Eur. Phys. J. A (2021) 57:155 https://doi.org/10.1140/epja/s10050-021-00455-2

Review

Spin and pseudo-gauges: from nuclear collisions to gravitational

Enrico Speranzaa, Nora Weickgenannt Institute for , 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 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 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 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 in curved .... 19 theory, an extension of conventional , 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 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 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. A.Itisworth J d λ JC mentioning that the four components of Pˆ μ coincide with Σ   μ ν ν ν h¯ μν the four generators of the spacetime translations [5]. The = d3x ψ† ih¯ (x ∂ − x ∂ ) + σ ψ, (2.16) operatorial structure of Pˆ μ derives from the creation and 2 annihilation operators inside the quantized fields ψ and ψ†. The action of Pˆ μ on a one-particle state |p, s, p being the where the hypersurface integration at constant x0 was used. ˆλ,μν ˆμν particle four-momentum and s the spin projection, is such Since JC is conserved, the quantity J is independent that2 of the hypersurface and it transforms as a rank-two antisym- μ μ (see App. A). Moreover, the six independent Pˆ |p, s=p |p, s μν (2.8) components of Jˆ are the generators of the Lorentz trans- (we use the symbol pμ for the particle momentum to dis- formations [5]. tinguish it from the momentum variable pμ of the Wigner By looking at Eqs. (2.13) and (2.16), it would be sugges- ˆ μν operator discussed in Sect. 4). tive to identify the terms involving TC as the orbital angular ˆλ,μν Under the action of infinitesimal Lorentz four- momentum density and SC as the spin density. In fact, for μ μ μ μν a scalar field the spin tensor vanishes and we are left only x → x = x + ζ xν, (2.9) with an orbital angular momentum-like contribution, so that ζ μν =−ζ νμ ˆλ,μν with constant, the total variation of the this intuitive interpretation of SC as a spin density would reads seem to work. However, there are two reasons why this iden-   1 μν tification cannot be done straightforwardly. The first reason δT ψ = ψ (x ) − ψ(x) = ζμν f ψ(x), (2.10) 2 is that the quantity that we call global spin defined as  μν μν =−i σ μν λ,μν with f 2 and ˆ ≡ Σ ˆ SC d λ SC (2.17) Σ μν i μ ν σ = [γ ,γ ]. (2.11) 2 is not a Lorentz tensor since, as can be seen from Eq. (2.15), ˆλ,μν Using Noether’s theorem, the invariance of the action under SC is not conserved (see App. A). This also means that the the transformations (2.9) yields the conservation of the canonical global spin is not independent of the choice of the ˆλ,μν hypersurface. Hence, the usage of the canonical spin tensor canonical total angular momentum tensor JC [4,5], does not lead to a covariant description of spin for free fields ∂ ˆλ,μν = . λ JC 0 (2.12) which, instead, is something one would like to require. We where will discuss a solution to this problem in Sects. 2.3, 2.5 and 2.4. The second reason is that the decomposition of orbital ˆλ,μν = μ ˆ λν − ν ˆ λμ + ˆλ,μν. JC x TC x TC SC (2.13) and spin angular momentum in Eq. (2.13) is ambiguous. It is μν λ,μν indeed always possible to define a new pair of tensors Tˆ Sˆ pgt The tensor C is called canonical spin tensor and it is ˆλ,μν defined as and Spgt connected to the canonical currents through the so-called pseudo-gauge transformations [1,6–8] ∂L ∂L ˆλ,μν = D μνψ − ψ¯ μν D SC f f ¯ ∂(∂λψ) ∂(∂λψ) ˆ μν ˆ μν 1 ˆ λ,μν ˆ ν,μλ ˆ μ,νλ T = T + ∂λ(Φ + Φ + Φ ), (2.18a) pgt C 2 2 In order to avoid divergences due to the vacuum expectation values, λ,μν λ,μν λ,μν μν,λρ ˆ = ˆ − Φˆ + ∂ρ ˆ . we implicitly assume the normal ordering of the operators. Spgt SC Z (2.18b) 123 155 Page 4 of 25 Eur. Phys. J. A (2021) 57 :155

Φˆ λ,μν ˆ μν,λρ ˆλ,μν = . The quantities and Z are arbitrary differentiable SB 0 (2.23) operators called superpotentials satisfying Φˆ λ,μν =−Φˆ λ,νμ and Zˆ μν,λρ =−Zˆ νμ,λρ =−Zˆ μν,ρλ. It is easy to check that Consequently, the angular momentum tensor can be cast in ˆ μν ˆλ,μν = μ ˆ λν − ν ˆ λμ + ˆλ,μν the new tensors Tpgt and Jpgt x Tpgt x Tpgt Spgt a purely orbital-like form. defined through Eqs. (2.18), lead to the same total charges as We report in passing that in the original works by Belin- in Eqs. (2.7) and (2.16), but in general different global spin, fante [6,7] the approach is slightly different. One can actu- once integrated over some hypersurface and provided that ¯ˆλ,μν ally define a Belinfante spin tensor SB by decomposing boundary terms vanish. In the case of pseudo-gauge trans- the Belinfante angular momentum in the following way formations with the same Φˆ λ,μν and different Zˆ μν,ρλ,the new global spins coincide. Furthermore, the new currents ˆλ,μν = μ ˆ λν − ν ˆ λμ JB x TB x TB are also conserved, = μ ˆ λν − ν ˆ λμ + ¯ˆλ,μν, x TC x TC SB (2.24) ∂ ˆ μν = ,∂ˆλ,μν = , μTpgt 0 λ Jpgt 0 (2.19) with like for the canonical currents in Eqs. (2.5) and (2.12). We   ¯ˆλ,μν 1 μ ˆρ,λν ν ˆρ,λμ note that, if we consider a pseudo-gauge transformation lead- S = x ∂ρ S − x ∂ρ S . (2.25) ˆ [μν] = B 2 C C ing to a symmetric energy-momentum tensor, i.e., Tpgt 0, the superpotential Φˆ λ,μν is not completely arbitrary, but con- However, in this paper, by Belinfante spin tensor we always strained by the relation mean the vanishing one (2.23). ∂ Φˆ λ,μν = ˆ [νμ], λ TC (2.20) 2.3 Hilgevoord–Wouthuysen currents as can be seen from Eq. (2.18a). In this section we show a different decomposition between Following Ref. [1], the viewpoint adopted in this paper is spin and orbital angular momentum first propsed by Hilgevo- to assign the physical meaning of energy, momentum and ord and Wouthuysen (HW) such that the global spin trans- spin densities to the energy-momentum and spin tensors. forms properly as a tensor under Lorentz transformations, Moreover, we assume that such densities can in principle unlike the canonical one [9–11]. The idea is based on the be measurable quantities. The act of performing the pseudo- fact that the Dirac spinor is also a solution of the Klein- gauge transformations can be understood as “relocalization” Gordon equation. The strategy is to start from the Klein- of energy, momentum and spin. The goal is then to find out Gordon Lagrangian, derive the currents through Noether’s which choice of the superpotentials gives a “physical” pair of theorem and use the Dirac equation as a subsidiary condi- tensors to be used in a specific context. In the next subsections tion. It follows that these currents will also be conserved for we discuss several choices of pseudo-gauge transformations. the Dirac theory. The Klein–Gordon Lagrangian for the spinor reads 2.2 Belinfante–Rosenfeld currents 1 2 ¯ μ 2 ¯ LKG = (h¯ ∂μψ∂ ψ − m ψψ) (2.26) As mentioned above, the canonical energy-momentum ten- 2m sor (2.6) is not symmetric. This inevitably leads to a con- and the corresponding equations of motion are ceptual problem in conventional general relativity where 2 μ 2 2 μ 2 ¯ (h¯ ∂μ∂ + m )ψ = 0,(h¯ ∂μ∂ + m )ψ = 0. (2.27) the energy-momentum tensor is assumed to be symmetric because defined as the variation of the action with respect to By requiring the invariance of the action under spacetime the metric, see a related discussion in Sect. 8. This issue was translations and Lorentz four-rotations, we obtain the Klein- overcome by Belinfante and Rosenfeld [6–8]. It is possible Gordon canonical energy-momentum and spin tensors using to perform a pseudo-gauge transformation (2.18) with the Noether’s theorem. These are respectively given by the cur- superpotentials given by rent defined in the first line of Eq. (2.6), with LKG instead of L , and by Eq. (2.14): Φˆ λ,μν = ˆλ,μν, ˆ μν,λρ = , D SC Z 0 (2.21) ¯ 2   such that the new energy-momentum tensor is symmetric ˆ μν = h ∂μψ∂¯ νψ + ∂νψ∂¯ μψ − μνL , THW g KG (2.28a) (corresponding to the symmetric part of the canonical energy- 2m λ,μν 2 Sˆ λ,μν ih¯ μν←→λ momentum tensor C ) and the new spin tensor vanishes, Sˆ = ψσ¯ ∂ ψ. (2.28b) i.e., HW 4m

←→ ←→ Notice that, in contrast to the canonical energy-momentum ˆ μν ih¯ ¯ μ ν ν μ μν μν T = ψ(γ ∂ + γ ∂ )ψ − g LD, (2.22) tensor derived from the Dirac Lagrangian, Tˆ is symmetric, B 4 HW 123 Eur. Phys. J. A (2021) 57 :155 Page 5 of 25 155 then is a tensor, which is what we were searching for. In Eq. (2.36) we used Eq. (2.17) and chose the hyperplane at constant x0. ∂ ˆλ,μν = , λ SHW 0 (2.29) which also follows after using the Klein–Gordon Eq. (2.27). ψ We now require that is also a solution of the Dirac equation. 2.4 de Groot-van Leeuwen-van Weert currents Multiplying (2.3a) and (2.3b)byγ λ on the left and on the γ λγ μ = λμ − right, respectively, and using the identity g Here we discuss another pair of currents which leads to σ λμ i , the Dirac equations can be written as the same global spin as in the HW formulation, but differ- ent energy-momentum and spin tensors. These currents are ∂λψ =− σ λμ∂ ψ + γ λψ, ih¯ h¯ μ m (2.30a) derived by de Groot, van Leeuwen and van Weert (GLW) in λ ¯ ¯ λμ ¯ λ −ih¯ ∂ ψ =−h¯ ∂μψσ + mψγ . (2.30b) Ref. [4] by performing the pseudo-gauge transformation

Using Eqs. (2.30) one can easily derive the Gordon decom- 2 position [12] λ,μν ih¯ λμ←→ν λν←→μ Φˆ = ψ(σ¯ ∂ − σ ∂ )ψ, (2.37) ←→   4m ¯ μ ih¯ ¯ μ ¯ μν ¯ μν μνλρ ψγ ψ = ψ ∂ ψ − i ψσ ∂νψ − ∂νψσ ψ . Zˆ = 0. (2.38) 2m (2.31)

With the help of Eqs. (2.30) and (2.31), the HW currents in Thus, we obtain Eqs. (2.28a) and (2.28b) become

