Particle polarization, spin tensor and the Wigner distribution in relativistic systems
aa r X i v : . [ nu c l - t h ] A ug Particle polarization, spin tensor and the Wignerdistribution in relativistic systems
Leonardo Tinti and Wojciech Florkowski
Abstract
Particle spin polarization is known to be linked both to rotation (angularmomentum) and magnetization of a many particle system. However, in the mostcommon formulation of relativistic kinetic theory, the spin degrees of freedomappear only as degeneracy factors multiplying phase-space distributions. Thus, it isimportant to develop theoretical tools that allow to make predictions regarding thespin polarization of particles, which can be directly confronted with experimentaldata. Herein, we discuss a link between the relativistic spin tensor and particle spinpolarization, and elucidate the connections between the Wigner function and averagepolarization. Our results may be useful for theoretical interpretation of heavy-iondata on spin polarization of the produced hadrons.
In relativistic heavy-ion collisions the produced matter is formed at extreme con-ditions of high temperature and density [1]. The exact details of the evolution ofthe resulting fireball are difficult to track, however, some generally accepted con-cepts are typically assumed now. In particular, the evidence has been found thatthe strongly interacting matter behaves as an almost perfect (low viscosity) fluid(for recent reviews see, for example, Refs. [2, 3]). During its expansion, when thesystem is diluted enough, the matter undergoes a cross-over phase transition from astrongly interacting quark-gluon plasma to a hadron gas (for systems with negligible
Leonardo TintiInstitut für Theoretische Physik, Johann Wolfgang Goethe-Universität, Max-von-Laue-Str. 1, D-60438 Frankfurt am Main, Germanye-mail: [email protected]
Wojciech FlorkowskiJagiellonian University, ul. prof. St. ÅĄojasiewicza 11, 30-348 KrakÃşw, Polande-mail: [email protected] baryon number density [4, 5]). Shortly thereafter, the system becomes too dilute tobe properly treated as a fluid, the interaction effectively ends, and produced particlesfreely stream to the detectors (freeze-out).The main purpose of relativistic hydrodynamics is to solve the four-momentumconservation equation, ∂ µ T µν =
0, and the (baryon) charge conservation equation, ∂ µ J µ =
0, under some realistic approximations (i.e., with suitably chosen initialconditions given by the Glauber model [6] or the theory of color glass condensate [7],and with a realistic equation of state obtained from an interpolation between thelattice QCD simulations and hadron resonance gas calculations).In this way one obtains the four-momentum and charge fluxes at freeze-out. Theseare space-time densities, which are not directly connected with the momentum dis-tributions (and polarization) measured by different experiments. The most commonway to link the stress energy tensor and the charge flux at the freeze-out to parti-cle spectra makes use of a classical intuition [8]; namely, one assumes that afterthe freeze-out the system is described well enough by the distribution functions f ( x , p ) for (non interacting) particles and the corresponding functions ¯ f ( x , p ) forantiparticles. Matching the stress energy tensor and charge current with the corre-sponding formulas from the relativistic kinetic theory, one can guess the form of thedistribution functions, and from that predict the final spectra.The purpose of this contribution is to extend this formalism to include particle spinpolarization (for particles with spin 1/2). In the next section we show the limitations ofthe traditional kinetic theory in relation to particle polarization degrees of freedom.In Sec. 3 we introduce the concept of the relativistic spin tensor and discuss itsrelation to particle polarization for a free Fermi-field. In Sec. 4 we show that theappropriate generalization of the distribution function is the Wigner distribution. Wesummarize and conclude in Sec. 5. Some useful expressions and transformationsare given in Appendix A. For complementary information we refer to the recentreviews [10, 11, 12, 13]. In the classical relativistic kinetic theory the charge current density J µ and the stressenergy tensor T µν have a rather simple connection with the phase-space distributionsof particles and antiparticles [9, 14], J µ = g S ( π ) ∫ d pE p p µ (cid:16) f ( x , p ) − ¯ f ( x , p ) (cid:17) , T µν = g S ( π ) ∫ d pE p p µ p ν (cid:16) f ( x , p ) + ¯ f ( x , p ) (cid:17) . (1)Here g S = S + S being the spin of (massive)particles. In order to properly take into account the polarizaton degrees of freedom,one can easily notice that the framework based on Eqs. (1) should be extended to a elation between particle polarization and the spin tensor 3 matrix formalism. This is so, since the standard kinetic theory essentially assumesequipartition of various spin states.In general, the expectation values of the charge current and the stress energytensor, determined in a generic state of the system described by the density matrix ρ ,cannot be expressed by the integrals of the form (1) with the integrands dependingon a single momentum variable. A generic density matrix can be written as ρ = Õ i P i | ψ i i h ψ i | , (2)where P i are classical (non-interfering) probabilities normalized to one, Í i P i = | ψ i i are generic quantum states. We assume that h ψ i | ψ i i = | ψ i i ’s are not necessarily eigenstates of the total energy, linear momentum, angularmomentum, or charge operators.Starting from the definition of the charge current operatorˆ J µ ( x ) = ¯ Ψ ( x ) γ µ Ψ ( x ) , (3)expressed by the non-interacting Fermi fields Ψ , Ψ ( x ) = Õ r ∫ d p ( π ) p E p h U r ( p ) a r ( p ) e − ip · x + V r ( p ) b † r ( p ) e ip · x i , (4)we obtain the normal-ordered expectation value J µ ( x ) = h : ˆ J µ ( x ) : i = tr (cid:16) ρ : ˆ J µ ( x ) : (cid:17) == Õ r , s ∫ d pd p ′ ( π ) p E p E p ′ h h a † r ( p ) a s ( p ′ )i ¯ U r ( p ) γ µ U s ( p ′ ) e i ( p − p ′ )· x − h b † r ( p ) b s ( p ′ )i ¯ V s ( p ′ ) γ µ V r ( p ) e i ( p − p ′ )· x + h a † r ( p ) b † s ( p ′ )i ¯ U r ( p ) γ µ V s ( p ′ ) e i ( p + p ′ )· x + h b r ( p ) a s ( p ′ )i ¯ V r ( p ) γ µ U s ( p ′ ) e − i ( p + p ′ )· x i . (5)It is easy to check that the stress energy tensor has an analogous structure. In general,none of the expectation values of the creation-destruction operators vanishes and theintegrals over three-momenta cannot be reduced to the Dirac delta functions.We use the Wigner representation of the Clifford algebra and we adopt the con-vention for the massive eigenspinors Note that in the Weyl representation of the Clifford algebra a different explicit formula for themassive eigenspinors is typically used, but final results remain the same. Leonardo Tinti, Woiciech Florkowski U r ( p ) = p E p + m (cid:18) φ r σ · p E p + m φ r (cid:19) , V r ( p ) = p E p + m (cid:18) σ · p E p + m χ r χ r (cid:19) , (6)with the two component vectors φ r and χ r being the eigenstates of the matrix σ z = diag ( , − ) , φ = (cid:18) (cid:19) , φ = (cid:18) (cid:19) ,χ = (cid:18) (cid:19) , χ = − (cid:18) (cid:19) , (7)therefore the normalization of the states | p , r i and the anticommutation relationsbetween the creation-destruction operators read { a s ( q ) , a † r ( p )} = { b s ( q ) , b † r ( p )} = ( π ) δ rs δ ( p − q ) , { a s ( q ) , b r ( p )} = { a s ( q ) , b r ( p )} = { a s ( q ) , b r ( p )} = { a s ( q ) , b r ( p )} = , | p , r i = p E p a † r ( p )| i ⇒ h q , s | p , r i = ( π ) δ r , s δ ( p − q ) . (8)Differently from the current density, the total charge has a similar structure