Spin tensor and pseudo-gauges: from nuclear collisions to gravitational physics

Abstract

The relativistic treatment of spin is a fundamental subject which has an old history. In various physical contexts it is necessary to separate the relativistic total angular momentum into an orbital and spin contribution. However, such decomposition is affected by ambiguities since one can always redefine the orbital and spin part through the so-called pseudo-gauge transformations. We analyze this problem in detail by discussing the most common choices of energy-momentum and spin tensors with an emphasis on their physical implications, and study the spin vector which is a pseudo-gauge invariant operator. We review the angular momentum decomposition as a crucial ingredient for the formulation of relativistic spin hydrodynamics and quantum kinetic theory with a focus on relativistic nuclear collisions, where spin physics has recently attracted significant attention. Furthermore, we point out the connection between pseudo-gauge transformations and the different definitions of the relativistic center of inertia. Finally, we consider the Einstein–Cartan theory, an extension of conventional general relativity, which allows for a natural definition of the spin tensor.

Relativistic spin hydrodynamics with torsion and linear response theory for spin relaxation

“…In leading order it gives the ideal limit of spin hydrodynamics with no production of entropy [13], which is in accord with the existence of globally rotating equilibrium with a constant, but non-zero thermal vorticity. Interestingly, this ideal spin hydrodynamics has been shown to be equivalent, by a pseudo-gauge transformation [35,36,38,40], to conventional hydrodynamics based on the symmetric energy-momentum tensor with no spin degrees of freedom, with certain non-dissipative second order transport coefficients involving fluid vorticity [13]. The equivalence provides an important conceptual bridge between the two different formulations in the strict regime of hydrodynamics.…”

“…This motivation has led to several theoretical studies of relativistic hydrodynamics with spin polarization, based on the second law of thermodynamics [11][12][13][14][15], equilibrium partition functions [15], quantum kinetic theory of relativistic fermions [16][17][18][19][20][21][22][23][24][25], holographic approach for strongly-coupled plasma [26][27][28], effective Lagrangian approach [29][30][31][32], and JHEP11(2021)150 quantum statistical density operators [33][34][35][36][37] (see also refs. [38][39][40] and references therein for a review). These works have shed light on different aspects of relativistic hydrodynamics including spin degrees of freedom, often referred to as (relativistic) spin hydrodynamics.…”

“…Here, we use it to specify the local state where the strict hydrodynamic description can apply. In refs [35,36,40],. however, the same term was used to refer to the state in which local entropy density is maximized with fixed local densities of conserved quantities, where the angular momentum "chemical" potential is not necessarily equal to the local thermal vorticity.…”

Spin waves in spin hydrodynamics

“…Various models based on the thermodynamic equilibrium of spin degrees of freedom [12][13][14][15] have shown good agreement with experimental data, for recent reviews and papers on this topic see, e.g., Refs. [16][17][18][19][20][21][22][23]. Nevertheless, the differential measurements of polarization [5,11] lack a clear explanation.…”

“…1. (Top, middle)The speed of the spin wave spin as a function of = ∕ corresponding to the (top) MJ and (middle) FD statistics, together with the asymptotic forms for small and large given in Eqs (19). and(20), respectively.…”