2 ˆ μν ˆ μν ih¯ ν μβ αμ ν T = T + (∂ ψσ¯ ∂β ψ + ∂αψσ¯ ∂ ψ) ¯ 2 ←→ ←→ HW C ˆ μν =−h ψ¯ ∂ μ ∂ νψ, 2m TGLW (2.39) 2 4m ih¯ μν ¯ λα←→   − g ∂λ(ψσ ∂ αψ), (2.32a) ¯ 2 ←→ ˆλ,μν = ih ψσ¯ μν ∂ λψ − ∂ μνλρψγ¯ 5ψ , 4m SGLW ρ (2.40) 2   4m ˆλ,μν ˆλ,μν h¯ μν λρ λρ μν S = S − ψσ¯ σ ∂ρψ + ∂ρψσ¯ σ ψ . HW C 4m (2.32b) where we used the Gordon decomposition (2.31). We note Equations (2.32) make apparent the connection between the that Eq. (2.40) differs from Eq. (2.28b) only by a total diver- ˆμν = ˆμν canonical and the HW spin tensor. In the language of relo- gence, hence for the global spin we have SGLW SHW. calization, Eqs. (2.32) tell us that the HW currents can be obtained from the canonical ones through a pseudo-gauge transformation with the superpotentials [13,14] 2.5 Alternative Klein–Gordon currents λ,μν [μν]λ λ[μ ν]ρ Φˆ = Mˆ − g Mˆ , (2.33) ρ There is a possible choice of currents where the energy- μνλρ h¯ μν λρ λρ μν Zˆ =− ψ(σ¯ σ + σ σ )ψ, (2.34) momentum tensor is the same as in the GLW case and the 8m spin tensor is the same as in the HW formulation. To find such currents we follow the same idea as HW, but we now where consider the Klein–Gordon Lagrangian for spinors built with 2 λμν ih¯ μν←→λ second-order derivatives of the fields Mˆ ≡ ψσ¯ ∂ ψ. (2.35) 4m λ,μν  Since Sˆ is conserved, the global HW spin defined from 2 HW  1 h¯ μ μ 2 L = − (∂μ∂ ψ)ψ¯ + ψ∂¯ μ∂ ψ − m ψψ¯ . Eq. (2.32b)as KG 2m 2   (2.41) ˆμν ≡ Σ ˆλ,μν = 3 ˆ0,μν SHW d λ SHW d x SHW Σ  2  h¯ 3 † μν h¯ 3 † [μ ν] L L = d x ψ σ ψ + d x ψ γ ∂ ψ, Since KG and KG in Eq. (2.26) differ only by a total 2 2m divergence, they yield the same equations of motion, namely (2.36) L Eqs. (2.27). If we apply Noether’s theorem using KG,we 123 155 Page 6 of 25 Eur. Phys. J. A (2021) 57 :155 obtain the following energy-momentum and spin tensors3 particle at rest. Furthermore, the components of Sˆμν are such that 2 ˆ μν h¯ ¯ ←→μ←→ν μν  T =− ψ ∂ ∂ ψ − g L , (2.42a) ˆ0i KG 4m KG p, s| S |p, s=0, (3.2) 2 λ,μν ih¯ μν←→λ p , | ˆij |p , ≡ijk p , | ˆk |p ,  , Sˆ = ψσ¯ ∂ ψ. (2.42b)  s S  s  s Snr  s (3.3) KG 4m where the operator on the right-hand side of Eq. (3.3)isalso L = Since using the equations of motion KG 0, we have expressed in second quantization. Thus, we establish a rela- ˆ μν = ˆ μν ˆμν TKG TGLW, as can be seen from Eq. (2.39). Furthermore, tion between the components of S in the particle rest frame ˆλ,μν = ˆλ,μν from Eq. (2.28b), we see that SKG SHW . The pseudo- and those of the nonrelativistic spin vector (3.1). In a compact gauge transformation to obtain Eqs. (2.42) is given by form, for a particle state in a general frame, we write 2 ˆ ˆμν λ,μν ih¯ λμ←→ν λν←→μ p, s| Pμ S |p, s=0. (3.4) Φˆ = ψ(σ¯ ∂ − σ ∂ )ψ, (2.43) 4m The equation above is called the Frenkel condition. 2 μνλρ ih¯ μνλρ Zˆ =  ψγ¯ 5ψ. (2.44) Now we want to relate the Frenkel theory to the pseudo- 4m gauges discussed in the previous sections. In particular, we want to find for which pseudo-gauge choice the tensor Sˆμν introduced in this section can be identified with the global 3 Spin vector spin. Consider first the canonical global spin in Eq. (2.17). We note that, Sˆ0i = 0 and In nonrelativistic , the spin vector oper- C ator Sˆ in first quantization is simply given by nr ˆij ≡ ijk ˆk , SC SC (3.5) ˆ h¯ Snr = σ , (3.1) 2 with  where σ = (σ ,σ ,σ ) are the Pauli matrices [15]. In this h¯ 1 2 3 Sˆk = d3x ψ† Skψ, (3.6) section we examine two approaches to generalize the defini- C 2 tion above to a covariant expression in quantum field theory: where the integration over the hypersurface is performed the first one is based on the Frenkel theory and the second one choosing a hyperplane with constant x0, and on the Pauli-Lubanski vector. Furthermore, we show how the   σ k spin vectors in these approaches can be related to the various Sk = 0 . k (3.7) spin tensors introduced in Sect. 2. 0 σ If we take the expectation value of a one-particle state at 3.1 Frenkel theory rest, Eq. (3.6) is the expression in second quantization of the ˆμν nonrelativistic spin operator (3.1). We see again that SC is Following the idea of Frenkel [16] we introduce an anti- not a tensor and it is clearly not compatible with the Frenkel ˆμν S which depends on the total four- condition since, in this case, Eq. (3.4) holds only when taking ˆ μ momentum P in Eq. (2.7). In the particle rest frame, the expectation value of a state of particle at rest. the spatial components of the four-momentum vanish, i.e., Let us now turn to the HW (or equivalently the GLW ˆ i μ P |p, s=0, where p = (m, 0) is the momentum of the or KG) global spin 4 in Eq. (2.36). In a general frame the ˆ0i components SHW do not vanish and 3 For a Lagrangian with second-order derivatives of spinorial fields, ˆij = ijk ˆk the energy-momentum and spin tensors derived by applying Noether’s SHW SHW (3.8) theoremaregivenby[4] with ∂L ∂L    μν KG ν ν ¯ KG 2 T = ∂ ψ + (∂ ψ) ˆk 3 † h¯ k h¯ kln l n ∂(∂μψ) ∂(∂ ψ)¯ S = d x ψ S +  γ ∂ ψ. (3.9) μ HW 2 2m ∂L ←→ ←→ ∂L + KG ∂ ∂ν ψ−(∂ν ψ)¯ ∂ KG − μν L , ρ ρ g KG ∂(∂ρ ∂μψ) ∂(∂ρ ∂μψ)¯ One can check by an explicit calculation using similar steps  ∂L ∂L as those outlined in App. B that, for a particle at rest, λ,μν =−i σ μν ψ − ψσ¯ μν S ¯ 2 ∂(∂λψ) ∂(∂λψ) p , | ˆ0i |p , = ,   s SHW  s 0 (3.10) ∂L ←→ ←→ ∂L + ∂ σ μν ψ + ψ¯ ∂ σ μν . ρ ρ ¯ ∂(∂ρ ∂λψ) ∂(∂ρ ∂λψ) 4 ˆμν = ˆμν = ˆμν In the following, since SHW SGLW SKG, we will only use the HW subscript for the sake of ease of notation. 123 Eur. Phys. J. A (2021) 57 :155 Page 7 of 25 155 the second addend in Eq. (3.9) vanishes and, from Eq. (3.6), Eq. (2.16)inEqs.(3.15) and (3.18), and taking the matrix we see that element of one-particle states, one obtains ij ij p , | ˆ |p , =p , | ˆ |p ,  .   μ  s SHW  s  s SC  s (3.11) p , s | Sˆ |p, s This means that the space components of the HW global =− 1 μναβ p, | ˆ ˆ |p,  ˆμν s Pν SC,αβ s spin reduce to the nonrelativistic spin vector and that SHW 2m behaves as a tensor in accordance with the Frenkel theory 1 μναβ   ˆ ˆ =−  p , s | Pν SHW,αβ |p, s (3.4), i.e., 2m  1 μναβ 3   † h¯ p, | ˆ ˆμν |p, = . =−  pν d x p , s | ψ (x) σαβ ψ(x) |p, s , s Pμ SHW s 0 (3.12) 2m 2 (3.20) Thus, the HW global spin gives a covariant generalization of the nonrelativistic spin operator. for the details of the derivation see App. B. The equation above shows that the contribution of the orbital parts of the 3.2 Pauli–Lubanski vector canonical and HW pseudo-gauge vanish when one-particle states are considered. For a related discussion, see Refs. [18, In this subsection, we generalize the nonrelativistic spin oper- ˆμ 19]. Finally, we note that the inverse relation involving the ator in Eq. (3.1) to a covariant vector S (even though strictly expectation value of one-particle states of the form speaking it is a pseudovector, we will use the term vector for simplicity). This vector is such that, in the particle rest frame, p, | ˆμν |p, =−1 μναβ p, | ˆ ˆ |p,  s SHW s s Pα Sβ s (3.21) it reduces to the form m is only valid for the HW, GLW and KG (and not for the p , | ˆμ |p , =( , p , | ˆ |p , )  s S  s 0  s Snr  s (3.13) canonical) pseudo-gauge since in this case Eq. (3.12) holds. or, covariantly, We can thus write ˆμν μναβ ˆ ˆ ˆμ p, s| Sv |p, s=− p, s| vα Sβ |p, s , (3.22) p, s| Pμ S |p, s=0. (3.14) ˆμν ˆμν μ ˆμν ˆμν where Sv = S for v = (1, 0) and Sv = S for In order to define the relativistic spin operator, we introduce C HW vμ = ˆ μ/ the Pauli-Lubanski vector [15,17] P m. We conclude this section with a brief remark about mass- μ 1 μναβ ˆ ˆ wˆ =−  Pν Jαβ (3.15) less particles. It is clear that the spin vector in Eq. (3.18)is 2 not defined for particles with vanishing mass. We note that ˆ the action of the commutator of the Pauli-Lubanski vectors where Jμν is the total angular momentum in Eq. (2.16). Using the commutation relations of the Poincaré algebra we obtain (3.16) on a massless one-particle state cannot be reduced to [15] the conventional commutation relations for spin operators (3.19), since the physical quantum number is now the parti- μ ν μναβ ˆ [ˆw , wˆ ]=−ih¯  wˆ α Pβ . (3.16) cle helicity [15]. It is possible to show that the action of the Pauli-Lubanski vector on a massless one-particle state |p,λ If we consider the action of the commutator on states at rest, with helicity λ =±1/2, is given by [15,20] then for the spatial components we have μ μ wˆ |p,λ=h¯ λPˆ |p,λ . (3.23) i j ijk0 [ˆw , wˆ ]=−ih¯  wˆ km. (3.17) Equation (3.23) shows that for massless particles the spin is Therefore, the relativistic spin operator can be defined as slaved to the momentum. μ μ wˆ Sˆ = , (3.18) m 4 Wigner operator since its spatial components follow the usual commutation relations for spin operators Quantum mechanics can be equivalently formulated with ˆ ˆ ˆ the so-called Wigner function [21], namely a distribution [Si , S j ]=ih¯ ijkSk, (3.19) function in phase-space which can be alternatively used to provided that they act onto states at rest. calculate expectation values of operators. In this sense, the We note that, since the total charges are pseudo-gauge Wigner function generalizes the concept of classical distri- invariant by construction, it follows that the relativistic spin bution function to the quantum case. However, caution must vector is also a pseudo-gauge invariant quantity. Inserting be taken when making this identification, since the Wigner 123 155 Page 8 of 25 Eur. Phys. J. A (2021) 57 :155