to theone used in the kinetic theory, ∫ d x J ( x ) = ∫ d p ( π ) "Õ r h a † r ( p ) a r ( p )i − Õ r h b † r ( p ) b r ( p )i (9)which directly comes from the normalization of the bispinors (6) U † r ( p ) U s ( p ) = V † r ( p ) V s ( p ) = E p δ rs , U † r ( p ) V s (− p ) = , (10)and the integral representation of the Dirac delta functions, δ ( ) ( p ± p ′ ) , used toperform the volume integrals. Following the same steps, one can compute the totalfour-momentum ∫ d x T µ ( x ) = ∫ d p ( π ) p µ "Õ r h a † r ( p ) a r ( p )i + Õ r h b † r ( p ) b r ( p )i . (11)Equations (9) and (11) should be compared to the analogous formulas obtained fromthe kinetic-theory definitions (1), elation between particle polarization and the spin tensor 5 ∫ d x J = ∫ d p ( π ) (cid:20) g s ∫ d x f ( x , p ) − g s ∫ d x ¯ f ( x , p ) (cid:21) , ∫ d x T µ = ∫ d p ( π ) p µ (cid:20) g s ∫ d x f ( x , p ) + g s ∫ d x ¯ f ( x , p ) (cid:21) . (12)It is important to note that for non-interacting, spin / particles the volumeintegrals are time independent. The collisionless Boltzmann equation for particlesreads ∂ µ ( p µ f ( x , p )) = , (13)hence, the expression p µ f is a conserved vector current (of course, the same propertyholds also for the non-interacting antiparticles described by the function ¯ f ( x , p ) ).The integral of the divergence (13) over a space time-region vanishes. Therefore,as long as the spatial boundary for the volume integral lies outside of the region inwhich the distribution function does not vanish, the integral E p ∫ d x f remains thesame at all times. Massive particles always have a positive E p , therefore, the volumeintegral itself is also time-independent .Moreover, the space integral equals the integral over the freeze-out hyper-surface [8], with d Σ µ being the generic hyper-surface element (note that d Σ µ = ( d x , , , ) for the volume integrals in the lab frame) g s ∫ d Σ µ p µ f ( x , p ) = g s E p ∫ d x f ( x , p ) = ( π ) E p dNd p . (14)The last term here is the invariant number of particles per momentum cell, which isconsistent with the formula for the total number of particles N = ∫ d x ∫ d p ( π ) g s f ( x , p ) = ∫ d x ∫ d p ( π ) E p E p g s f ( x , p ) . (15)We note that the factor ( π ) (to be replaced by ( π ~ ) if the natural units are notused) is included in the momentum integration measure rather than in the definitionof the phase-space distributions, and ∫ d p / E p is the Lorentz-covariant momentumintegral.In the general case, it can be proved that the expectation value of the numberoperator is a non-negative quantity and the sum over the spin states is proportionalto the (anti)particle number density in momentum space, hence, N = ∫ d p ( π ) Õ r h a † r ( p ) a r ( p )i (16)for particles, and ¯ N = ∫ d p ( π ) Õ r h b † r ( p ) b r ( p )i (17) Massless particles have a positive energy for any non vanishing momentum and the situation forthem is quite similar. Leonardo Tinti, Woiciech Florkowski for antiparticles. For more information see Appendix A.Consequently, even if the expectation values of the charge current and the stress-energy tensor cannot be written as momentum integrals of the phase-space distri-butions, it is possible to have consistent distribution functions for the particles andantiparticles in the sense that they can reproduce the correct invariant momentum(anti)particle densities, g s ∫ d x f ( x , p ) = Õ r h a † r ( p ) a r ( p )i , g s ∫ d x ¯ f ( x , p ) = Õ r h b † r ( p ) b r ( p )i . (18)For heavy-ion collisions, this implies that for any given J µ and T µν at freeze-out,one can construct a pair of the distribution functions ( f , ¯ f ) that provide the sametotal current, energy, and linear momentum, ∫ d p ( π ) (cid:20) g s ∫ d x f ( x , p ) − g s ∫ d x ¯ f ( x , p ) (cid:21) == ∫ d p ( π ) "Õ r h a † r ( p ) a r ( p )i − Õ r h b † r ( p ) b r ( p )i , (19) ∫ d p ( π ) p µ (cid:20) g s ∫ d x f ( x , p ) + g s ∫ d x ¯ f ( x , p ) (cid:21) == ∫ d p ( π ) p µ "Õ r h a † r ( p ) a r ( p )i + Õ r h b † r ( p ) b r ( p )i . (20)We note that the distributions functions obtained from the conditions (19) and (20)provide the correct total charge and four-momentum if used in Eqs. (1) to define thecurrent density and the energy-momentum tensor. However, very different distribu-tion functions may provide (after integration) the same macroscopic quantities. Incertain cases, to remove such an ambiguity, one can use additional physical insights.For example, the specific forms of the distribution functions can be introduced forsystems being in (local) thermodynamic equilibrium or close to such a state.In any case, it is important to note that the total charge is sensitive to the imbalancebetween particles and antiparticles, and the total four momentum is sensitive to themomentum distribution of both particles and antiparticles. Therefore, the conditions(19) and (20) provide an important constraint on the distribution functions. However,the right-hand sides in (19) and (20) are sums over the spin states. Being insensitiveto polarization, they are not useful to check if a given extension of kinetic theoryreproduces the average polarization in a satisfactory manner.In the next section, we will argue that the relativistic spin tensor is sensitiveto particle polarization in a very similar way as the charge current is sensitiveto particle-antiparticle imbalance and the stress-energy tensor controls the averageparticle momentum. elation between particle polarization and the spin tensor 7 In this section we introduce and discuss in more detail one of the main objectsof our interest, namely, the relativistic spin tensor. Although, it is less well knowncompared to the tensors analyzed in the previous section, we are going to demonstratethat its intuitive understanding as a quantity related to particle’s polarization is indeedcorrect.The Noether theorem links the symmetries of the action to conserved chargesand, to a lesser extent, conserved currents. If the action A contains only first orderderivatives of the fields φ a ( x ) , we can write [15] A[ φ a ] = ∫ d x L( φ, ∂ µ φ, x ) , (21)where L is the Lagrangian density. If the action is invariant with respect to aninfinitesimal transformation x µ → ξ µ = x µ + ǫδ x µ ,φ a ( x ) → α a ( ξ ) = φ a ( x ) + ǫδφ a ( x ) + ǫδ x µ ∂ µ φ a ( x ) , (22)one can extract a conserved current Q µ = (cid:20) ∂ L ∂ ( ∂ µ φ a ) ∂ ν φ a − L δ µν (cid:21) δ x ν − ∂ L ∂ ( ∂ µ φ a ) (cid:16) δφ a ( x ) + δ x ν ∂ ν φ a ( x ) (cid:17) ,∂ µ Q µ = , (23)where the summation over repeated indices is understood. Because of the vanishingdivergence, the integral over a space-time region of (23) vanishes. Hence, if the fieldflux at the space boundary vanishes , the space integral of Q is a constant of motion.For instance, considering the action of a free, massive, spin / spinor field Ψ , A = ∫ d x (cid:20) i Ψ ( x ) γ µ ↔ ∂ µ Ψ ( x ) − m ¯ Ψ ( x ) Ψ ( x ) (cid:21) , (24)one has the internal symmetry under a global phase change of the fields with: δ x µ ≡ δ Ψ = i ǫ Ψ , and δ ¯ Ψ = − i ǫ ¯ Ψ . The corresponding current in this case is thecharge current J µ , J µ ( x ) = ¯ Ψ ( x ) γ µ Ψ ( x ) . (25)The invariance under space-time translations (with δ x µ being a constant and α ( ξ ) − φ ( x ) = − δ x µ ∂ µ φ ) yields the canonical stress-energy tensor T µν c as the conservedcurrent. The conserved charge in this case is the total four-momentum of the system The space-time region might be finite, with the fields going to zero at the boundary or infinite, aslong as the fields decay fast enough to have a vanishing flux at infinity. Leonardo Tinti, Woiciech Florkowski T µν c ( x ) = i Ψ ( x ) γ µ ↔ ∂ ν Ψ ( x ) − g