary elastic scattering, using both the de Groot-van Leeuwen-van Weert and Kadanoff-Baym method, with the latter retaining the effect of quantum statistics. The resulting collision term features both local and nonlocal contributions, with the latter providing a relaxation mechanism for the spin degrees of freedom of the quasiparticles. The local-equilibrium distribution function is derived from the requirement that the local part of the collision term vanishes. From quantum kinetic theory, dissipative spin hydrodynamics is then constructed via the method of moments, extended to particles with spin. The system of moment equations is closed via the Inverse-Reynolds Dominance (IReD) approach, resulting in a set of equations of motion describing the evolution of both ideal and dissipative degrees of freedom. The application to polarization phenomena relevant to heavy-ion collisions is discussed.
The recently proposed Wigner function for a particle in an infinite lattice (Hinarejos M, Bañuls M C and Pérez A 2012 New J. Phys. 14 103009) is extended here to include an internal degree of freedom as spin. This extension is made by introducing a Wigner matrix. The formalism is developed to account for dynamical processes, with or without decoherence. We show explicit solutions for the case of Hamiltonian evolution under a position-dependent potential, and for evolution governed by a master equation under some simple models of decoherence, for which the Wigner matrix formalism is well suited. Discrete processes are also discussed. Finally, we discuss the possibility of introducing a negativity concept for the Wigner function in the case where the spin degree of freedom is included.
The recently proposed Wigner function for a particle in an infinite lattice (Hinarejos M, Bañuls M C and Pérez A 2012 New J. Phys.14 103009) is extended here to include an internal degree of freedom as spin.This extension is made by introducing a Wigner matrix.The formalism is developed to account for dynamical processes, with or without decoherence.We show explicit solutions for the case of Hamiltonian evolution under a position-dependent potential, and for evolution governed by a master equation under some simple models of decoherence, for which the Wigner matrix formalism is well suited.Discrete processes are also discussed.Finally, we discuss the possibility of introducing a negativity concept for the Wigner function in the case where the spin degree of freedom is included.
Abstract A formalism enabling computation of the one-particle density of a freely rotating assembly of identical particles that vibrate about their equilibrium positions with amplitudes much smaller than their average distances is presented. It produces densities as finite sums of products of angular and radial functions, the length of the expansion being determined by the interplay between the point-group and permutational symmetries of the system in question. Obtaining from a convolution of the rotational and bosonic components of the parent wavefunction, the angular functions are state-dependent. On the other hand, the radial functions are Gaussians with maxima located at the equilibrium lengths of the position vectors of individual particles and exponents depending on the scalar products of these vectors and the eigenvectors of the corresponding Hessian as well as the respective eigenvalues. Although the new formalism is particularly useful for studies of the Wigner molecules formed by electrons subject to weak confining potentials, it is readily adaptable to species (such as ʻballiums’ and Coulomb crystals) composed of identical particles with arbitrary spin statistics and permutational symmetry. Several examples of applications of the present approach to the harmonium atoms within the strong-correlation regime are given.