h¯ μ ˆ ˆνμ function can in general take on complex values. In quantum ∂ F + pνS = 0. (4.7e) field theory the Wigner operator for the Dirac field is defined 2 as [4,22–24]  The energy-momentum and spin tensors discussed in Sect. 2 4 ˆ d y − i p·y ¯ can be easily expressed in terms of the functions (4.6). In Wκχ(x, p) = e h¯ ψχ (x ) ψκ (x ) , (4.1) (2πh¯ )4 1 2 particular, Eqs. (2.6), (2.22), (2.28a) and (2.39) can be written as with x = x + y/2, x = x − y/2 and κ, χ denote here 1 2  Dirac indices. Using the Dirac equation for free fields (2.3a) ˆ μν = 4 νVˆ μ, and (2.3b), one derives the equation of motion for the Wigner TC d pp (4.8)  operator [4,23,24] μν 1 ν μ μ ν     Tˆ = d4 p (p Vˆ + p Vˆ ), (4.9) h¯ B 2 γ · p + i ∂ − m Wˆ (x, p) = 0, (4.2)    ¯ 2   2 ˆ μν = 1 4 μ ν + h ∂μ∂ν − μν∂2 Fˆ ,   THW d p p p g Applying the operator [γ · p + i h¯ ∂ + m] to Eq. (4.2) and m 4 2 (4.10) separating real and imaginary part we obtain  μν μν 1 μ ν   Tˆ = Tˆ = d4 pp p Fˆ , (4.11) h¯ 2 GLW KG m p2 − m2 − ∂2 Wˆ (x, p) = 0, (4.3) 4 respectively. In the equations above, the free equation of ˆ p · ∂ W(x, p) = 0, (4.4) motion (4.7a) and the on-shell condition (4.3)wereusedto eliminate the term proportional to gμν with the Dirac and the respectively. We recognize in Eq. (4.3) the modification of Klein–Gordon Lagrangian in the energy-momentum tensors the on-shell condition defining the particle spectrum and in (4.8), (4.9) and (4.11). We note that for the HW energy- Eq. (4.4) a Boltzmann-like equation. momentum tensor, the term proportional to gμν cannot be It is convenient to decompose the Wigner operator in terms eliminated with the equations of motion. The spin tensors in of a of the generators of the Clifford algebra Eqs. (2.14), (2.28b), (2.40) can be now expressed as    ˆ = 1 Fˆ + γ 5Pˆ + γ μVˆ + γ 5γ μAˆ + 1σ μνSˆ , W i μ μ μν ˆλ,μν h¯ λμνρ 4 ˆ 4 2 S =−  d p Aρ, (4.12) C 2 (4.5)  λ,μν λ,μν h¯ λ μν Sˆ = Sˆ = d4 pp Sˆ , (4.13) with the coefficients given by HW KG  2m  ˆλ,μν h¯ 4 λ ˆμν h¯ λμνα ˆ Fˆ = ( ˆ ), S = d p p S −  ∂αP . (4.14) Tr W (4.6a) GLW 2m 2 Pˆ =−i Tr(γ 5Wˆ ), (4.6b) We now want to relate the spin vector to the Wigner oper- Vˆ μ = (γ μ ˆ ), Tr W (4.6c) ator. In accordance with Ref. [18], we define the spin vector μ μ Aˆ = Tr(γ γ 5Wˆ ), (4.6d) as ˆμν μν ˆ ˆ μ 1 μναβ ˆ S = Tr(σ W), (4.6e) Π (p) =−  pν jαβ (p), (4.15) 2m ˆ where the traces are meant over the Dirac indices. We substi- where jαβ is related to the total angular momentum in ˆ ˆ tute Eq. (4.5) into Eq. (4.2) and decompose real and imagi- Eq. (2.16) through Jαβ = d4 p jαβ . Let us justify Eq. (4.15) nary part to obtain the equations of motion for the coefficient using the Frenkel theory. To do so we define the quantity functions. We write here only the two equations we will use  ¯ in this section (for the complete set of equations of motion ˆμν ( ) = h Σ λSˆμν( , ) sHW p d λ p x p for the coefficients when also interactions are considered see 2m Σ  0 Sect. 6.1) p μν = h¯ d3x Sˆ (x, p). (4.16) 2m p · Vˆ − mFˆ = 0, (4.7a) ˆμν μ ˆ h¯ ˆνμ ˆ μ It is easy to check that sHW is a Lorentz tensor, since p F − ∂νS − mV = 0, (4.7b) λ ˆμν 2 p ∂λS = 0 which, in turn, is a consequence of Eq. (4.4). h¯ μ ˆ 1 μναβ ˆ ˆμ The global spin can be written as − ∂ P +  pνSαβ + mA = 0, (4.7c)  2 2 μν μν ˆ ˆ = 4 ˆ ( ). p · A = 0, (4.7d) SHW d p sHW p (4.17) 123 Eur. Phys. J. A (2021) 57 :155 Page 9 of 25 155   From Eq. (4.7e) we get 3 ˆ 3 j ˆ = x0∂0 d x ∂i F + d xxi ∂ j ∂ F  2  μi h¯ 0 3 i ˆ pμsˆ =− p d x ∂ F = 0, (4.18) = 3 ∂ j Fˆ = , HW 2m gij d x 0 (4.24) where boundary terms were neglected, and  where we integrated by parts the second integral in the second 2 α = β = μ0 h¯ 3 ˆ line and neglected boundary terms. For i, j we have pμsˆ =− d xp· ∂F = 0, (4.19)  HW 2m 3 ˆ lij = ∂0 d x (xi ∂ j − x j ∂i )F = 0, (4.25) which follows again from Eq. (4.4). Therefore, we deduce that which follows after again integration by parts of both terms. μi Using the canonical currents (4.8) and (4.12), Eq. (4.15) pμsˆ = 0. (4.20) HW can also be written as ˆμν Equation (4.20) shows that, for free fields, sHW satisfies a ˆ μ 1 μναβ form of the Frenkel condition. The difference with Eq. (3.4) Π (p) =−  pνsˆC,αβ (p) is that here pμ is the conjugate variable of yμ in the Wigner 2m h¯ λ ˆμ transform (4.1) and not in general the total momentum of = dΣλ p A (x, p) 2m Σ the system (or of the particle). In other words the spin in  0 p μ the rest frame of the momentum operator is equivalent to = h¯ d3 x Aˆ (x, p), (4.26) defining the spin in the rest frame of the momentum variable 2m pμ. Therefore, we can define the spin vector based on the variable pμ and the HW spin tensor (4.13)as where from the first to the second step we used Eq. (4.7d). It is possible to show that Eqs. (4.26) and (4.21) are equal by using the equation of motion (4.7c) and the Boltzmann Πˆ μ( ) ≡− 1 μναβ ˆ ( ) p pνsHW,αβ p equation p ·∂Pˆ = 0 which follows from Eq. (4.4). Note that, 2m  however, the inverse relation h¯ μναβ λ ˆ =−  pν dΣ pλSαβ (x, p) 4m2 Σ μν 1 μναβ ˆ  sˆ (p) =−  pαΠβ (p) (4.27) 0 HW m p μναβ 3 ˆ =−h¯  pν d x Sαβ (x, p), (4.21) 4m2 is only valid for the HW, GLW and KG (but not canonical) spin tensor. Finally, if we use the Belinfante pseudo-gauge where in the last line we chose the hypersurface integration in Eq. (4.15), the spin vector has only the contribution of at constant x0. We stress that, although expressed in terms of the orbital part of the angular momentum, since the spin ten- the HW spin tensor, Eq. (4.21) is pseudo-gauge invariant and sor vanishes. Thus, using the Belinfante energy-momentum is equal to Eq. (4.15). To show this, we only need to prove tensor (4.9), we can express the spin vector (4.15)as that the HW orbital part of the spin vector obtained using  Eq. (4.10), μ 1 μναβ λ ˆ Πˆ (p) =−  pν dΣ pλx[αVβ](x, p)   4m Σ ¯ 2  − 1 μναβ 3 Fˆ + h , 0 pν d xx[α pβ] p0 lαβ (4.22) p μναβ 3 ˆ 2m2 4 =−  pν d xxαVβ (x, p), (4.28) 2m vanishes, where    which can be shown to be equal to Eqs. (4.21) and (4.28) 3 2 ˆ lαβ = d xx[α ∂β]∂0 − gβ]0∂ F. (4.23) after employing Eq. (4.7b), integrating by parts, and utilizing p · ∂Sˆμν = 0 which follows from Eq. (4.4). Note that both terms in Eq. (4.22) inside the square brackets As a concluding remark of this section, we mention that the ˆ μ are tensors. The first one vanishes when contracted with the operator Π (p) should not be confused with the spin vector 4 ˆ μ Levi–Civita tensor. For the second one, we have lαβ = 0, defined in Sect. 3. In fact, one can see that d p Π (p) is which is what must hold, since the HW and GLW pseudo- in general different from Sˆμ in Eq. (3.18). However, when gauges have the same global orbital angular momentum taking the matrix element on one-particle states, we obtain [compare, e.g., Eqs. (4.10) and (4.11)]. To see this, let us con- the relation  sider each component of lαβ separately. For α = 0, β = i, p, | ˆμ |p, = 4 p, | Πˆ μ( ) |p,  . we have s S s d p s p s (4.29)  We emphasize that Eq. (4.29) is only valid for free fields. In = 3 ( ∂ ∂ − ∂2 + ∂2)Fˆ l0i d x x0 0 i xi 0 xi this case one can prove that the momentum of the Wigner 123 155 Page 10 of 25 Eur. Phys. J. A (2021) 57 :155 function is set on-shell by the spacetime integration [18,19]. operator Oˆ , For more details see Appendix. B. ˆ ˆ O ≡O=Tr(ρˆLE,0 O). (5.5) Let us now follow the same steps to construct the density 5 Pseudo-gauges and statistical operator operator using the canonical currents. We immediately see that the constraint on the angular momentum is not redundant So far, we have only dealt with operators and their matrix anymore because we have a nonvanishing spin tensor, i.e., element of one-particle states. A natural question we can (ρˆ ˆ μν) = μν, now ask is how different pseudo-gauges affect the thermo- Nμtr C TC NμTC (5.6) dynamic description of a system [25–28]. To address this ˆμ,λν λ ˆ μν ν ˆ μλ ˆμ,λν Nμtr(ρˆC J ) = Nμtr[ˆρC (x T − x T + S )] question, in this section we study the consequences of the C C C C = μ,λν. pseudo-gauge transformations on the statistical operator ρˆ Nμ JC (5.7) [27]. In statistical quantum field theory a possible way to determine ρˆ is by using the method proposed by Zubarev Therefore, Eq. (5.7) reduces to the effective independent con- [29] and later rediscussed in Refs. [30–33]. The local equi- straint on the spin tensor μ,λν μ,λν librium density operator ρˆLE is obtained by maximizing ˆ Nμtr(ρˆC S ) = Nμ S , (5.8) the entropy s =−tr(ρˆ log ρ)ˆ imposing constraints on the C C energy-momentum and total angular momentum to be equal and the canonical local equilibrium density operator is given to the actual ones. Let us first start with the Belinfante case by   in which the constraints are given by  ρˆ = 1 − Σ ( ˆ μνβ − 1 ˆμ,λνΩ ) , C,LE exp d μ TC C,ν SC C,λν Z Σ 2 (ρˆ ˆ μν) = μν, C Nμtr B TB NμTB (5.1) (5.9) ˆμ,λν λ ˆ μν ν ˆ μλ μ,λν Nμtr(ρˆB J ) = Nμtr[ˆρB(x T − x T )]=Nμ J , λν B B B B with Z = Trρˆ , . The quantity Ω is called spin potential (5.2) C C LE C and corresponds to the Lagrange multiplier related to the β μν μ,λν conservation of the total angular momentum. The fields C where on the right-hand sides the quantities TB and JB and ΩC are solutions of the equations are the actual densities. The vector N in Eqs. (5.1) and (5.2) is the normal to some hypersurface Σ that we defined by ˆ μν μν NμTr ρˆC,LE(βC ,ΩC ) T = NμT (βC ,ΩC ), (5.10) a proper foliation of the spacetime. Notice that Eq. (5.1) C C ρˆ (β ,Ω ) ˆμ,λν = μ,λν(β ,Ω ). implies that Eq. (5.2) is redundant: once we have the con- NμTr C,LE C C SC Nμ SC C C (5.11) straint on the energy-momentum tensor we automatically have that on the total angular momentum. Thus, the local It is clear that, in general, Eqs. (5.3) and (5.9) are not equilibrium density operator reads equal. In order to compare the two density operators, we    perform in Eq. (5.9) the pseudo-gauge transformation with ρˆ = 1 − Σ ˆ μνβ , B,LE exp d μTB B,ν (5.3) the superpotentials in Eq. (2.21) and we obtain Z B Σ    = ρˆ β where Z B Tr B,LE and B,ν is the Lagrange multiplier ρˆ = 1 − Σ ˆ μνβ C,LE exp d μ TB C,ν associated to momentum conservation. We stress that the ZC Σ  Lagrange multiplier depends on the choice of the pseudo- 1 ˆμ,λν 1 ˆλ,μν ˆν,μλ − (ΩC,λν − C,λν)S + χC,λν(S + S ) , gauge because it has to be a solution of the constraint at local 2 C 2 C C equilibrium (5.12)

ρˆ (β ) ˆ μν = μν(β ), NμTr B,LE B TB NμTB B (5.4) where we made use of the integration by parts and neglected boundary terms. Furthermore we defined which are four equations for the four unknowns βB,ν.The operator (5.3), however, being time dependent, is not the real λν 1 λ ν ν λ  =− (∂ β − ∂ β ), (5.13) density operator in the Heisenberg picture. The true statistical C 2 C C ρˆ , operator LE 0 is assumed to be the one in Eq. (5.3) evaluated λν 1 λ ν ν λ Σ χ = (∂ β + ∂ β ). (5.14) at some specific time with corresponding hypersurface 0 C 2 C C where the system is known to be in local thermodynamic equilibrium [29]. The true statistical operator is what one The tensor in Eq. (5.13) is called thermal . We note needs for the calculation of the ensemble average of any that, since the canonical spin tensor (2.14) is antisymmetric 123 Eur. Phys. J. A (2021) 57 :155 Page 11 of 25 155