µν L ≡ i Ψ ( x ) γ µ ↔ ∂ ν Ψ ( x ) . (26)In the last passage, we have made use of the equations of motion of the fields.Finally, we consider the invariance under the Lorentz group, i.e., boosts androtations. The representation of the Lorentz group is the source of spin in quantumfield theory, it is then expected that the conserved currents in this case are sensitiveto spin polarization. In general, for an infinitesimal Lorentz transformation one has δ x µ = ω µν x ν with constant ω µν = − ω νµ . The fields will change, according to theirrepresentation, following the rule δφ a + δ x µ ∂ µ φ a = − i / ω µν ( Σ µν ) ab φ b . One obtainsthen the conserved angular-momentum flux density, M λ,µν c = x µ T λν c − x ν T λµ c − i ∂ L ∂ ( ∂ λ φ a ) ( Σ µν ) ab φ b . (27)We note that the comma between the first index and the last two is used to emphasizethe fact that M λ,µν c = − M λ,νµ c (different orders of the indices and conventions areused by different authors).The first two terms in (27) depend on the canonical stress-energy tensor alreadyobtained from the space-time translational invariance of the action and representthe orbital part of the angular momentum. The last term in (27) defines the spincontribution to the angular momentum and is called a canonical spin tensor. With Σ µν = i h γ µ , γ ν i , (28)the canonical spin tensor reads S λ,µν c ( x ) = i Ψ ( x ) n γ λ , h γ µ , γ ν i o Ψ ( x ) . (29)Using the anticommutation relations { γ µ , γ ν } = g µν , it is straightforward to checkthat S λ,µν c is totally antisymmetric under the exchange of any indices.Differently from the total charges (i.e., quantities obtained by the volume inte-grals), the conserved density currents given by the Noether theorem are not uniquelydefined. Whatever the conserved current Q µ is originally derived, if one buildsfrom the fields a tensor C αµ = −C µα , called a superpotential, the new current Q ′ µ = Q µ + ∂ α C αµ is equally conserved and provides the same conserved totalcharge. In the particular case of the angular momentum flux and the stress-energytensor there is a class of well-known transformations that leave the conserved to-tal charges invariant (i.e., the generators of the Poincaré group). They are calledpseudo-gauge transformations and have the form [19] T ′ µν = T µν + ∂ λ (cid:16) G λ,µν − G µ,λν − G ν,λµ (cid:17) , S ′ λ,µν = S λ,µν − G λ,µν − ∂ α Ξ αλ,µν . (30) elation between particle polarization and the spin tensor 9 The tensors T µν and S λ,µν on the right-hand side of Eq. (30) can be either thecanonical ones or the already transformed ones. The auxiliary tensor G λ,µν must beantisymmetric in the last two indices, while Ξ αλ,µν should be antisymmetric in boththe first two and the last two indices.For any pair of the stress-energy and spin tensors, the following relations hold ∂ µ T µν = ,∂ λ S λ,µν = − (cid:16) T µν − T νµ (cid:17) , (31)and the total four-momentum P µ and angular momentum J µν read P µ = ∫ d x T µ , J µν = ∫ d x (cid:16) x µ T ν − x ν T µ + S µν (cid:17) . (32)By construction the last integrals are equal to those obtained with the canonical ten-sors, therefore Eqs. (31) can be equally well considered as the local four-momentumand angular momentum conservation equations. In this work we are not going todiscuss which pair is the most appropriate or convenient to represent the physicaldensities of the (angular) momentum of a physical system. This point is reviewed inRef. [20].A very special case of the transformations defined by Eqs. (30) is the Belinfantesymmetrization procedure. In this case, one starts with the canonical tensors T µν c and S λ,µν c , and takes G λ,µν = S λ,µν c and Ξ αλ,µν =
0. As a result, one obtains a vanishingnew spin tensor S λ,µν B =
0, and the angular momentum conservation becomes justthe requirement that the antisymmetric part of T µν B vanishes. There is an apparentparadox here, namely, that one starts with ten independent equations for 16+24=40degrees of freedom in (31) and ends up with only four equations for 10 degrees offreedom; the vanishing divergence of a symmetric rank two tensor.It is possible to resolve this paradox by writing the result of the Belinfantesymmetrization in a less deceitful way, namely ∂ µ T { µν } B = , T [ µν ] B = ⇒ ∂ λ S λµν c = − (cid:16) T µν c − T νµ c (cid:17) , P µ = ∫ d x T { µ } B , J µν = ∫ d x (cid:16) x µ T { ν } B − x ν T { µ } B (cid:17) . (33)The middle line of Eqs. (33) emphasizes an important point — although the symmet-ric part T { µν } B of the Belinfante symmetrized stress-energy tensor T µν B is separately This is so in the case of an arbitrary original spin tensor which is antisymmetric only in the lasttwo indices. For the canonical spin tensor that is totally antisymmetric, the number of independentcomponents is 16+4=20.0 Leonardo Tinti, Woiciech Florkowski conserved and both the total four-momentum and angular momentum can be ex-pressed through T { µν } B , the requirement that the antisymmetric part of T [ µν ] B vanishesshould be treated as a complementary set of equations. Indeed, starting with thecanonical tensors obtained for the Dirac field (26) and (29), and performing theBelinfante symmetrization, one obtains T µν B = i Ψ ( x ) γ µ ↔ ∂ ν Ψ ( x ) − i ∂ λ (cid:16) Ψ ( x ) n γ λ , h γ µ , γ ν i o Ψ ( x ) (cid:17) , (34)a formula which is not manifestly symmetric under a µ ↔ ν exchange. In order toshow that (34) is indeed symmetric, one has two options: i) Solve exactly the Euler-Lagrange equations of motion for the fields (possiblefor a free field) and directly check the symmetry of (34). ii)
Make use of the angular momentum conservation for the canonical tensors,accepting them as another set of equations.Consequently, although one needs a rank two, symmetric, and conserved tensor inorder to make a comparison with kinetic theory (since the stress-energy tensor inkinetic theory is symmetric by construction, see Eq (1)), one can always considerthe equation ∂ λ S λµν c = − ∂ λ S λµν c = − (cid:16) T µν c − T νµ c (cid:17) , (35)which remains valid. We note that the equations for the fields are usually far frombeing trivial and the same property holds for the symmetrization procedure that startsfrom some generic T µν and non-vanishing S λ,µν . For instance, a different Lagrangiandensity having the same Euler-Lagrange equation of motion for the fields, generallylead to different canonical tensors. In any case all of these tensors lead to the sameconserved charges and provide the same number of equations. For a modern andmore detailed discussion over the different possible choices of T µν and S λ,µν , andtheir physical consequences we refer to Refs. [21, 22].If one excludes the particular case of quantum anomalies , very similar argumentsto those presented above hold also in the quantum case for the operators built fromfundamental fields. In particular, the canonical spin tensor in the quantum case stillreads as in Eq. (29), with the only addition that one has to renormalize it (make normalordering) to avoid infinities related to the vacuum. The macroscopic, classical, spintensor is the expectation value of the quantum counterpart. For a generic (pure ormixed) state of the system ρ we have S λµν c = tr (cid:16) ρ : ˆ S λµν c : (cid:17) = i (cid:16) ρ : ¯ Ψ ( x ) n γ λ , h γ µ , γ ν i o Ψ ( x ) : (cid:17) . (36)This form is probably the most intuitive guess for a macroscopic object embedding theparticle’s polarization degrees of freedom, because in the total angular momentumoperator Neither in a free theory nor in the standard model there are anomalies in the conservation lawsfor the four-momentum and angular momentum. However, one has to check on a case by case