Eur. Phys. J. A (2021) 57:155 https://doi.org/10.1140/epja/s10050-021-00455-2 Review Spin tensor and pseudo-gauges: from nuclear collisions to gravitational physics Enrico Speranzaa, Nora Weickgenannt Institute for Theoretical Physics, Goethe University, Max-von-Laue-Str. 1, 60438 Frankfurt am Main, Germany Received: 30 June 2020 / Accepted: 31 March 2021 / Published online: 3 May 2021 © The Author(s) 2021 Communicated by Laura Tolos Abstract The relativistic treatment of spin is a fundamen- 5 Pseudo-gauges and statistical operator ........ 10 tal subject which has an old history. In various physical con- 6 Spin-polarization effects in relativistic nuclear colli- texts it is necessary to separate the relativistic total angular sions ......................... 11 momentum into an orbital and spin contribution. However, 6.1 Spin hydrodynamics and quantum kinetic theory 12 such decomposition is affected by ambiguities since one can 7 Pseudo-gauge transformations and the relativistic always redefine the orbital and spin part through the so-called center of inertia .................... 14 pseudo-gauge transformations. We analyze this problem in 7.1 External and internal components of angular detail by discussing the most common choices of energy- momentum ................... 15 momentum and spin tensors with an emphasis on their phys- 7.2 Center of inertia and centroids ......... 15 ical implications, and study the spin vector which is a pseudo- 7.3 Belinfante pseudo-gauge ............ 16 gauge invariant operator. We review the angular momentum 7.4 Center of inertia as reference point: canonical decomposition as a crucial ingredient for the formulation of pseudo-gauge .................. 16 relativistic spin hydrodynamics and quantum kinetic theory 7.5 Center of mass as reference point: HW, GLW with a focus on relativistic nuclear collisions, where spin and KG pseudo-gauges ............. 16 physics has recently attracted significant attention. Further- 7.6 Massless particles and side jumps ....... 17 more, we point out the connection between pseudo-gauge 8 Einstein–Cartan theory ................ 18 transformations and the different definitions of the relativis- 8.1 Riemann–Cartan geometry ........... 18 tic center of inertia. Finally, we consider the Einstein–Cartan 8.2 Tetrads and spinors in curved spacetime .... 19 theory, an extension of conventional general relativity, which 8.3 Local Poincaré transformations and conserva- allows for a natural definition of the spin tensor. tion laws ..................... 19 9 Conclusions ..................... 20 A Lorentz transformation properties of hyper-surface- Contents integrated quantities ................. 21 B Matrix element of the Pauli–Lubanski vector .... 23 1 Introduction ..................... 1 References ........................ 24 2 Spin tensor and pseudo-gauge transformations ... 2 2.1 Canonical currents ............... 2 2.2 Belinfante–Rosenfeld currents ......... 4 1 Introduction 2.3 Hilgevoord–Wouthuysen currents ........ 4 2.4 de Groot-van Leeuwen-van Weert currents ... 5 The decomposition of the relativistic total angular momen- 2.5 Alternative Klein–Gordon currents ....... 5 tum into an orbital and spin part is a long-standing prob- 3 Spin vector ...................... 6 lem both in quantum field theory and gravitational physics 3.1 Frenkel theory .................. 6 [1]. The definitions of the energy-momentum and spin ten- 3.2 Pauli–Lubanski vector ............. 7 sors used to construct the total angular momentum density 4 Wigner operator ................... 7 suffer from ambiguities. In fact, one can always redefine them through the so-called pseudo-gauge transformations a e-mail: [email protected] (corresponding author) such that the total charges (i.e., the total energy, momentum 123 155 Page 2 of 25 Eur. Phys. J. A (2021) 57 :155 and angular momentum) do not change. As long as one is inition of spin tensor arises by allowing the spacetime to a only interested in the total charges, this ambiguity is clearly have a nonvanishing torsion. Brief conclusions are given in of no importance. However, in many physical contexts, it is Sect. 9. important to separate the orbital and spin angular momentum We use the following notation and conventions: a · b = μ of the system. The question we would like to address can be a bμ, a[μbν] ≡ aμbν − aνbμ, gμν = diag(+, −, −, −) for 0123 stated as follows: is there a particular choice for the angu- the Minkowski metric, and =−0123 = 1. lar momentum decomposition for a specific system which gives a “physical” local distribution of energy, momentum and spin? Here, by “physical” we mean that such decompo- 2 Spin tensor and pseudo-gauge transformations sition can have some consequences on experimental observ- ables. Although in conventional Einstein’s general relativity The spin tensor is one of the fundamental