ˆλ,μν + ˆν,μλ = under the exchange of all indices, then SC SC 0 6 Spin-polarization effects in relativistic nuclear and the last term in the exponent in Eq. (5.12) vanishes. By collisions comparing the two statistical operators (5.12) and (5.9), we βν = βν infer that they are equal if C B, and The formalism discussed so far is a powerful tool to study spin dynamics of relativistic many-body systems. In this λν λν Ω =  , (5.15) section we will focus on applications to the physics of relativistic heavy-ion collisions (HICs). In HICs strongly- interacting matter is created by colliding atomic nuclei at where we removed the subscript C, since the βν fields in the energies much higher than the nuclear mass rest energies. different pseudo-gauges have to be the same. Under such extreme conditions, quarks and gluons are decon- We can generalize what discussed so far by considering a generic pseudo-gauge transformation. If we start from the fined and form a new phase of quantum chromodynamics canonical statistical operator in Eq. (5.9) and perform the (QCD) matter called the quark-gluon plasma (QGP). An transformation in Eq. (2.18), we obtain extremely important feature of the QGP produced in HICs is that it shows a strong collective behavior and its spacetime    evolution can be very accurately described using relativistic ρˆ = 1 − Σ ˆ μνβ − 1 ˆμ,λνΩ C,LE exp d μ Tpgt C,ν Spgt C,λν ZC Σ 2 hydrodynamics, see, e.g., [35]. Besides being a nearly perfect 1 μ,λν 1 λ,μν ν,μλ relativistic fluid, the QGP exhibits other surprising properties − (Ω ,λν −  ,λν)Φˆ + χ ,λν(Φˆ + Φˆ ) 2 C C 2 C connected to its fluid nature.  ˆ λν,μρ Noncentral HICs have large global angular momentum − (∂ρΩ ,λν)Z . (5.16) C which is estimated to be on the order of thousands of h¯ .Itis expected that part of it is transferred to the QGP as vorticity which, in turn, generates particle spin polarization [36–39]. After comparing with the statistical operator constructed with μν μ,λν This mechanism resembles the Barnett effect, where a ferro- the currents Tˆ and Sˆ , pgt pgt magnet gets magnetized when spinning around an axis [40]. Recent experimental studies showed that some hadrons emit-    ted in noncentral collisions (e.g., Lambda baryons) exhibit ρˆ = 1 − Σ ˆ μνβ pgt,LE exp d μ Tpgt pgt,ν a spin alignment along the direction of the global angu- Zpgt Σ  lar momentum. This gives the evidence that the QGP has − 1 ˆμ,λνΩ , Spgt pgt,λν (5.17) a strong vortical structure [2,41,42]. The global polarization 2 (namely the polarization along the global angular momen- tum) turns out to be in very good agreement with models proposed in Refs. [39,43–49]. For recent reviews see, e.g., where Zpgt = Trρˆpgt,LE, we can readily get the conditions βμ = βμ Ωλν = [50,51]. The assumption of these models is that local ther- for the equivalence of the states, namely: C pgt, C Ωλν ,Eq.(5.15) has to hold, and modynamic equilibrium is reached at some early stage of the pgt process (QGP formation) and kept until hadronization, where the fluid becomes a kinetic hadronic system. At freeze-out, χ λν = ,∂ρΩλν = . 0 0 (5.18) when scatterings cease, particles become polarized only if the thermal vorticity defined in Eq. (5.13) (computed with Equation (5.15) together with Eq. (5.18) are the conditions relativistic hydrodynamics) is different from zero. The for- for global equilibrium [34]. Therefore, we conclude that, in mula for the spin vector used to describe the Lambda global general, the local equilibrium statistical operator is a pseudo- polarization is based on an educated ansatz for the distribu- gauge dependent quantity. Only in global equilibrium the tion function [44]. Such formula is given by the expectation statistical operators derived with different pseudo-gauges value of Eq. (4.26) [or equivalently Eq. (4.21)] with respect are equal, provided that the thermodynamic fields are the to the local equilibrium state. After carrying out an expansion same. An important consequence of what showed in this sec- in gradients, one obtains at first order [44] tion is that the expectation value of a generic operator Oˆ   λ h¯ 2 Σ dΣλ p fF (1 − fF )αβ (x) will depend on the pseudo-gauge if the statistical operator Πˆ μ( )=− μναβ FO  , p pν λ 8m dΣλ p f describes a many-body state away from global equilibrium. ΣFO F Only in global equilibrium the expectation values calculated (6.1) with different pseudo-gauges will be the same. Therefore, studying the expectation values of observables away from where the integration is carried out over the freeze-out hyper- αβ global equilibrium is in principle a possible way to under- surface ΣFO, fF is the Fermi–Dirac distribution and  stand which pseudo-gauge best describes the system. the thermal vorticity (5.13). 123 155 Page 12 of 25 Eur. Phys. J. A (2021) 57 :155

The models which were able to describe so accurately 6.1 Spin hydrodynamics and quantum kinetic theory the data [50,52], however, fail when it comes to explain the longitudinal Lambda polarization, i.e., the projection The equations of motion of conventional relativistic hydro- of the spin along the beam direction [53]. More specifi- dynamics are the conservation of energy and momentum cally, the Lambda longitudinal polarization is measured as μν ∂μT = 0. (6.2) a function of the azimuthal angle of the transverse momen- tum and it exhibits a very similar pattern to that of the The main idea to extend hydrodynamics to include the elliptic flow of the azimuthal particle spectra [35]. The dynamics of spin degrees of freedom is by promoting the predictions of Ref. [52] for the longitudinal polarization conservation of the total angular momentum show a correct sin(2φ) behavior, where φ is the azimuthal ∂ λ,μν = [νμ] angle, but with an opposite sign in the amplitude with λ S T (6.3) respect to the experimental data [53]. Unfortunately, this as a new equation of motion, where the spin tensor plays mismatch between theory and experiments, which we will the role of spin density, in the same logic as the energy- call “sign puzzle”, does not yet have a definitive theoreti- momentum tensor is related to energy and momentum den- cal explanation, although many attempts have been recently sity [27,62–68]. Relativistic hydrodynamics is in principle a made [54–61]. A crucial feature of the models in Refs. classical theory, while spin is inherently a quantum feature of [39,43–49] is that they assume local equilibrium also of matter. Therefore, the natural starting point for a consistent spin degrees of freedom. However, the “spin puzzle” sug- treatment of spin in hydrodynamics is quantum field theory. gests that spin degrees of freedom may undergo a non- In practice, we establish a connection to quantum field theory trivial dynamics related to the conversion between orbital by defining our densities T μν and Sλ,μν as ensemble average and spin angular momentum which is not well understood of quantum operators yet. μν ˆ μν λ,μν ˆλ,μν In the past few years, the study of spin dynamics has T =T , S =S . (6.4) attracted considerable attention. Many works have focused The set of Eqs. (6.2) and (6.3) is called spin hydrodynam- on spin hydrodynamics, an extension of relativistic hydrody- ics. The unknowns of this system of equations will be the namics where spin degrees of freedom are included. There are Lagrange multipliers associated to energy and momentum several approaches in the literature which, actually, were not conservation βμ = uμ/T (uμ is the fluid velocity and T the specifically developed to address the “spin puzzle”: one can temperature), and to the total angular momentum conserva- promote the total angular momentum conservation as a new tion, the spin potential Ωμν, introduced in Sect. 5. More- hydrodynamic equation of motion with a suitable definition over, if dissipation effects are considered, extra equations of the spin tensor [27,62–68], use the Lagrangian formal- of motion for the dissipative quantities should be provided. ism [69–72] or the holographic duality [73]. There has been In order to compute Eq. (6.4), one has to choose a specific intense activity also on the description of nonequilibrium pseudo-gauge. dynamics of spin polarization during the collision process Relativistic hydrodynamics can be derived, for example, using the Wigner-function formalism in the free-streaming from the Boltzmann equation by applying the method of case [74–81], and including particle collisions [68,82,83]. moments [100]. Therefore, having a quantum kinetic theory It is worth to mention that the Wigner-function fomalism framework is important to derive spin hydrodynamics. On a has been widely used also for the description of anomalous microscopic level, angular momentum conservation implies chiral transport in the QGP, see, e.g., Refs. [84–94]. An that the conversion between orbital and spin angular momen- important question to be addressed is also whether spin equi- tum can occur only if particles collide with a finite impact librates fast enough for the time scales of HICs. Calculations parameter. Hence, we need a kinetic picture where the non- of spin equilibration time were recently carried out using per- locality of the collisions is consistently taken into account. turbative QCD [95,96], Nambu–Jona–Lasinio model [97], In order to formulate a transport theory from quantum field and effective vertex for the interaction with thermal vortic- theory, we use the Wigner-function formalism. We define the ity [98,99]. In the rest of this section we will discuss the Wigner function as the normal-ordered ensemble average of newly developed spin hydrodynamics and quantum kinetic the Wigner operator (4.1)[4,22–24], theory as promising approaches for a solution of the “spin puzzle” and for a deeper understanding of spin effects in W(x, p) ≡Wˆ (x, p). (6.5) HICs. In particular, we will focus on the impact of different pseudo-gauge choices on the formulation of spin hydrody- Since we need to introduce collisions, the right-hand side of Eq. (4.2) will be modified as [4] namics.     h¯ γ · p + i ∂ − m W = h¯ C, (6.6) 2 123 Eur. Phys. J. A (2021) 57 :155 Page 13 of 25 155