basisif this is so while dealing with a generic quantum field theory.elation between particle polarization and the spin tensor 11 ˆ J µν = ∫ d x Ψ † ( x ) (cid:18) i x µ ↔ ∂ ν − i x ν ↔ ∂ µ + i γ n γ , h γ µ , γ ν i o(cid:19) Ψ ( x ) , (37) ˆ S ,µν c describes the last term in the round brackets, depending on the gamma matrices.It is the only term that mixes the components of the spinor fields — the part stemmingfrom T µν c depends on the gradients, hence, can be interpreted as the relativistic QFTanalogue of x × p , the orbital angular momentum of classical particles.In general, similarly to the stress-energy tensor and the current, the macroscopicspin tensor (36) depends on both space-time coordinates and two momentum vari-ables. However, its volume integral can be written in terms of a single momentumvariable and the expectation values of creation/destruction operators, much like wehave seen in the previous section. The space integral of S ijc can be expected tobe related to the sum of the spin polarizations of particles. Defining the vector ofmatrices Σ i in the following way Σ i = i ε ijk [ γ j , γ k ] = (cid:18) σ i σ i (cid:19) , ∀ i ∈ { , , } (38)with σ i being the × Pauli matrices, one has ε ijk ∫ d x S jkc = Õ r , s ∫ d x ∫ d pd p ′ ( π ) p E p E p ′ h h a † r ( p ) a s ( p ′ )i U † r ( p ) Σ i U s ( p ′ ) e i ( p − p ′ )· x − h b † r ( p ) b s ( p ′ )i V † s ( p ′ ) Σ i V r ( p ) e i ( p − p ′ )· x + h a † r ( p ) b † s ( p ′ )i U † r ( p ) Σ i V s ( p ′ ) e i ( p + p ′ )· x + h b s ( p ) a r ( p ′ )i V † s ( p ) Σ i U r ( p ′ ) e − i ( p + p ′ )· x i (39)that can be rewritten as Õ r , s ∫ d p ( π ) E p h h a † r ( p ) a s ( p )i U † r ( p ) Σ i U s ( p )− h b † r ( p ) b s ( p )i V † s ( p ) Σ i V r ( p ) + h a † r ( p ) b † s (− p )i U † r ( p ) Σ i V s (− p ) e i E p t + h b s (− p ) a r ( p )i V † r (− p ) Σ i U s ( p ) e − i E p t i . (40)Making use of the direct formulas for the massive eigenspinors Note that in the Weyl representation of the Clifford algebra a different explicit formula for themassive eigenspinors is typically used, but the general conclusions remain the same.2 Leonardo Tinti, Woiciech Florkowski U r ( p ) = p E p + m (cid:18) φ r σ · p E p + m φ r (cid:19) , V r ( p ) = p E p + m (cid:18) σ · p E p + m χ r χ r (cid:19) , (41)with the two component vectors φ r and χ r being the eigenstates of the matrix σ z = diag ( , − ) , φ = (cid:18) (cid:19) , φ = (cid:18) (cid:19) ,χ = (cid:18) (cid:19) , χ = − (cid:18) (cid:19) , (42)and the standard relations between the Pauli matrices { σ i , σ j } = δ i , j , [ σ i , σ j ] = i ε ijk σ k , (43)one can rewrite the matrix elements in (40) in the following way U † r ( p ) Σ i U s ( p ) = m φ r σ i φ s + p i E p + m φ r ( p · σ ) φ s , V † s ( p ) Σ i V r ( p ) = m χ s σ i χ r + p i E p + m χ s ( p · σ ) χ r , U † r ( p ) Σ i V s (− p ) = (cid:16) V † s (− p ) Σ i U r ( p ) (cid:17) ∗ = − i Õ j , k ε ijk p j φ r σ k χ s . (44)The first two terms do not correspond exactly to the polarization in the i ’th directionof a particle or an antiparticle in an eigenstate of four-momentum p µ , but they arevery closely linked to it (we discuss this point in more detail in the next section).In any case, the volume integral of the canonical spin tensor is strongly relatedto the polarization state of the fundamental excitations of the fields, before anyphenomenological approximation. Because of this, it is a reasonably good candidateto study, if one wants to extend the standard treatment of hydrodynamics to includepolarization degrees of freedom.In general, the awkward mixed terms involving creation and destruction operatorsare present in Eq. (39). They introduce time dependence in the volume integral (39)and have no clear interpretation in terms of particle-antiparticle degrees of freedom.Regarding the time dependence, this is not completely unexpected since the canonicalspin tensor is not conserved, see Eq. (35), hence the volume integral depends on thetime at which it is done. As explained in Appendix A, the expectation values of themixed terms h a † b † i and h ab i do not vanish only if the quantum state of the system elation between particle polarization and the spin tensor 13 is a superposition of states, and among them, some states differ in the number ofparticles/antiparticles by a single particle-antiparticle pair. How much relevant arethese kind of states in a heavy-ion collision environment is yet to be understood.If one considers non-canonical spin tensors obtained through a pseudo-gaugetransformation (30), it is possible to remove the time dependent part. For instance,the transformation proposed in [9] has the form ˆ Ξ αβ,µν = , ˆ G λ,µν = − m ¯ Ψ ( x ) (cid:18)h γ λ , γ µ i ↔ ∂ ν − h γ λ , γ ν i ↔ ∂ µ (cid:19) Ψ ( x ) , (45)which results in a conserved spin tensor S λ,µν = S λµν c − G λ,µν , ∂ λ S λ,µν = . (46)Hence the flux of the spin density, described by S λ,µν , across the freeze-outhypersurface is equal to the volume integral of S ,µν at later times. The latter canbe computed following the same steps used for the canonical tensor. After lengthybut straightforward calculations one obtains a time independent formula that is stillstrongly related to the polarization degrees of freedom, ε ijk ∫ d x S , jk == Õ r , s ∫ d p ( π ) (cid:20) h a † r ( p ) a s ( p )i (cid:18) E p m φ r σ i φ s − p i m ( E p + m ) φ r ( p · σ ) φ s (cid:19) −h b † r ( p ) b s ( p )i (cid:18) E p m χ s σ i χ r − p i m ( E p + m ) χ s ( p · σ ) χ r (cid:19) (cid:21) . (47)It is important to note that the term in the brackets, despite reducing to the polarizationof a two component spinor in the non relativistic limit ( m → ∞ ), does not correspondto the polarization of a relativistic particle. Therefore the last integral must not beconfused with the the average (relativistic) polarization multiplied by the averagenumber of (anti)particles. For a discussion of the spin tensor in the context ofrelativistic thermodynamics we refer to [23, 24], see also Becattini’s review in thismonograph [13]. We have already shown that the spin tensor is a macroscopic object sensitive to thepolarization of the excitations of a free quantum field. In this section, we show thatthe relativistic Wigner distribution is the appropriate extension of the distributionfunction, that takes into account the polarization degrees of freedom. In particular,the Wigner distribution of a generic state of the system can be linked to the average polarization of particles with fixed momentum, which is probably the most importantthing from the point of view of comparisons of theory predictions with experimentaldata.Our starting point is a relativistic polarization pseudovector, namely, the rela-tivistic counterpart of the expectation value h ψ | σ | ψ i used for a two-component,non-relativistic spinor. An important property to take into account is that the latter isa constant of motion for free particles (the free hamiltonian commutes with the Paulimatrices). Thus, the most straightforward way to generalize the concept of h ψ | σ | ψ i is to look at the classical (non-quantum) relativistic generalization of the internalangular momentum and to apply the same reasoning to the operators in QFT.For a classical (extended) object the angular momentum reads j = x × p + s , with s being the intrinsic angular momentum . The immediate relativistic generalizationis [25, 26] j µν = x µ p ν − x ν p µ + s µν , (48)with an antisymmetric tensor s µν = − s νµ . It is easy to notice that the components ( / ) Í j , k ε ijk j jk describe the angular momentum, while the components j i areneeded for relativistic covariance, to have the correct transformation rules