quantities we will the answer to this question is that the energy-momentum consider in this paper. In this section we give a definition density is measurable through geometry, this issue is cur- starting from Noether’s theorem and discuss the pseudo- rently debated in quantum field theory where spin degrees of gauge transformations which allow a redefinition of energy- freedom are considered. We stress that, in this paper, we do momentum and spin tensors. We review some of the most not aim at finding a physical decomposition which is valid commonly used pseudo-gauge choices in the literature and for all possible systems. Rather, we address this problem explore their physical implications. In this paper we will by studying different contexts, indicating for each of them focus on the Dirac theory. the implications of various pseudo-gauge choices and which decomposition appears to be a physical one. 2.1 Canonical currents The paper is structured as follows. In Sect. 2 we introduce the basic concepts: the spin tensor and pseudo-gauge trans- Let us consider the Lagrangian density for the free Dirac field formations, and discuss some of the most common choices ψ(x) with mass m in the literature. In Sect. 3, we generalize the concept of ih¯ ¯ μ←→ ¯ nonrelativistic spin vector to the relativistic case, discussing LD(x) = ψ(x)γ ∂ μψ(x) − mψ(x)ψ(x), (2.1) 2 the Frenkel theory and the Pauli-Lubanski pseudovector. In ←→μ −→μ ←−μ μ Sect. 4 we consider the Wigner-operator formalism. Further- where ∂ ≡ ∂ − ∂ and γ are the Dirac matrices. more, in Sect. 5 we study the effect of different pseudo-gauge The corresponding action is given by choices in thermodynamics using the method of Zubarev to = 4 L ( ). construct the statistical operator. We show that the local equi- A d x D x (2.2) librium state is in general pseudo-gauge dependent, unlike The equations of motion associated to the Lagrangian (2.1) the global equilibrium one. This leads to the consequence are the Dirac equation for the field and its adjoint that expectation values of observables are in general sen- sitive to different pseudo-gauges for many-body systems μ (ih¯ γ ∂μ − m)ψ(x) = 0, (2.3a) away from global equilibrium. In relativistic heavy-ion colli- ¯ μ←− sions, spin-polarization phenomena related to medium rota- ψ(x)(ih¯ γ ∂ μ + m) = 0, (2.3b) tion have recently attracted significant interest. This is due to experimental observations showing that certain hadrons respectively. Consider the infinitesimal spacetime transla- emitted in noncentral collisions are indeed produced with a tions μ μ μ μ finite spin polarization [2]. The Wigner operator turns out x → x = x + ξ , (2.4) to be particularly suitable to study spin effects in heavy-ion ξ μ collisions and a brief review on some recent applications with a constant parameter. The canonical energy-mo- ˆ μν( ) in the field is given in Sect. 6. In Sect. 7 we make a con- mentum tensor TC x is defined as the conserved current nection between pseudo-gauges and the relativistic center of obtained using Noether’s theorem by requiring the invariance 1 mass which, unlike in the nonrelativistic case, suffers from of the action under the transformations in Eq. (2.4)[4,5]: ambiguities in the definition. In particular, we show that the ∂ ˆ μν = , μTC 0 (2.5) redefinition of the center of inertia can be related to a differ- ent decomposition of orbital and spin angular momentum of where a relativistic system which, in turn, is associated to a specific ∂L ∂L ˆ μν = D ∂νψ + ∂νψ¯ D − μνL spin vector [3]. In the end, in Sect. 8 the physical mean- TC g D ∂(∂μψ) ∂(∂μψ)¯ ing of the energy-momentum tensor in general relativity is discussed. We outline an extension of conventional general 1 For the sake of ease of notation we will omit the x dependence when relativity, called Einstein-Cartan theory, where a natural def- there is no risk of confusion. 123 Eur. Phys. J. A (2021) 57 :155 Page 3 of 25 155 ih¯ ¯ μ←→ν μν h¯ ¯ λ μν = ψγ ∂ ψ − g LD. (2.6) = ψ{γ ,σ }ψ 2 4 h¯ μν λμνα ¯ ˆ =− ψγαγ5ψ. (2.14) Note that TC is not symmetric. Equation (2.5)impliesthat, 2 for any three-dimensional space-like hypersurface Σλ,the total four-momentum operator is given by Inserting Eq. (2.13)into(2.12), we obtain a (non)conservation law for the spin tensor ˆ μ = Σ ˆ λμ = ¯ 3 ψ†∂μψ P d λ TC ih d x (2.7) λ,μν [νμ] Σ ∂ ˆ = ˆ . λ SC TC (2.15) where we assumed that boundary terms vanish and used the ˆ μν Dirac equation in Eq. (2.3). In the second equality we chose Note that, since TC is not symmetric, the spin tensor is not 0 ˆ μν conserved. The conserved charge associated to Eq. (2.12)is the hyperplane at constant x . Since TC is conserved, the total charge is independent of the choice of the hypersurface the total angular momentum integration and it transforms properly as a four-vector under ˆμν = Σ ˆλ,μν Lorentz transformations, for the proof see App.