≡ μ ≡−( / )μναβ where with CF 2CF and C A 1 m pνCS,αβ . In order  4   to obtain a more intuitive understanding of spin-related quan- d y − i p·y C ≡ e h¯ J (x )ψ(¯ x ) (6.7) tities, we introduce spin as an additional variable in phase (2πh¯ )4 2 1 space [65,67,68,101–103]. We define the distribution func- ¯ and J =−(1/h¯ )∂LI /∂ψ, with LI being the interaction tion Lagrangian. Applying the decomposition (4.5)toEq.(6.6) 1 f(x, p, s) ≡ F¯ (x, p) − s · A(x, p) , (6.14) and separating real and imaginary part, we obtain the equa- 2 tions of motion for the coefficient functions. Thus, we get and the integration measure   1 dS(p) ≡ d4s δ(s · s + 3)δ(p · s), (6.15) p · V − mF = hD¯ F , (6.8a) κ(p) h¯ √  ∂ · A + mP =−hD¯ P , (6.8b) with κ(p) ≡ 3π/ p2 such that 2  μ h¯ νμ μ μ p F − ∂νS − mV = hD¯ , (6.8c) F¯ = ( ) f( , , s), 2 V dS p x p (6.16a)  h¯ μ 1 μναβ μ μ − ∂ P +  pν Sαβ + mA =−hD¯ A, (6.8d) Aμ = ( ) sμf( , , s). 2 2 dS p x p (6.16b) h¯ [μ ν] μναβ μν μν ∂ V −  pαAβ − mS = hD¯ , (6.8e) 2 S The distribution (6.14) can be parameterized as for the real part, and f(x, p, s) = mδ(p2 − m2 − h¯ δm2) f (x, p, s), (6.17) where f (x, p, s) is a function without singularity at p2 = h¯ ∂ · V = 2hC¯ F , (6.9a) m2 + h¯ δm2 and δm2(x, p, s) is a correction to the mass- · A = ¯ , p hCP (6.9b) shell condition for free particles arising from interactions. h¯ μ νμ μ The final Boltzmann equation to be solved is thus given by ∂ F + pνS = hC¯ , (6.9c) 2 V p · ∂ f = m C[f], (6.18) μ h¯ μναβ μ p P +  ∂νSαβ =−hC¯ , (6.9d) 4 A C[f]≡ 1 ( −s· ) C[f] where 2 CF C A . The collision term contains [μ ν] h¯ μναβ μν both local and nonlocal contributions and has been recently p V +  ∂αAβ =−hC¯ S . (6.9e) 2 explicitly calculated in Ref. [68]. It was demonstrated that, using the standard form of the equilibrium distribution func- = (γ˜ C) for the imaginary part, where we defined Di Re Tr i , tion [44,62,65] Ci = Im Tr (γ˜i C), i = F, P, V, A, S, γ˜F = 1, γ˜P =−iγ5,   μ μ 5 μν 1 h¯ μν γ˜V = γ , γ˜A = γ γ , γ˜S = σ . Following Ref. [68], f (x, p, s) = exp −β(x) · p + Ωμν (x)Σs , eq ( π ¯ )3 we employ an expansion in powers of h¯ for the coefficient 2 h 4 functions of the Wigner function and the collision term in (6.19) Eqs. (6.8) and (6.9), e.g., for the scalar part we write and the total angular momentum conservation in binary scat- ( ) ( ) terings, the conditions under which the collision term van- F = F 0 + h¯ F 1 + O(h¯ 2∂2). (6.10) ishes are indeed those of global equilibrium discussed in We stress that, since in the equations of motion (6.6) a gradi- Sect. 5.InEq.(6.19) we defined the spin-dipole-moment μν μναβ ent is always accompanied by a factor of h¯ , this is effectively tensor Σs ≡−(1/m) pαsβ . also a gradient expansion. We now make the assumption that Once the quantum kinetic theory is established, one can spin effects are at least of first order in the h¯ -gradient expan- evaluate the hydrodynamic quantities (6.4). To do so, a sion. As a consequence it can be shown that [68] pseudo-gauge choice has to be made. Let us first consider the canonical currents. Substituting Eq. (6.11) into Eq. (4.8) μ 1 μ V = p F¯ + O(h¯ 2∂2), (6.11) and Eq. (6.16b) into Eq. (4.12), we obtain 5 m  μν = μ ν ( , , s) + O(¯ 2∂2), where we defined TC dPdS p p f x p h (6.20) ¯ h¯ μ (1) F ≡ F − p DV,μ. (6.12) 5 − μν (L + L ) m2 The term g D I in the energy-momentum tensor does not in general vanish when using the equations of motion, but it is propor- The relevant Boltzmann equations then read tional to an interaction term. However, we study kinetic theory of dilute systems where such contribution to the energy-momentum tensor can · ∂F¯ = , · ∂Aμ = μ, p mCF p mCA (6.13) be neglected [4]. 123 155 Page 14 of 25 Eur. Phys. J. A (2021) 57 :155  λ,μν m λμνα S =−h¯  dPdSsα f (x, p, s) Notice from Eqs. (6.20) and (6.25) that, under our assumption C 2  of spin as a first order quantity, the canonical and HW energy- 2   m 1 λ μν μ νλ ν λμ momentum tensor at O(h¯ ∂) are equal. The HW hydrody- = h¯ dPdS p Σs + p Σs + p Σs 2 p2 namic equations of motion can be written with the help of × f (x, p, s), (6.21) the Boltzmann Eq. (6.18)as  4 2 2 where dP = d pδ(p − m ). Note that Eq. (6.21)is μν ν ∂μT = dPdS(p) p C[ f ]=0, (6.27) indeed exact. 6 Using the Boltzmann Eq. (6.18), the hydro- HW  dynamic equations of motion corresponding to the tensors in λ,μν h¯ μν [νμ] ∂λ S = dPdS(p) Σs C[ f ]=T . (6.28) Eqs. (6.20) and (6.21)aregivenby HW 2 HW  ∂ μν = ( ) ν C[ ]= , Equation (6.28) shows the relation between the antisymmet- μTC dPdS p p f 0 (6.22)    ric part of the HW energy-momentum tensor and the collision λ,μν h¯ μν [μ ν]λ ∂λ S = dPdS(p) Σs C[ f ]+p Σs ∂λ f (x, p, s) kernel of the Boltzmann equation. When only local collisions C 2 μν are considered then Σs is a collisional invariant, leading to = [νμ], TC (6.23) a conserved spin tensor and a vanishing antisymmetric part of the HW energy-momentum tensor in Eq. (6.28). In general, respectively. Equation (6.22) relates the conservation of when we take into account the nonlocality of the collisions, μ energy and momentum to the collisional invariant p .On the spin tensor is not conserved and orbital angular momen- the other hand, from Eq. (6.23), which can be viewed as the tum can be converted into spin through the antisymmetric definition of the antisymmetric part of the energy-momentum part of the HW energy-momentum tensor which arises at tensor, the relation of the divergence of the spin tensor to a O(h¯ 2∂2). In global equilibrium, the HW energy-momentum collisional invariant is not apparent. Furthermore, in global tensor is again symmetric and the spin tensor is conserved. equilibrium, after expanding Eq. (6.19)uptoO(h¯ ∂) and Therefore, the HW pseudo-gauge turns out to be a consis- recalling that C[f]=0, Eq. (6.23) becomes tent choice to describe the conversion between orbital and  spin angular momentum of a relativistic fluid. Finally, since 2 [μν] 1 h¯ [ν μ]λ ρ −β·p 3 3 the HW and KG spin tensors are the same and, moreover, T , = dP p  p λρ e + O(h¯ ∂ ). C eq (2πh¯ )3 2 they differ from the GLW spin tensor by a divergenceless (6.24) term [see Eqs. (4.13) and (4.14)], the antisymmetric parts of the HW, KG and GLW energy-momentum tensors are also We see from Eqs. (6.23) and (6.24) that the antisymmetric equal. The differences between these three pseudo-gauges part of the canonical energy-momentum tensor is different arise at O(h¯ 2∂2), as can be seen from Eqs. (4.10), (4.11), from zero, and hence the spin tensor is not conserved, even in (4.13) and (4.14) (note that P(0) = 0[68]). The physical the case of vanishing collisions or global equilibrium. Since implications of these differences require further investiga- the physical picture is that spin changes only due to particle tion. As a final remark for this section, we mention that in scatterings until global equilibrium is reached, the interpre- λ,μν Ref. [68] it was shown that, in the nonrelativistic limit, the tation of SC as a spin density is not consistent. equations of motion with the HW pseudo-gauge reduce to the In Ref. [68], it was shown that the HW choice carries well-known form of hydrodynamics with internal degrees of interesting physical implications. Starting from the canonical freedom [104]. currents and performing the pseudo-gauge transformations in Eqs. (2.33) and (2.34), one obtains up to first order in the h¯ -gradient expansion 7 Pseudo-gauge transformations and the relativistic  center of inertia μν = ( ) μ ν ( , , s) + O(¯ 2∂2), THW dPdS p p p f x p h (6.25)    In Newtonian mechanics the center of mass has an unam- λ,μν λ 1 μν h¯ [μ ν] S = h¯ dPdS(p)p Σs − p ∂ f (x, p, s) HW 2 4m2 biguous definition: it is the unique point obtained by the mean of all points weighted by the local mass. However + O(h¯ 2∂2). (6.26) this is not true anymore for a relativistic system, since the energy (or inertial mass) depends on the velocity. Follow- 6 In Ref. [68] the spin tensor is defined as the spin tensor in this paper ing Ref. [3], we discuss several possible definitions for the divided by h¯ such that the total angular momentum reads J λ,μν = μ λν ν λμ λ,μν relativistic center of mass and study their physical meanings x T − x T + hS¯ . This implies that the h¯ factor in Eq. (6.21) should not be counted in the h¯ -gradient expansion since it is not accom- when internal angular-momentum degrees of freedom are panied by a gradient. included (see also a recent related work Ref. [105]). In par- 123 Eur. Phys. J. A (2021) 57 :155 Page 15 of 25 155 ticular, we show that, since a pseudo-gauge transformation 7.2 Center of inertia and centroids can be seen as a rearrangement of the splitting between spin and orbital angular momentum, the different choices of spin A natural definition of the center of inertia of a system is the tensors discussed in the previous sections can be related to mean of all points weighted by the local energy. Given an different definitions of the relativistic center of mass. energy-momentum tensor T μν in a certain pseudo-gauge we define the center of inertia as  7.1 External and internal components of angular μ 1 μ 1 μ μ q ≡ d3xx T 00 = (x0 P + L 0), (7.8) momentum P0 P0  μν ≡ μν − μν μ = 3 0μ Let us consider for simplicity classical fields. In special rel- where L J S and P d xT . The defini- 0 = 0 ativity, any angular momentum can be decomposed into an tion above implies q x . Using the conservation of the internal and external part with respect to a reference point energy-momentum tensor, we obtain for the time derivative μ of the center of inertia X ,  μν μν μν μ 1 μ J = L + S , (7.1) ∂ q = d3xT 0, (7.9) X X 0 P0 where the external component is given by provided that, as usual, boundary terms can be neglected. If μν [μ ν] we require that the center of inertia moves along a straight L ≡ X P (7.2) X line, then we need to impose μν and the internal component SX describes the about μν μ μν ∂ν T = 0, (7.10) X .ThetermSX is not necessarily related to spin in the corresponding quantum theory. On the other hand, we can since in this case also decompose the total angular momentum into generators  μ μ 2 μ 1 3 μν of boosts Kn and generators of rotation Jn depending on (∂0) q = d x ∂ν T = 0. (7.11) μ P0 the four-velocity n of the frame in which the generators are defined, We call Eq. (7.9) together with Eq. (7.10) the relativistic center-of-mass theorem. The condition (7.10) is trivially ful- μν = [μ ν] − μναβ , J Kn n nα Jnβ (7.3) filled for symmetric energy-momentum tensors, e.g., for the μ ≡ μν μ ≡−1 μναβ Belinfante, HW, GLW and KG pseudo-gauge. We note from with Kn J nν and Jn 2 nν Jαβ . Combining Eqs. (7.1) and (7.3) we obtain Eq. (2.15) that the condition (7.10) is also fulfilled for the canonical case, since the canonical spin tensor is totally anti- μν [μ [μ ν] μναβ J = (K + K )n −  nα(J , ,β + J , ,β ), (7.4) symmetric. Hence, all energy-momentum tensors discussed n,L X n,SX n L X n L X in this paper are consistent with the relativistic center-of- where mass theorem. μ μ μ Obviously the center of inertia (7.8) is not covariant. It K ≡ (P · n)X − (X · n)P , (7.5a) n,L X can be generalized in a covariant way by introducing the so- μ μν μ K , ≡ S nν, (7.5b) called centroid qn which is identical to the center of inertia n SX X μ μ μναβ in a generic frame moving with four velocity n , J ≡− Xν Pαnβ , (7.5c) n,L X μ 1 μ 1 0 μ μν μναβ q = (x P + L nν), (7.12) J , ≡−  nν SX,αβ . (7.5d) n · n n SX 2 P n 0 ≡ · Consequently, if we want to identify the internal angular where xn x n is the time in the given frame. Notice also · = 0 momentum with the generators of rotation in the frame char- that qn n xn . acterized by the four-velocity nμ, we should impose the con- Now we want to express the orbital angular momentum as dition μν = [μ ν], L qn P (7.13) μν S nν = 0 (7.6) X which implies the choice of the centroid as reference point, μ ≡ μ μν = μν in order to remove the contribution from the boost generators X qn , and hence, from Eq. (7.2), L Lqn .Asa to the internal part in Eqs. (7.5). In this case, Eq. (7.5d) can consequence, from Eq. (7.1), we identify the global spin with μν = μν be inverted to obtain the internal angular momentum in terms the internal angular momentum S Sqn . We also note that of the rotation generators the condition (7.6) for the spin tensor is needed to ensure the μν μναβ validity of the center-of-mass theorem (7.11) and to make =− α , ,β . SX n Jn SX (7.7) the centroid a Lorentz vector, since it allows us to write it in 123 155 Page 16 of 25 Eur. Phys. J. A (2021) 57 :155 terms of conserved quantities provided that the operators act on a single-particle state with momentum pμ and with S defined in Eq. (3.7). μ 1 0 μ μν q = (x P + J nν). (7.14) n P · n n P·n 1 ν 7.5 Center of mass as reference point: HW, GLW and KG Requiring q0 = q0 = x0, we obtain x0 = x0 − L0 nν n n P0 P0 pseudo-gauges and thus 0 μ ν0 μ μν μ x P L nν P L nν For a system with finite mass m there is a preferred reference q = − + , (7.15) n P0 P0(P · n) P · n frame for defining physical quantities in a covariant way in terms of the Poincaré generators. This frame is given by the in order to obtain the worldline of qn parametrized by the original time coordinate x0 and to compare to Eq. (7.8). comoving frame of the system, denoted by the four-velocity μ ≡ μ/  We stress that writing the orbital part in terms of the cen- n P m (the subscript indicates that the corresponding troid as in Eq. (7.13) is not possible for all pseudo-gauges. quantities are evaluated in the rest frame). Clearly, in this In the next subsections we will study whether we can estab- frame the mass is given by μ lish a correspondence between different choices of n and 0 m ≡ P = P · n. (7.22) different expressions for Sμν. The corresponding centroid, that we call the center of mass, μ 7.3 Belinfante pseudo-gauge is obtained by using n in Eq. (7.12), i.e.,   μ 1 μ 1 μν Since in the Belinfante case the spin tensor vanishes, we have q = τ P + J Pν , (7.23) μν = μν m m L B J . Thus 0 μ 1 μ μ where we defined the proper time τ ≡ x and already q = (x0 P + J 0) (7.16) μν 0 imposed Pμ S = 0 in accordance with Eq. (7.6). Note that P μ and q is a Lorentz vector. In a similar way, the spin of a massive particle is defined as the proper internal angular momentum, μ 1 0 μ μν μ q = (x P + J nν). (7.17) i.e., choosing the reference vector n and the reference point n P · n n μ q in Eqs. (7.5). This leads to 7.4 Center of inertia as reference point: canonical μ μ μ K = mq − τ P , (7.24a) pseudo-gauge ,L μ = , K,S 0 (7.24b) The canonical global spin fulfills the condition (7.6)inthe μ = , J,L 0 (7.24c) frame specified by nμ = (1, 0), i.e., we have Si0 = 0(see C μ 1 μναβ Sect. 3). We can then use the canonical currents to evaluate J =−  Pν Sαβ . (7.24d) ,S 2m Eq. (7.8) and obtain In the last equation, we identify the Pauli-Lubanski vector 1 μ 1 μ μ qμ = (x0 Pμ + L 0) = (x0 Pμ + J μ0). w = mJ (cf. with Sect. 3.2), and we verify that it is 0 C 0 (7.18) ,S P P indeed identical to the generator of rotations defined in the We can now define the global spin center-of-mass frame. μν ≡ μν − [μ ν] One can show that the HW (hence the GLW and KG) Sq J q P (7.19) global spin (2.36) is obtained as the difference between the i0 = which, as the canonical spin, fulfills Sq 0inanyframe total angular momentum and the orbital angular momentum μν μν and is not a tensor. We stress that Sq is different from S with respect to the center of mass, i.e., μν C as LC cannot be expressed in terms of the center of inertia μν μν μν [μ ν] μ S = S ≡ J − q P , (7.25) q , even though the canonical currents were used to calcu- HW q μ late q . Defining the spatial components of the total angular which is clearly a Lorentz tensor. The HW spin vector is then ij ≡ ijk k momentum J J we obtain given by