changingthe frame of reference.The polarization pseudovector Π µ is proportional to the dual of the angularmomentum, contracted with the four-momentum Π µ = − m ε µνρσ j νρ p σ = − m ε µνρσ s νρ p σ . (49)Here we have made use of the definition (48), removing the contribution of theorbital momentum since it is proportional to the four-momentum and vanishes aftercontraction with the Levi-Civita symbol. In any inertial reference frame comovingwith the system (hence for p = ) the polarization pseudovector has a vanishing timecomponent, and the space components are just the intrinsic angular momentum s .Since Π µ is the contraction of an antisymmetric object with the totally antisym-metric ε µνρσ and the four-momentum p µ , one needs only tree components to fullydescribe it, for instance, the space ones. Having these comments in mind, we obtain = p µ Π µ = E p Π − p · Π ⇒ Π = p · Π E p . (50)It is particularly useful to write the polarization pseudovector in the comovingframe, Π com . , in terms of the polarization in the lab frame. It will serve us later to At the classical level, the latter corresponds to rotation with respect to an internal axis of theextended object A very similar definition is used for the Pauli-Lubanski pseudovector. It follows the same con-struction procedure, but without mass in the denominator. Beside different physical dimensions, itis a very close concept which is well defined in the massless case. Since we focus on massive fieldsherein, we are not going to analyze it. It is useful to notice, however, that using the Pauli-Lubanskidefinition one can follow the same steps in the massless case, obtaining the helicity distributioninstead of the polarization one.elation between particle polarization and the spin tensor 15 make a direct connection between the relativistic polarization operator and the nonrelativistic one, h ψ | σ | ψ i .A boost Λ p from the lab frame to the comoving frame is characterized by thespeed β = k p k/ E p and the Lorentz gamma factor γ = / p − β = E p / m . Thezeroth component of Π µ must vanish after such a boost, as immediately followsfrom Eq. (50), Π Λ p −→ Π . = γ (cid:18) Π − β Π · p k p k (cid:19) = γ (cid:18) Π − Π · p E p (cid:19) ≡ . (51)On the other hand the non-trivial spatial part reads Π com . = Π − Π · p k p k p + γ (cid:18) Π · p k p k − β Π (cid:19) p k p k = Π − Π · p k p k " − E p m − k p k E p ! p = Π − Π · p k p k (cid:20) E p − mE p ≡ k p k E p ( E p + m ) (cid:21) p = Π − Π · p E p ( E p + m ) p . (52)In relativistic quantum field theory one has the operator analogue of the polariza-tion (49) for a massive Dirac field, namely, the operator ˆ Π µ = − m ε µνρσ : ˆ J νρ :: ˆ P σ : . (53)We note that one has to use normal ordering for the two operators separately, becauseotherwise the expectation value of ˆ Π µ in single-particle states would vanish . Themost general one-particle state | ψ i for a free field reads | ψ i = Õ r ∫ d p ( π ) E p ψ ( p , r )| p , r i , (54)with the normalization = h ψ | ψ i = Õ r ∫ d p ( π ) E p ψ ∗ ( p , r ) ψ ( p , r ) , (55)in which we made use of the normalization of the states h p , r | q , s i = E p ( π ) δ rs δ ( p − p ′ ) . (56) If one applies the normal ordering : ˆ J νρ ˆ P σ : at the operator level there are two destructionoperators on the left-hand side, which annihilate any single particle state. One would need at leasttwo (anti)particle states to have a non-vanishing expectation value.6 Leonardo Tinti, Woiciech Florkowski The polarization vector then reads h ψ | ˆ Π µ | ψ i = − m ε µνρσ Õ r , r ′ ∫ d pd p ′ ( π ) E p E p ′ ψ ∗ ( p ′ , r ′ ) ψ ( p , r )×× h p ′ , r ′ | : ˆ J νρ :: ˆ P σ : | p , r i . (57)Using the anticommutation relations { a s ( q ) , a † r ( p )} = ( π ) δ rs δ ( p − q ) , (58)and taking into account the definition | p , r i = p E p a † r ( p )| i , (59)it is relatively straightforward to prove that : ˆ P σ : | p , r i = p σ | p , r i , (60)where ˆ P µ is the total four-momentum operator, : ˆ P µ : = Õ s ∫ d q ( π ) q µ h a † s ( q ) a s ( q ) + b † s ( q ) b s ( q ) i . (61)The situation is slightly more complicated for the expectation value of the angularmomentum operator h p ′ , r ′ | : ˆ J νρ : | p , r i . One can check that a non-zero contributionreads h ψ | ˆ Π µ | ψ i = − m ε µνρσ Õ r , r ′ ∫ d pd p ′ ( π ) E p E p ′ ψ ∗ ( p ′ , r ′ ) ψ ( p , r ) p σ ×× h p ′ , r ′ | : ˆ J νρ : | p , r i == − m ε µνρσ Õ r , r ′ ∫ d x ∫ d pd p ′ ( π ) E p E p ′ ψ ∗ ( p ′ , r ′ ) ψ ( p , r ) p σ ×× i U † r ′ ( p ′ ) γ n γ , h γ ν , γ ρ i o U r ( p ) e − i ( p − p ′ )· x == − m ε µ ij σ ε ijk Õ r , r ′ ∫ d p ( π ) ψ ∗ ( p , r ′ ) ψ ( p , r ) E p p σ E p U † r ′ ( p ) Σ k U r ( p ) . (62)In particular, the space part of the polarization h ψ | ˆ Π | ψ i reads h ψ | ˆ Π | ψ i = m Õ r , s ∫ d p ( π ) ψ ∗ ( p , r ) ψ ( p , s ) E p U † r ( p ) (cid:18) σ σ (cid:19) U s ( p ) == Õ r , s ∫ d p ( π ) ψ ∗ ( p , r ) ψ ( p , s ) E p (cid:20) φ r σ φ s + φ r ( p · σ ) φ s m ( E p + m ) p (cid:21) . (63) elation between particle polarization and the spin tensor 17 At this point, with the help of (52), it is possible to highlight the link betweenthe expectation value we have just computed and the non-relativistic polarization h ψ | σ | ψ i . We first define the spin momentum-dependent density matrix f rs ( p ) = ψ ∗ ( p , r ) ψ ( p , s ) E p , (64)which is a two-by-two Hermitian matrix in the indices r , s for every value of themomentum p and describes a polarized state. It is normalized to one, i.e., its trace overthe r , s indices is unitary while integrated with the measure ∫ d p /( π ) (becauseof the normalization of the wave function (55)). Taking into account a momentumeigenstate of the form f rs = ( π ) H rs δ ( p − ˜ p ) , one finds the polarization Õ r , s H rs (cid:20) φ r σ φ s + φ r ( ˜ p · σ ) φ s m ( E ˜ p + m ) ˜ p (cid:21) , (65)with H rs = H ∗ sr and tr ( H ) = . Making use of (52), one finds that the polarization inthe comoving frame reads Õ r , s H rs φ r σ φ s , (66)which is, indeed, the polarization of a non-relativistic spinor. Hence, it is limitedbetween − / and / in each direction.It is important to note, however, that the polarization in the lab frame is notlimited. For instance H = (cid:18) (cid:19) ⇒ h ˆ Π i = © « ˜ p z ˜ p x m ( E ˜ p + m ) ˜ p z ˜ p y m ( E ˜ p + m ) + ˜ p z m ( E ˜ p + m ) ª®®®¬ , (67)which has, manifestly, arbitrarily large components as long as ˜ p z , , maintainingthe expected polarization ( , , / ) in the comoving frame for a z polarized state. Fora more general case, i.e., if f rs /∝ δ ( p − ˜ p ) , one must keep the relativistic corrections.If one considers the defining relation (64), the spin density matrix has an imme-diate physical interpretation. The trace Õ r (cid:20) f rr ( p )( π ) (cid:21) (68)is the probability density to obtain p in a momentum measurement, while the averagepolarization in the comoving frame reads This is actually forbidden, since the wave function ψ is a regular distribution in momentum.However, one can have the spin density matrix factorized in a Hermitian × matrix times andarbitrarily sharp gaussian in the momentum. Such a strongly delocalized state is, for all practicalpurposes, equivalent to a momentum eigenstate.8 Leonardo Tinti, Woiciech Florkowski Õ rs [ f rs ( p ) ( φ r σ φ s )] Í t [ f tt ( p )] . (69)We thus see that the spin density matrix is sufficient to characterize the most importantexperimental observables for a free spin / particle. It is understood that all thesteps can be repeated for a single antiparticle wave-function to obtain the antiparticlepolarization, which depends on the antiparticle spin density matrix ¯ f rs ( p ) . The onlysignificant difference