Sq = J − q × P. (7.20) SHW = J − q × P, (7.26) If we want to go from a classical to a quantum framework where Pμ as well as qμ are promoted to be operators, then which corresponds to Eq. (3.9) and, in the rest frame, Eq. (7.20) is given by [3] = = ,  SHW J SC (7.27)

ˆ 3 h¯ † 2 Sq = d x ψ m S + imp × γ + (p · S)p ψ, (7.21) 2(p0)2 consistently with Eq. (3.11)[3]. 123 Eur. Phys. J. A (2021) 57 :155 Page 17 of 25 155

μ →˜μ = μ +Δμ Δμ It is worth mentioning that there is another option for the position qP qP qP withashift , then we need reference point, namely the mean position [3] to redefine a new global spin as Sμν → S˜μν = Sμν −Δ[μ Pν] in order for the total angular momentum to be the same. μ 1 0 μ μ q˜ ≡ (P q + mq ), (7.28) ˜μν = 0 + The condition on the new global spin Pμ S 0 holds if P m μ PμΔ = 0. A solution to this condition can be found such which is not a vector. The internal angular momentum with μ μ [μ ν] that Δ is not proportional to P leading to Δ P = 0. respect to this point corresponds to the internal angular This implies that the definition of orbital and spin angular momentum momentum is ambiguous [110–113]. Thus, in contrast to the μν ≡ μν −˜[μ ν], massive case, there is no possibility to determine the spin in Sq˜ J q P (7.29) a frame-independent way. In other words, the HW pseudo- with spin vector gauge, which is related to the particle rest frame, does not exist. This fact is also apparent from Eqs. (2.28b) and (3.9) Sq˜ = J − q˜ × P (7.30) as a factor of m is present in the denominator. It may seem μ ˜μ After quantizing by promoting P and q to operators [3], natural to use the canonical spin tensor instead. However, the spin vector reads Eq. (3.6) does not yield a familiar definition for the spin  of a massless state since it should be slaved to the particle ˆ 3 h¯ † S ˜ = d x ψ m S + i p × γ momentum. Interestingly, we note that the quantum spin vec- q p0 2  tor in Eq. (7.21) has a smooth massless limit which yields 1 + (p · S)p ψ, (7.31) the familiar form related to the particle helicity p0 +  m p · S p p ˆ 3 h¯ † Sq,m=0 = d x ψ ψ = h¯ λ , (7.34) where p is again the three-momentum of the one-particle 2 |p| |p| |p| state. Equation (7.31) corresponds to the spin vector derived where we considered the expectation value of a one-particle by Foldy and Wouthuysen in Ref. [106]. state |p,λ with helicity λ =±1/2. Hence, we can generalize We conclude this subsection with a physical remark. μν 1 μναβ The canonical currents describe position and spin emerging S = h¯ λ  pαn¯β , (7.35) q,m=0 p ·¯n directly from the Dirac equation and thus contain the rapid μ oscillation (“Zitterbewegung”) in the motion of a Dirac par- where n¯ ≡ (1, 0) in any frame. This coincides with the ticle. However, this oscillation is not measurable [107–109] global spin used in Ref. [111] and, as pointed out in Sect. 7.4, and should be removed to obtain physical quantities. Equa- corresponds to defining the position as the center of iner- μ tions (7.21), (3.9) and (7.31) are expressions calculated from tia q , which is not a Lorentz vector. As a consequence, a various definitions of the relativistic center of mass. As these Lorentz transformation leads to a shift in the position known definitions are mean positions, the spin vectors (7.21), (7.26) as the side-jump effect [110–115]. Consider a Lorentz trans- and (7.31) do not contain the rapid oscillation [3]. formation Λ, the total angular momentum is a tensor and will transform as 7.6 Massless particles and side jumps μν μν μ ν αβ J → J = Λα Λβ J . (7.36) We summarize the results obtained in this section for the On the other hand, the spin (7.35) transforms as massive case as follows: the splitting of the total angular μν μν 1 μναβ  momentum into orbital and spin part S → S = h¯ λ  pαn¯β q,m=0 q,m=0 p ·¯n μν = [μ ν] + μν J qn P S (7.32) = λ 1 μναβ p  − Δ[μpν] h¯   αnβ can be fixed by requiring p · n μ ν αβ [μ ν] μν = Λ Λ S − Δ p (7.37) nμ S = 0 (7.33) α β q,m=0 μ μ μ ν μ μ ν [μ ν] as a supplementary condition, which determines qn accord- with p ≡ Λν p and n ≡ Λν n¯ . Here the term Δ p μ μ ing to Eq. (7.14). For finite mass, choosing n = n as a is an anomalous contribution to the Lorentz transformation frame vector yields a unique covariant decomposition corre- of the global spin for Eq. (7.35) to be preserved after the sponding to the HW pseudo-gauge. Lorentz transformation. In order to ensure Eq. (7.36), For vanishing mass, however, the absence of a rest frame 2 μν leads to additional complications. As P = 0, imposing the J μν = q[μpν] + S μ μ μν q,m=0 condition (7.33) with n ∝ P , Pμ S = 0, does not deter- = ΛμΛν αβ mine the splitting uniquely. Consider a redefinition of the α β J (7.38) 123 155 Page 18 of 25 Eur. Phys. J. A (2021) 57 :155 the center of inertia qμ has to transform as and Rosenfeld as [6–8]

1 δ AM μ μ μ ν μ Tμν = , (8.1) q → q = Λν q + Δ . (7.39) g δgμν  where the matter action A = d4xL , with L the The anomalous contribution of a Lorentz transformation for M M √ M Lagrangian, gμν is the metric tensor and g = −det(gμν). the center of inertia Δμ can be found by contracting Eq. (7.37) μ In the literature, the expression for the energy-momentum with n¯ν. Choosing Δ to be purely spatial in the frame at tensor above is often considered to be the fundamental one rest with the observer after the Lorentz transformation, i.e., because it is defined as the source of the gravitational field. n¯ · Δ = 0, we obtain in this frame It is important to note that since gμν is a symmetric tensor, μναβ   μ  pνnαn¯β Tμν in Eq. (8.1) is also symmetric and indeed reduces in Δ = h¯ λ . (7.40) (p ·¯n)(p · n) to the energy-momentum tensor discussed in Sect. 2.2. Following these considerations, it is usually μ The physical implications of the anomalous shift Δ can claimed that the “physical” energy-momentum tensor must be seen in a binary particle scattering p1i + p2i → p1 f + be symmetric. However, we observe that in conventional gen- p2 f . Consider first the frame, called “no-jump frame”, which eral relativity spinorial degrees of freedom are not taken into we assume to coincide with the center-of-momentum frame, account and we can regard the absence of an antisymmetric where the two initial particles collide in one point and the part of the energy-momentum tensor as the consequence of final particles are emitted from the same point. If we see the this fact. In the following we shall briefly review an extension scattering in a boosted frame in a direction parallel to the of general relativity called Einstein–Cartan theory, where one μ initial momenta, then we have to compute the shift Δ in allows the spacetime geometry to have a nonvanishing tor- Eq. (7.40) for each particle. For the two incoming particles, sion, an additional property of the geometry which μ since the spatial components of n are parallel to the three- couples to spin. In such theory the energy-momentum tensor Δμ = Δμ = momenta, then 1i 2i 0. For the final particles, since can gain an antisymmetric part [116–118]. the momenta are not parallel to the spatial components of nμ Δμ Δμ anymore, we have that 1 f and 2 f are different from zero. 8.1 Riemann–Cartan geometry This means that the particles in the final state are emitted in Δμ Δμ a position shifted by an amount 1 f and 2 f , respectively, The Einstein–Cartan theory is based on the so-called Riemann– from the point where the initial particles collided. This is the Cartan spacetime [116–120]. Let us consider a four-dimensi- side-jump effect. onal differentiable manifold, whose spacetime points are We stress that the side jump effect occurs for massless labeled with xμ. In order to specify the geometrical struc- particles due to the absence of a covariant definition of the ture, we introduce a spacetime dependent symmetric metric μν μν μα μ μ center of mass and, hence, of a covariant spin. For massive g = g (x) (such that g gαν = δν , with δν the Kro- particles, instead, it is always possible to define a covariant necker delta) and the notion of parallel transport of vectors. center of mass which leads to the HW, GLW or KG spin (at Consider an infinitesimal displacement xμ + dxμ from the least for free fields or in case of local interactions). Therefore, point xμ, then a vector Bμ changes by in the massive case it is natural to use the HW, GLW or KG μ =−Γ˜ μ ( ) α β , pseudo-gauge where the spin is defined in the particle rest dB αβ x B dx (8.2) frame and no anomalous shift has to be taken into account. In where Γ˜ α is the affine connection. In contrast to conven- the case of nonlocal interactions, the HW spin tensor turns out βμ tional Einstein’s general relativity, we allow the affine con- not to be conserved anymore [see Eq. (6.28)] and the situation nection to have an antisymmetric part of the form might be different. However, for a full understanding of this issue, further studies are needed. μ 1 μ F ≡ Γ˜ , (8.3) αβ 2 [αβ] which is called . If we constrain the affine con- 8 Einstein–Cartan theory nection in such a way that the vanishes (metric compatibility), i.e., impose local Minkowski struc- So far, we have studied different energy-momentum and spin ture, we can write the affine connection as tensors in flat spacetime. However, as in gravitational physics μ μ μ Γ˜ = Γ − K , (8.4) the energy-momentum tensor is directly measurable through αβ αβ αβ geometry, it is also interesting to review the role of the den- where sities in curved spacetime. In conventional general relativity μ 1 μν Γ = g (∂α gβν + ∂β gαν − ∂ν gαβ ) the energy-momentum tensor is defined following Belinfante αβ 2 123 Eur. Phys. J. A (2021) 57 :155 Page 19 of 25 155

ab is the conventional Christoffel symbol and where Rμν is the Riemann-Cartan tensor μ μ μ μ Kαβ =−Fαβ + Fβα− F αβ (8.5) ab = ∂ ω ab − ∂ ω ab + ω bω ac − ω bω ac. Rμν μ ν ν μ μc ν νc μ (8.11) is the contorsion tensor. Obviously, if torsion vanishes we The inclusion of torsion will also lead to a modification of recover the usual Riemann spacetime. While curvature can the field equations [116]. be regarded as a “rotation field strength” connected to the loss of parallelism of parallel transported vectors, torsion can be interpreted as a “translation field strength” which is manifest 8.3 Local Poincaré transformations and conservation laws in the closure failure of parallelograms [120]. The fundamental idea of the Einstein-Cartan theory is to 8.2 Tetrads and spinors in curved spacetime promote the global Poincaré symmetry of the action to a gauge symmetry [116–118]. This approach is analogous to the Yang-Mills formulation of gauge theory. Given the metric gμν defined on the manifold, we can always define a tangent space at each spacetime point and establish In order to give as an intuitive explanation as possible of μ the Einstein–Cartan theory, we will start by considering a flat a local flat orthonormal ea(x) = e a(x)∂μ which are called tetrads or vierbein. 7 Hence, their compo- Minkowski spacetime. In this case, the tetrads will simply be μ a nents e a and the reciprocal eμ are such that μ = δμ e a a (8.12) μ a = δμ, μ b = δb e a eν ν e a eμ a (8.6) and what is discussed in Sect. 2 holds, expect here we restrict and to classical fields. The global Poincaré transformations (2.4) and (2.9), which we write in a compact form a b gμν = eμ eν gab, (8.7) μ μ μ μν μ x → x = x + ζ xν + ξ , (8.13) where gab = (+1, −1, −1, −1) is the Minkowski metric. μν Spacetime indices are raised or lowered with g ,tetrad will induce a functional variation of the spinor indices with gab, and transvection is done by appropriate   1 contraction with the tetrads, e.g., for a vector Bμ, we define δψ = ψ( ) − ψ( ) = ζ ab − Ξ a∂ ψ( ), μ x x fab a x (8.14) Ba = e a Bμ. The intuitive picture is that we are assigning 2 at each spacetime point an observer which measures lengths with Ξ a = ξ a + ζ a δb xμ.UsingEq.(8.14) we obtain and time with respect to the local flat coordinate system e . b μ a through Noether’s theorem the canonical currents. Let us We now introduce a classical spinor ψ which will play the now promote the (4 + 6) infinitesimal parameters of the role of matter field. In the locally flat spacetime all the famil- Poincaré transformations to be functions of spacetime, ξ a(x) iar properties of the spinors hold, in particular they transform μ μ and ζ ab(x). If we now calculate the variation of the action under a Lorentz transformation of the tetrads e → Λ be a a b (2.2), δ A, with respect to these new local Poincaré transfor- as ψ → U(Λ)ψ with U −1γ aU = Λa γ b and all the conven- b mations and make use of the spinor variation in Eq. (8.14), tional relations of the Dirac γ -matrices apply. The covariant we obtain [116] derivative of a spinor is defined as      1 ab 4 1 ab μ a μ Dμψ ≡ ∂μ − ωμ fab ψ, (8.8) δ A = d x (∂μζ )J − (∂μξ )T 2 2 Cab Ca    =−i σ σ 4 1 ab μ a a b μ where fab 2 ab and ab is given by Eq. (2.11). The = d x (∂μζ )S − (∂μΞ − ζ δμ)T , ab ba Cab b Ca quantity ωμ =−ωμ is the 2   (8.15) ω b = 1 −Ω b + Ω b − Ωb − α b ν , μa μa a μ μa Kμν eα e a (8.9) 2 where we made use of the conservation laws for the canonical a a α with Ωμν ≡ ∂[μeν] and Kμν being the contorsion in currents (2.5) and (2.12). In order to make δ A vanish and μ Eq. (8.5). The commutator of the covariant derivatives is thus obtain local Poincaré invariance, we introduce e a(x) ab given by and ωμ (x) as gauge fields in the Lagrangian and couple them to the spinors such that 1 ab [Dμ, Dν]ψ =− R f ψ, (8.10) μν ab ∂L 2 μ a a a b ,δ ∂μΞ − ζ δ , a TCa eμ b μ ∂eμ 7 We use the Greek letters to denote the conventional holonomic space- ∂L (8.16) , , ... = , , , 1 μ ab ab time indices and the Latin letters a b 0 1 2 3 for the anholo- ,δω ∂μζ . ab SCab μ nomic tangent-space indices. ∂ωμ 2 123 155 Page 20 of 25 Eur. Phys. J. A (2021) 57 :155

a The relations above are supposed to be valid only in the case respectively, where e = det(eμ ). We can ensure the validity of weak fields, as a coupling of this form will necessarily of Eqs. (8.22)–(8.25) by applying the so-called minimal cou- modify the canonical currents and their conservation laws pling when generalizing the special-relativistic Lagrangian to which have were used to obtain Eq. (8.15). From this dis- the Einstein–Cartan theory cussion we deduce an important result: if we demand local μ μ L(ψ, δ ∂μψ) → e L(ψ, e Dμψ). (8.26) Poincaré invariance, then we see that special relativity is a a not adequate anymore and a deformation of the flat space- We stress that in the Einstein–Cartan theory the currents μ ab time due to e a(x) and ωμ (x) is needed to compensate which arise by taking the variations with respect to the gauge a ab the change of the action due to the variation of the spinor fields eμ and ωμ reduce in flat spacetime to the canonical field. As a consequence, we also have to adjust the derivative currents. operator by replacing it with the covariant derivative. These We conclude this section by mentioning that it is in princi- new fields encode geometrical properties of the spacetime ple possible to generalize also quadratic Lagrangians (such and they indeed represent the gravitational interaction. Such as the squared Dirac Lagrangian) to curved spacetime and geometry turns out to be the Riemann–Cartan geometry. thus connect the HW tensors to the Einstein–Cartan theory We can now relax the assumption of weak gravitational [121,122]. field limit implied in the derivation above. In order to do so, let us assume that in general the Lagrangian density has the following functional dependence 9 Conclusions

a ab L = L[g ,γ ,γ ,ψ,∂μψ, e ,ω ], (8.17) ab a 5 μ μ The relativistic decomposition of the total angular momen- where the Minkowski metric gab and the Dirac matrices tum is an old problem which embraces many branches of γa,γ5 are constant and defined in the local orthonormal physics. In this review we focused on some formal aspects frame. For simplicity in the Lagrangian (8.17)weomitto and applications which are decades old and on some oth- write the dependence on the adjoint spinor field. In order ers which have recently attracted considerable attention. We for the action corresponding to the Lagrangian (8.17)tobe showed in different contexts which choice appears to be a invariant under local Poincaré transformations, the following physical one. In particular, we reviewed some of the latest condition has to hold [116–118]: results regarding the description of spin dynamics in relativis-   tic fluids in relation to the physics of the QGP in heavy-ion δL ∂L μ collisions. In this case, the HW pseudo-gauge has important δQ + Dμ δQ + Ξ L = 0, (8.18) δQ ∂(∂μ Q) advantages, namely the spin tensor is not conserved only when nonlocal particle scatterings are considered, and orbital a ab angular momentum can be converted into spin through the where Q = (ψ, eμ ,ωμ ). It is possible to prove that the general variations of the spinor and gauge fields read antisymmetric part of the energy-momentum tensor. Eventu- ally, when global equilibrium is reached, the HW spin tensor   is conserved and no orbital-to-spin angular momentum con- 1 ab a δψ = ζ fab − Ξ Da ψ, (8.19) 2 version can occur. Moreover, we discussed how different def- a a a b b a initions of the relativistic center of inertia are related to the δe = DμΞ − ζ e +Ξ F , (8.20) μ b μ bμ pseudo-gauge transformations. In this context, we showed δω ab = ζ ab+Ξ c ab, μ Dμ Rcμ (8.21) that, unlike in the massless case, for massive particles it is always possible to define the spin in a covariant way using (cf. with Eqs. (8.14) and (8.16)). From Eq. (8.18), requir- the HW, GLW or KG global spin (at least for free fields or ing that the functions multiplying the independent quantities in the case of local interactions). The fact that the defini- DμΞ a, Dμζ ab, Ξ a,ζab vanish, after using the equations of tion of the spin of a massless particle is inherently nonco- motion for ψ we obtain variant leads to the side-jump effect. Finally, we discussed the Einstein-Cartan theory, an extension of general relativ- ∂L ∂L μ = = ψ − μ L, ity which allows a physical definition through geometry of eTCa a Da e a (8.22) ∂eμ ∂(∂μψ) an asymmetric energy-momentum tensor and a spin tensor ∂L ∂L which reduce to the conventional canonical currents in flat μ = = ψ, eSCab 2 ab fab (8.23) spacetime. ∂ωμ ∂(∂μψ) The ultimate question one would like to address is how μ μ 1 μ ( ) = b + bc , to extract information from experiments on which pseudo- Dμ eTCa Faμ eTCb Raμ eSCbc (8.24) 2 gauge describes the system and whether it is unique. A pos- ( μ ) = , Dμ eSCab eTC [ba] (8.25) sible answer can be found exploiting the fact that, in gen- 123 Eur. Phys. J. A (2021) 57 :155 Page 21 of 25 155 eral, the expectation value of an observable onto a state for ful collaborations. We are particularly grateful to A. Sadofyev also which local equilibrium can be defined, does depend on the for extremely valuable comments on the manuscript. This work was pseudo-gauge. Furthermore, in relativistic spin hydrodynam- supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through the Collaborative Research Center CRC- ics the values of the fields may be different with respect to TR 211 “Strong-interaction matter under extreme conditions”—project which pseudo-gauge one uses to decompose the total angular number 315477589 - TRR 211. E.S. acknowledges support also by the momentum. In particular, in the Belinfante case, one does not COST Action CA15213 “Theory of hot matter and relativistic heavy- need to introduce the spin potential as an additional dynam- ion collisions (THOR)”. ical field and the spin dynamics may be different than in Funding Information Open Access funding enabled and organized by other pseudo-gauges. On the other hand, in the context of Projekt DEAL. gravitational physics, the way one is supposed to measure Data Availability Statement This manuscript has associated data in a energy, momentum and spin densities is through spacetime data repository. [Authors’ comment: There is no data because this is a geometry. purely analytic work.] One may expect that the formalism and the problems cov- ered in the present review will be relevant in the near future Open Access This article is licensed under a Creative Commons Attri- bution 4.0 International License, which permits use, sharing, adaptation, since they are shared in different fields, some of which are and distribution and reproduction in any medium or format, as long as you will be under active experimental investigation. In heavy-ion give appropriate credit to the original author(s) and the source, pro- collisions the development of dissipative spin hydrodynam- vide a link to the Creative Commons licence, and indicate if changes ics and quantum kinetic theory appears to be an urgent task were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indi- in order to understand the nontrivial dynamics of polariza- cated otherwise in a credit line to the material. If material is not tion, especially in light of the recent and future experimental included in the article’s Creative Commons licence and your intended program [50]. Some important questions one would like to use is not permitted by statutory regulation or exceeds the permit- address are whether spin can equilibrate fast enough for time ted use, you will need to obtain permission directly from the copy- right holder. To view a copy of this licence, visit http://creativecomm scales relevant to nuclear collisions and how this nonequilib- ons.org/licenses/by/4.0/. rium dynamics can modify the expression for spin polariza- tion commonly used to describe the Lambda global polariza- tion data. A Lorentz transformation properties of Understanding how angular momentum can be split into hyper-surface-integrated quantities an orbital and spin part is also crucial for the description of gauge fields. In addition to the complications discussed In this appendix we show that, given a generic rank-(n + 1) in this paper, there is the question of whether it is possible λμ ···μ tensor Bˆ 1 n (x), the quantity defined as to find a gauge-invariant way to decompose the total angu-  lar momentum. A fundamental description of the spin and ˆμ ···μ ˆ λμ ···μ B 1 n = dΣλ B 1 n (x) (A.1) orbital angular momentum of light is still controversial, see, Σ e.g., works related to optics in Refs. [123,124]. Recently, λμ ···μ a gauge-invariant measure of the spin of the photon called transforms as a rank-n tensor only if Bˆ 1 n (x) is a con- ˆ λμ ···μ zilch current has been studied also in the context of quan- served quantity, i.e., ∂λ B 1 n (x) = 0, and suitable bound- tum kinetic theory and nuclear collisions [94,125]. Further- ary conditions are fulfilled. This also means that the quantity more, in hadron physics, the angular momentum decompo- in Eq. (A.1) is independent of the choice of the hypersurface sition is of utmost importance to understand the contribution integration. The correct properties under Lorentz transfor- of quarks and gluons to the spin of the nucleon [126]. For mations are not in general guaranteed because, in the inte- μ ···μ related works about the angular momentum decomposition grated quantity Bˆ 1 n , only the fields in the integrand have with a focus on chiral physics, see Refs. [127,128] in which to be transformed and the integration hypersurface must be connections to nuclear collisions are also discussed. Address- unchanged. In other words, the experimenter is not trans- ing the problem of the nucleon spin will also be at the core formed. This is the point of view of an active transformation of the experimental effort of the future electron–ion collider which we will adopt in the proof below. We could instead per- [129,130]. Finally, we would like to mention applications form a passive transformation, namely change the observer, of spin dynamics in cosmology, where the Einstein-Cartan i.e., the hypersurface, and leave the fields unchanged. The theory is often taken as a starting point [131,132], and in proof is independent of whether we do an active or passive condensed matter systems like, e.g., spintronics [133]. transformation. Since Eq. (A.1) is expressed in a covariant way in terms of a scalar product, if we changed both the fields Acknowledgements We thank F. Becattini, W. Florkowski, D. H. and the hypersurface together, it would look like it has the Rischke, R. Ryblewski, A. Sadofyev, X.-L. Sheng, L. Tinti, G. Tor- proper transformation behavior. However, this is not what rieri, D. Wagner and Q. Wang for enlightening discussions and fruit- one would demand from a Lorentz transformation, as either 123 155 Page 22 of 25 Eur. Phys. J. A (2021) 57 :155