is an overall − sign, because of the corresponding sign in thespin part of the normal ordered angular momentum operator, and an exchange of the r and s indices in the χ bispinors compared to the φ for the particles .The appropriate generalization of the classical distribution function is expected,therefore, to produce in some limit a multi-particle generalization of the spin densitymatrix that provides both the spectrum in momentum of the produced particles andtheir average polarization. As we have already anticipated, the desired object is theWigner distribution. Its most convenient definition is ˆ W AB ( x , k ) = ∫ d v ( π ) e − ik · v Ψ † B ( x + v / ) Ψ A ( x − v / ) , (70)that is, it is a four by four matrix obtained from thecomponents of two fields, Ψ ( y ) † B Ψ A ( z ) , Fourier transformed with respect to the relative distance v = z − y ,and with x = ( z + y )/ being the middle space-time point. The inversion on therelative position of the matrix elements between the left and right hand side of thelast equation is neede in order to use the ordinary rules in matrix multiplicationswith respect to the A , B indices. In the remander of this work we will omit the matrixindices, understanding the matrix nature, and we will just write, eg, U r U † r withoutindices in a similar way, understanding the fact that it is a × matrix.An important property of the Wigner distribution (70) is that it is a Hermitian ma-trix representing physical observables (at least in principle). Moreover, one expectsthat the usual causality rules apply to it. It is worth mentioning that some authorsuse a different sign convention or use ¯ Ψ B Ψ A in the definition of W ( x , k ) [16, 17, 18],thus making the alternatively defined matrix a non-Hermitian one, so one must checkwhich version of the Wigner distribution is actually used while comparing differentworks. In any case a matrix multiplication with γ and an eventual multiplication bya constant is enough to switch notation.By the correspondence principle, the classical distribution is the expectation valueof the renormalized operator W ( x , k ) = tr (cid:16) : ˆ W ( x , k ) : (cid:17) . (71) Which can be expected, since the conventional two component spinors χ in the negative frequencysolutions of the Dirac equation are taken with the opposite eigenvalue of σ z , compared to thepositive frequency solutions.elation between particle polarization and the spin tensor 19 Making use of the definition (70), and assuming some minimal smoothness of theintegrals , one can rewrite the expectation value of any bilinear form in the Diracfields using integration over the momentum k of the trace of the macroscopic Wignerdistribution (71) tr (cid:18) ρ : ¯ Ψ ( x ) γ ν · · · γ ν n i ↔ ∂ µ · · · i ↔ ∂ µ m Ψ ( x ) : (cid:19) = ∫ d k k µ · · · k µ m tr (cid:16) W ( x , k ) γ γ ν · · · γ ν n (cid:17) . (72)Here, the trace on the left-hand side is the usual trace over the quantum states, while tr on the right-hand side denotes the trace over the matrix indices. To derive theformula (72) we have used the integral representation of the Dircac delta δ ( v ) = ∫ d k /( π ) exp {− ik · v } and performed the integration by parts ∫ d v k µ e − ik · v [· · · ] = ∫ d v (cid:16) i ∂ µ ( v ) e − ik · v (cid:17) [· · · ] = ∫ d v k e − ik · v (cid:16) − i ∂ µ ( v ) (cid:17) [· · · ] . (73)In the next step, we can use the definition of the Dirac fields to expand (71) and toexplicitly represent it in terms of the expectation values of the creation/destructionoperators W ( x , k ) = Õ rs ∫ d v ( π ) e − ik · v ∫ d pd q ( π ) p E p E q h h a † r ( p ) a s ( q )i U † r ( p ) U s ( q ) e i ( p − q )· x e i ( p + q ) · v + − h b † r ( p ) b s ( q )i V † s ( q ) V r ( p ) e i ( p − q )· x e − i ( p + q ) · v ++ h b s ( q ) a r ( p )i V † s ( q ) U r ( p ) e − i ( p + q )· x e i ( p − q ) · v ++ h a † r ( p ) b † s ( q )i U † r ( p ) V s ( q ) e i ( p + q )· x e i ( p − q ) · v i , (74)it is important to remind that both sides must be a matrix, therefore the eigenspinorsare not contracted but must be read, eg, ( U † r ) B ( U s ) A , according to (70). The lastformula (74) allows us to make two important observations. The first one is that, afterperforming the d v integral, each of the four sectors is proportional to the Dirac deltafunction δ ( k ± ( p ± q )/ ) , with a different combination of the + and − signs in eachsector. The momenta p µ and q µ are both on the mass shell but their combination, ingeneral, is not. The pure particle/antiparticle contributions have δ ( k ± ( p + q )/ ) which can be on shell if and only if p = q . The mixed terms, however, include δ ( k ± ( p − q )/ ) where ( p − q )/ is never on shell and always space-like. This is In order to exchange the order of the integrations and integrate by parts.0 Leonardo Tinti, Woiciech Florkowski the reason why we call k µ a wave number vector, in order not to confuse it with thefour-momentum of some particle-like degree of freedom.The second and possibly the most important thing to notice is that k µ W ( x , k ) isconserved, i.e., k µ ∂ µ W ( x , k ) = , as one can check directly by applying k µ ∂ µ tothe right-hand side of (74) and using the integration by parts in (73) to convert thewavenumber vector k into a derivative with respect to v . This is a consequence ofthe Dirac equation for the fields, which implies that the Klein-Gordon equation issatisfied as well.Because of the conservation of k µ W ( x , k ) , one can use the same mathematicalframework as that already used for the conserved fluxes such as T µν and the classicalexpression p µ f ( x , p ) Here we can see the reason of the choice to define the Wigneroperator as Hermitian. Being k µ W ( x , k ) Hermitian too (an observable) is expectedto follow causality rules. As a consequence of the Gauss theorem, the flux over thefreeze-out hypersurface (or any other surface following the freeze-out) is equal tothe volume integral ∫ d Σ µ k µ W ( x , k ) = ∫ d x k W ( x , k ) . (75)The analysis of the volume integral corresponds to a considerable simplification inthe treatment of the Wigner distribution, like for the other quantum objects we haveseen. Integrating directly (74) one obtains ∫ d x k W ( x , k ) = Õ rs ∫ d p ( π ) k E p (cid:2) δ ( k − p ) h a † r ( p ) a s ( p )i U † r ( p ) U s ( p ) + − δ ( k + p ) h b † r ( p ) b s ( p )i V † s ( p ) V r ( p ) i . (76)The mixed terms vanish exactly, since the volume integral provides a δ ( p + q ) .Therefore, the Dirac function δ ( k ± ( p − q )/ ) becomes δ ( k ) δ ( k ± p ) in this case.The appearance of k makes these terms vanishing, as k δ ( k ) δ ( k ± p ) ≡ .The flux of the Wigner distribution (75) has many interesting properties. Theycan be identified while looking at its explicit form given by (76). It includes an on-shell positive-frequency contribution for the particles and a negative-frequency (withnegative momentum) contribution for the antiparticles. With k being on the mass-shell, one can divide by k since k k k ≥ m – something that cannot be done for thefull distribution (74). Having in mind the normalization tr ( U † r ( p )· U s ( p )) = U † r · U s = E p δ rs = V † r · V s = tr ( V † r ( p ) V s ( p )) , and the exact relations (see Appendix A) Õ r h a † r ( p ) a r ( p )i( π ) = dNd p , Õ r h b † r ( p ) b r ( p )i( π ) = d ¯ Nd p , (77) The hypothesis of an isolated system is important too. Being the integrand an observable, theflux over the light-cone starting from the spatial boundary of an isolated system must be vanishing.Causality prevents the Wigner distribution to flow out of the light cone, as it would be a superluminalsignal transfer, and the hypothesis of an isolated