 μ ···μ the fields or the observer should be changed. To show this, Bˆ 1 n transforms like a rank-n tensor, we must also have we closely follow the proof given in Ref. [134] and we stress  that it is valid for both quantum and classical fields. μ μ ˆσ ···σ μ μ 4 Λσ 1 ···Λσ n B 1 n = Λσ 1 ···Λσ n d x δ(N · x) Let us choose a hypersurface at x0 = 0, hence dΣλ = 1 n 1 n 4 δ( · ) = ( , ) ˆ ρσ ···σ d x N x Nλ, where Nλ 1 0 . Thus, Eq. (A.1) becomes × Nρ B 1 n (x). (A.7)  μ ···μ μ ···μ Bˆ 1 n = d3x Bˆ 0 1 n (x) Therefore, to conclude the proof, we only need to find the  conditions for which Eqs. (A.6) and (A.7) are equal, which is 4 ˆ λμ ···μ the same as requiring that the following difference vanishes: = d x δ(N · x)Nλ B 1 n (x)  ˆμ ···μ −1 μ −1 μ ˆ  σ ···σ 4 ˆ λμ ···μ B 1 n − (Λ )σ 1 ···(Λ )σ n B 1 n = d x [∂λθ(N · x)]B 1 n (x), (A.2)  1 n 4  ˆ ρμ ···μ = d x ∂ρ[θ(N · x) − θ(N · x)]B 1 n (x) where we introduced the step function such that θ(N ·x) = 1  4  ˆ ρμ ···μ for N · x ≥ 0 and θ(N · x) = 0forN · x < 0 and used the =− d x[θ(N · x)−θ(N · x)]∂ρ B 1 n (x), (A.8) relation where we used the relation (A.3) and, in the last step, we Nλδ(N · x) = ∂λθ(N · x). (A.3) integrated by parts and safely neglected boundary terms in Note that in Eq. (A.2) we clearly cannot neglect boundary the spatial as well as in the temporal direction, since in the θ( · ) = θ(  · ) terms in the temporal direction after integrating by parts, far future or far past we have N x N x . Finally, as θ(N · x) = 1 when x0 →+∞. We now study how Eq. (A.8) vanishes only if ˆμ ···μ μ B 1 n transforms under a Lorentz transformation Λν .As ˆ λμ ···μ ∂λ B 1 n (x) = 0, (A.9) previously discussed, we perform a transformation of the ˆ λμ ···μ fields under the integral. Since B 1 n (x) is a tensor, it which is what we wanted to proof. Note that what we also will transform as showed is that the integral in Eq. (A.2) is independent of  λμ ···μ λ μ μ ρσ ···σ − the choice of the hypersurface using the divergence theo- Bˆ 1 n (x) = Λ Λ 1 ···Λ n Bˆ 1 n (Λ 1x). (A.4) ρ σ1 σn rem, provided that Eq. (A.9) holds and suitable boundary conditions are fulfilled. In fact, the second line of Eq. (A.8) Therefore, the act of the Lorentz transformation on Eq. (A.2) ρμ ···μ is the difference of the integral of Bˆ 1 n (x) calculated is given by at two different hypersurfaces with normal vectors Nμ and   Nμ, respectively, which is transformed to a volume integral ˆ  μ ···μ 4 λ μ μ ˆ ρσ ···σ −1 B 1 n = d x δ(N · x)NλΛ Λ 1 ···Λ n B 1 n (Λ x). ρ σ1 σn in the last line. What discussed in this section is clearly not restricted to the form of the hypersurface chosen for the proof (A.5) and can be easily generalized to a generic shape. Consider a region of spacetime enclosed between two space-like hyper- We now change the integration variable x = Λ−1x and surfaces Σ , Σ corresponding to two different values of the define N  = Λ−1 N so that N · x = N  · x. Hence, Eq. (A.5) 1 2 parameter t used for the foliation of the spacetime, t , t , becomes 1 2 respectively (t can be, e.g., x0). Using the divergence theo-  rem and the fact that we can neglect terms at the boundaries, ˆ  μ ···μ 4   α λ μ μ ˆ ρσ ···σ  B 1 n = d x Λλα N ΛρΛσ 1 ···Λσ n B 1 n (x ) we have 1 n     λμ ···μ λμ ···μ × δ(N · x ) Σ ˆ 1 n − Σ ˆ 1 n  d λ B d λ B Σ1 Σ2 μ μ 4     ˆ ρσ ···σ   = Λσ 1 ···Λσ n d x δ(N · x )Nρ B 1 n (x ) 1 n ˆ λμ ···μ  = dV∂λ B 1 n , (A.10) V μ μ 4   ˆ ρσ ···σ = Λσ 1 ···Λσ n d x δ(N · x)Nρ B 1 n (x), 1 n where V is the four-dimensional volume. The right-hand side (A.6) of the equation above vanishes if (A.9)isvalid. We conclude that, since the energy-momentum and total where in the first equality we made use of the invariance of angular momentum tensors are conserved quantities, the 4 λ d x under the transformation Λ, in the second one ΛλαΛρ = associated charges in Eqs. (2.7) and (2.16) transform prop- gαρ , and in the last one we only relabeled the integration erly as tensors under Lorentz transformations. On the other variable by dropping the prime index. On the other hand, if hand, since the canonical spin tensor is not conserved [see 123 Eur. Phys. J. A (2021) 57 :155 Page 23 of 25 155

†  i (k−k)·x   †  Eq. (2.15)], the associated global spin in Eq. (2.17) will not × u  (k )ur (k)e h¯ p s | a  (k )ar (k) |p, s r  r transform as a tensor. 3 3  1 μναβ 3 d kd k †  =−  pν d x xαkβ u  (k )u (k) ( π ¯ )3  r r 2m 2 h k0k0 i (k−k)·x  ( ) ( )   × e h¯ p p δ 3 (k − p)δ 3 (k − p )δ δ   B Matrix element of the Pauli–Lubanski vector 0 0 rs r s = 0. (B.7) In the first part of this appendix we show the details of the derivation of Eq. (3.20). We first introduce the plane-wave In deriving the result above we made use of Eq. (B.1), the expansion of the Dirac field antisymmetry of the Levi-Civita tensor together with the rela- tion  2 3 1 d k − i · ψ( ) =  ( ) h¯ k x ( )   †  x ur k e ar k p s | a  (k )a (k) |p, s 3 0 r r 2(2πh¯ ) = k r 1  †  † =0| as (p )a  (k )ar (k)a (p) |0 i k·x † r s +vr (k)e h¯ b (k) , (B.1)  ( ) ( )   r = p p δ 3 ( − p)δ 3 ( − p )δ δ   , 0 0 k k rs r s (B.8) where ur (k) and vr (k) are the standard spinors for parti- which follows from cles and antiparticles, respectively. Moreover, ar (k) is the † annihilation operator and br (k) is the creation operator of { ( ), †(p)}=p0δ( − p)δ ar k as k rs (B.9) respectively a particle and antiparticle state with momentum μ k and spin projection r. We start by proving that and ar (k) |0=0. Using Eq. (B.7), (B.6) simplifies to Eq. (B.2). p, s| Sˆμ |p, s Consider now the Pauli–Lubanski vector written in terms of the canonical and HW pseudo-gauge. Using Eqs. (2.6) and 1 μναβ   ˆ ˆ =−  p , s | Pν Jαβ |p, s (2.28a) and following similar steps which lead to Eq. (B.7), 2m  one can prove that the orbital parts cancel when taking 1 μναβ 3   † h¯ =−  pν d x p , s | ψ (x) σαβ ψ(x) |p, s , 2m 2 the matrix element of one-particle states. For the canonical (B.2) global spin contribution, from Eq. (2.14)wehave where Sˆi0 = 0, C  ¯ |p, = †(p) |  . ˆij 3 † h ij s as 0 (B.3) S = d x ψ (x) σ ψ(x), (B.10) C 2 is the one-particle state of momentum p and spin projection s (for antiparticle states the proof would follow similar steps). where we performed the spacetime integration at constant 0 μ μ It is convenient to note that x .Usingu¯r (k)γ ur (k) = 2k δrr , one can prove that the μναβ ˆ ˆ μναβ ˆ ˆ contribution to Eq. (B.2) with α = l, β = 0 vanishes (l =  Pν Jαβ =  Jαβ Pν, (B.4) 1, 2, 3), which in turn implies since the total charges obey the Poincaré algebra   ˆμ 1 μναβ   ˆ ˆ ˆ ˆ ˆ ˆ p , s | S |p, s=−  p , s | Pν SC,αβ |p, s . [Pν, Jαβ ]=i(gβν Pα − gαν Pβ ). (B.5) 2m (B.11) Using Eqs. (2.16), (B.4) and Pˆ μ |p, s=pμ |p, s, we get Furthermore, following similar steps as in Eq. (B.7), one can μ p, s| Sˆ |p, s show that the contribution to the Pauli–Lubanski vector given  by the expectation value of the second term in the last line =− 1 μναβ p 3 p, | ψ† ν d x s of Eq. (2.36) cancels when taking the matrix element. Thus, 2m  h¯ we also have × 2ihx¯ α∂β + σαβ ψ |p, s . (B.6) 2   ˆμ 1 μναβ   ˆ ˆ p , s | S |p, s−  p , s | Pν SHW,αβ |p, s . We now calculate the orbital part 2m (B.12)  ih¯ μναβ 3   † −  pν d x p , s | ψ (x)xα∂β ψ(x) |p, s m We conclude the appendix by proving Eq. (4.29). Using   3 3  Eq. (4.4) and plugging Eq. (B.1) into Eq. (4.1), we obtain 1 μναβ 3 d kd k =−  pν d x xαkβ ( π ¯ )3  [18,19] 2m 2 h k0k0 123 155 Page 24 of 25 Eur. Phys. J. A (2021) 57 :155  λ ˆ dΣλ p Wκχ(x, p) 31. F. Becattini, L. Bucciantini, E. Grossi, L. Tinti, Eur. Phys. J. C 75,  191 (2015). arXiv:1403.6265 32. M. Hongo, Ann. 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