system prevents any signal to flow inside of thelight cone. The flux over the light cone is therefore vanishing.elation between particle polarization and the spin tensor 21 it is immediate to verify that the trace of the flux (76) reads ∫ d Σ µ k µ W ( x , k ) = δ ( k − E k ) E k dNd p ( k ) + δ ( k + E k ) E k d ¯ Nd p (− k ) . (78)In other words, the positive-frequency contribution to the flux is directly expressed bythe (invariant) spectrum of particles with momentum k ; while the negative-frequencycontribution is given by the invariant spectrum of antiparticles with momentum − k .The structure of (76) is quite rich. It does not include the (anti)particle’s spectraonly but depends on the polarization states. By construction, the expectation values h a † r ( p ) a s ( p )i and h b † r ( p ) b s ( p )i have a very similar structure to the one-particle spindensity matrix (64). They are both Hermitian matrices with respect to the indices r , s for all values of p . Moreover, their diagonal elements h a † r ( p ) a r ( p )i and h b † r ( p ) b r ( p )i are always non-negative. The only difference is the normalization. Instead of beingnormalized to they are normalized to the average number of particles h N i andantiparticles h ¯ N i Õ r ∫ d p h a † r ( p ) a r ( p )i( π ) = ∫ d p dNd p = h N i , Õ r ∫ d p h b † r ( p ) b r ( p )i( π ) = ∫ d p d ¯ Nd p = h ¯ N i . (79)They provide therefore the desired generalization of the one-particle spin densitymatrix to the multi-particle case. The flux of the Wigner distribution (75) directlydepends on them, it is therefore not surprising that one can get the average polar-ization density in momentum space from it. Making use of both (75) and (76) wefind m tr (cid:20) (cid:18)∫ d Σ µ k µ W ( x , k ) (cid:19) γ γ i γ (cid:21) == m Õ r , s ∫ d p " δ ( k − p ) h a † r ( p ) a s ( p )i( π ) U † r ( p ) (cid:18) σ i σ i (cid:19) U s ( p ) + δ ( k + p ) h b † r ( p ) b s ( p )i( π ) V † s ( p ) (cid:18) σ i σ i (cid:19) V r ( p ) == Õ r , s ( δ ( k − E k ) h a † r ( k ) a s ( k )i( π ) (cid:20) φ r σ i φ s + φ r ( k · σ ) φ s m ( E k + m ) k i (cid:21) + δ ( k + E k ) h b † r (− k ) b s (− k )i( π ) (cid:20) χ s σ i χ r + χ s ( k · σ ) χ r m ( E k + m ) k i (cid:21) ) , (80) which can be immediately recognized as the average polarization of particles withmomentum k (multiplied by the (non-invariant) spectrum dN / d p ( k ) for the positivefrequency) minus the average polarization of antiparticles of momentum − k (timesthe spectrum d ¯ N / d p (− k ) ). Since the spectra can be calculated from the flux of theWigner distribution, one can obtain the average polarizations of particles, h Π ( p )i ,and antiparticles, h ¯ Π ( p )i , for any momentum p , h Π ( p )i = Õ r , s h a † r ( p ) a s ( p )i Í t h a † t ( p ) a t ( p )i (cid:20) φ r σ φ s + φ r ( p · σ ) φ s m ( E p + m ) p (cid:21) , h ¯ Π ( p )i = − Õ r , s h b † s ( p ) b r ( p )i Í t h b † t ( p ) b t ( p )i (cid:20) χ r σ χ s + χ r ( p · σ ) χ s m ( E p + m ) p (cid:21) . (81)Making use of (52) one can compute the polarization in the comoving frame asusual.It is straightforward to check that the trace tr h m (cid:16)∫ d Σ µ k µ W ( x , k ) (cid:17) γ γ γ i corresponds to the momentum density of Π ( i.e., it is equal to the time componentof the polarization vector – particle contribution for the positive frequency minus theantiparticle contribution for the negative frequency). The same structure of the traceis obtained with γ replaced by γ i . One can summarize all these results by makinguse of the definitions h Π ( p )i = h Π ( p )i · p E p , h ¯ Π ( p )i = h ¯ Π ( p )i · p E p , (82)to complete the covariant h Π µ ( p )i with the correct time component . Thus, thecompact form of the previous results on the average polarization reads m tr (cid:20) (cid:18)∫ d Σ λ k λ W ( x , k ) (cid:19) γ γ µ γ (cid:21) == δ ( k − E k ) dNd p ( k ) h Π µ ( k )i − δ ( k + E k ) dNd p (− k ) h ¯ Π µ (− k )i . (83)The last equation, in conjunction with the exact result in (78), is enough to grantthat the Wigner at the freeze-out hypersurface is sufficient to predict all the relevantexperimental spectra of produced (anti)particles.In the last section we have seen that the spin tensor is a macroscopic observablesensitive to the microscopic polarization states. Looking at the relaitively simpletrace on the left-hand side of (83), one may think if there is macroscopic objectrelated to this. Taking into account the exact conversion rules (72), one finds that the d k integral of the left-hand side reads Compare with Eq. (50) for a quick check.elation between particle polarization and the spin tensor 23 m ∫ d k tr (cid:20) (cid:18)∫ d Σ λ k λ W ( x , k ) (cid:19) γ γ µ γ (cid:21) == ∫ d Σ λ h : i m ¯ Ψ (cid:18) ↔ ∂ λ γ µ γ (cid:19) Ψ : i . (84)It is, therefore, the flux of (the expectation value of) the rank pseudotensor i m ¯ Ψ ( x ) (cid:18) ↔ ∂ λ γ µ γ (cid:19) Ψ ( x ) , (85)which we may call the polarization flux pseudotensor, given its relation to theintegral of the polarization pseudovector density. It is straightforward to check thatits divergence in the λ index vanishes. Hence, it is conserved, as one could expectsince its flux is time-independent. It does not correspond to any spin tensor, butit provides a valid alternative as a macroscopic object sensitive to the micorscopicpolarization states. In this work, we have extended the standard kinetic-theory formalism to include spinpolarization for particles with spin 1/2. This has been achieved by using the spintensor and the Wigner function. Our results can be used for the interpretation of theheavy-ion data describing spin polarization of the emitted hadrons.
Acknowledgements
We would like to thank F. Becattini, B. Friman, R. Ryblewski, and E. Speranzafor insightful discussions. L.T. was supported by the Deutsche Forschungsgemeinschaft (DFG,German Research Foundation) through the CRC-TR 211 “Strong-interaction matter under extremeconditions” - project number 315477589 âĂŞ TRR 211. W.F. was supported in part by the PolishNational Science Center Grants No. 2016/23/B/ST2/00717. ppendix A
Expectation values of creation and destructionoperators
In this appendix we show details of the calculations of the expectation values ofcreation and destruction operators. In particular, we find an interesting and intuitivelink between the average (anti)particle number and the quantum fluctuations requiredfor the mixed terms ( h a † b † i and h ab i ) to be non-vanishing.The starting point is the density matrix (2), which reads ρ = Õ i P i | ψ i i h ψ i | . (A.1)All P i ’s are classical probabilities Õ i P i = . (A.2)The states | ψ i i are proper quantum states, that is, they are normalized to one h ψ i | ψ i i = , ∀ i . (A.3)The expectation value O of any quantum operator ˆ O is a weighted average of theexpectation values in the pure states, with the classical weights P i , O = tr (cid:16) ρ ˆ O (cid:17) = Õ i P i tr (cid:16) | ψ i i h ψ i | ˆ O (cid:17) , (A.4)therefore, our problem reduces to the expectation value in a generic pure state.The trace must be taken over a complete set of independent states (not necessarilyquantum states that are normalized to one). Since we are interested in the expectationvalues of the creation and destruction operators of four-momentum and polarizationeigenstates, the most convenient states are N -particle and ¯ N -antiparticle ones. The In general one needs the renormalized operators. For free fields this is just the normal ordering,that is, removing the vacuum expectation value. We always assume massive free Dirac fields andnormal ordering in this section. 256 A Expectation values of creation and destruction operators trace is defined as an integration over the momentum degrees of freedom and a sumover discrete polarizations, namely tr (cid:16) · · · (cid:17) = Õ r ∫ d p ( π ) E p h p , r | · · · | p , r i ++ Õ r , s ∫ d p ( π ) E p d q ( π ) E q h p , r , q , s | · · · | p , r , q , s i + · · · , (A.5)and so on, until exausting all the combinations of N particles and ¯ N antiparticles. Inthe last formula the standard definition is used, | p , r i = p E p a † r ( p )| i , (A.6)along with the analogous expressions for antiparticles and multiparticle states. Theanticommutation relations have the form { a r ( p ) , a † s ( p ′ )} = { b r ( p ) , b † s ( p ′ )} = ( π ) δ rs δ ( p − p ′ ) , (A.7)with the normalization h p , r | q , s i = E p ( π ) δ rs δ ( p − p ′ ) . (A.8)It is convenient to introduce the compact notation for multiparticle states | p , r ; ¯ q , ¯ s i = | p , r , p , r , · · · p N , r N ; ¯ q , ¯ s , ¯ q , ¯ s , · · · ¯ q ¯ N , ¯ s ¯ N i , ∫ [ dp ] N [ d ¯ q ] ¯ N = ∫ d p ( π ) E p · · · d p N ( π ) E p N d ¯ q ( π ) E ¯q · · · d ¯ q ¯ N ( π ) E ¯q ¯ N , (A.9)where the bar is used to distinguish antiparticle from particle variables. In this waythe trace (A.5) can be written in a more compact form as tr (cid:18) · · · (cid:19) = Õ N , ¯ N Õ r , ¯ s ∫ [ dp ] N [ d ¯ q ] ¯ N h p , r ; ¯ q , ¯ s | · · · | p , r ; ¯ q , ¯ s i . (A.10)This compact notation is useful to write the generic quantum state | ψ i , | ψ i = Õ N , ¯ N Õ r , ¯ s ∫ [ dp ] N [ d ¯ q ] ¯ N α N , ¯ N ( p , r ; ¯ q , ¯ s ) | p , r ; ¯ q , ¯ s i , (A.11)where the complex functions α N , ¯ N ( p , r ; ¯ q , ¯ s ) are partial N -particle- ¯ N -antiparticlewave functions in momentum space. The normalization reads Expectation values of creation and destruction operators 27 = h ψ | ψ i = Õ N , ¯ N Õ r , ¯ s ∫ [ dp ] N [ d ¯ q ] ¯ N α ∗ N , ¯ N ( p , r ; ¯ q , ¯ s ) α N , ¯ N ( p , r ; ¯ q , ¯ s ) == Õ N , ¯ N k α N , ¯ N k , (A.12)with k α N , ¯ N k being a short-hand notation for the (non-negative ) sum of integrals k α N , ¯ N k = Õ r , ¯ s ∫ [ dp ] N [ d ¯ q ] ¯ N α ∗ N , ¯ N ( p , r ; ¯ q , ¯ s ) α N , ¯ N ( p , r ; ¯ q , ¯ s ) . (A.13)The tensor product | ψ ih ψ | , that is, the projector on the quantum state | ψ i reads | ψ ih ψ | = Õ N , ¯ N Õ r , ¯ s Õ N ′ , ¯ N ′ Õ r ′ , ¯ s ′ ∫ [ dp ] N [ d ¯ q ] ¯ N [ dp ′ ] N [ d ¯ q ′ ] ¯ N ×× α ∗ N ′ , ¯ N ′ ( p ′ , r ′ ; ¯ q ′ , ¯ s ′ ) α N , ¯ N ( p , r ; ¯ q , ¯ s ) | p , r ; ¯ q , ¯ s ih p ′ , r ′ ; ¯ q ′ , ¯ s ′ | . (A.14)Making use of the normalization relations between the states, it is possible to writethe trace in a pure state | ψ i of an operator ˆ O in the compact form tr (cid:16) | ψ i h ψ | ˆ O (cid:17) = Õ N , ¯ N Õ r , ¯ s Õ N ′ , ¯ N ′ Õ r ′ , ¯ s ′ ∫ [ dp ] N [ d ¯ q ] ¯ N [ dp ′ ] N [ d ¯ q ′ ] ¯ N ×× α ∗ N ′ , ¯ N ′ ( p ′ , r ′ ; ¯ q ′ , ¯ s ′ ) α N , ¯ N ( p , r ; ¯ q , ¯ s ) h p ′ , r ′ ; ¯ q ′ , ¯ s ′ | ˆ O| p , r ; ¯ q , ¯ s i . (A.15)There is a couple of results that can be immediately inferred from the last formula.The first one is that the expectation values of a † b † and ba , for any momentumand polarization combination, can be non vanishing if and only if the quantumstate of the system is in a superposition of states with different particle content.More precisely, only the quantum interference between states that differ exactly by aparticle-antiparticle pair can give a non-vanishing contribution ( understanding thatthe integral over the partial wavefunctions can still simplify and give a vanishingresult).The second observation is that the expectation value of a † a and b † b can besimplified. The only combinations that can give a contribution are the ones betweenstates with exactly the same number of particles and the same number of antiparticles.In the following computations, we consider only the term a † a , understanding thatthe very same transformations hold for antiparticles.As a particular case of (A.15) one can write the expectation value of a † r ( p ) a s ( p ′ ) tr (cid:16) | ψ i h ψ | a † r ( p ) a s ( p ′ ) (cid:17) = Õ N , ¯ N Õ t , t ′ Õ ¯ u , ¯ u ′ ∫ [ dk ] N [ dk ′ ] N [ d ¯ q ] ¯ N [ d ¯ q ′ ] ¯ N ×× α ∗ N , ¯ N ( k ′ , t ′ ; ¯ q ′ , ¯ u ′ ) α N , ¯ N ( k , t ; ¯ q , ¯ u )h k ′ , t ′ ; ¯ q ′ , ¯ u ′ | a † r ( p ) a s ( p ′ )| k , t ; ¯ q , ¯ u i . (A.16) Being the sum of integrals of a real non-negative weight of the forms z ∗ z .8 A Expectation values of creation and destruction operators It is relatively simple to obtain the final formula by making use of the standardanticommutation relations · · · a † r ( p ) a s ( p ′ ) q E k j a † t j ( k j ) · · · = · · · a † r ( p ) q E k j (cid:16) { a s ( p ′ ) , a † t j ( k j )} − a † t j ( k j ) a s ( p ′ ) (cid:17) · · · = · · · q E k j (cid:16) a † t j ( k j ) a † r ( p ) a s ( p ′ ) + a † r ( p )( π ) δ st j δ ( k j − p ′ ) (cid:17) · · · = · · · hq E k j a † t j ( k j ) (cid:16) a † r ( p ) a s ( p ′ ) (cid:17) + p E p ′ ( π ) δ st j δ ( k j − p ′ ) a † r ( p ) i · · · == · · · hq E k j a † t j ( k j ) (cid:16) a † r ( p ) a s ( p ′ ) (cid:17) ++ s E p ′ E p ( π ) δ st j δ ( k j − p ′ ) p E p a † r ( p ) · · · (A.17)In other words, even if a † r ( p ) a s ( p ′ ) doesn’t commute with the creation operators, it ispossible to "move it to the right". However, each time we do that we have to add a newstate, with a delta between the j ’th degrees of freedom and the destruction operator a s ( p ′ ) , a numerical factor ( π ) p E p ′ / E p and a substitution of the momentum andpolarization at the j ’th place with the ones related to the creation operator a † r ( p ) .After moving to the right all the particle creation operators, a † r ( p ) a s ( p ′ ) commuteswith the creation operators of the antiparticles (if present).In the end, after making use of the normalization of the eigenstates we find tr (cid:16) | ψ i h ψ | a † r ( p ) a s ( p ′ ) (cid:17) = ++ p E p E p ′ Õ ¯ N , N > N Õ j + Õ t − t j Õ ¯ u ∫ [ dk ] ( N − j ) [ d ¯ q ] ¯ N ×× α ∗ N , ¯ N ( k − k j , p , t − t j , r ; ¯ q , ¯ u ) α N , ¯ N ( k − k j , p ′ , t − t j , s ; ¯ q , ¯ u ) . (A.18)The notation Í t − t j ∫ d [ k ] ( N − j ) means that the integral and the sum is over all theparticle degrees of freedom except for the j ’th. In the similar way α N , ¯ N ( k − k j , p , t − t j , r ; ¯ q , ¯ u ) is a shorthand notation for the (partial) wavefunction with the j ’th degreesof freedom fixed to the momentum p and polarization r .The formula (A.18) has many interesting consequences. Besides the expectedvanishing expectation value for purely antiparticle states, one can immediately checkthat the expectation value of a † r ( p ) a r ( p ) is non-negative, since it is a series of integralsand sums of squares. Moreover, as one could expect, it is linked to the average numberof particles. Indeed, the expression Õ r ∫ d p ( π ) tr (cid:16) | ψ ih ψ | a † r ( p ) a r ( p ) (cid:17) = Õ N N Õ ¯ N k α N , ¯ N k , (A.19) Expectation values of creation and destruction operators 29 exactly gives the average number of particles in the state | ψ i because of the nor-malization (A.12). More interestingly, the expectation value of a † r ( p ) a s ( p ) (samemomentum, different polarization) performs the role of a momentum dependentspin density matrix. The momentum integral of the trace is proportional to the av-erage number of particles, but the matrix itself is sensitive to polarization in the r , s indices and can be used to obtain the average number of particles, per momentumcell, for some polarization states.All these arguments do not change if one reinserts the classical probabilities P i from (A.1) and deals with mixed states. The classical fluctuations do not change theproperties of the spin density matrix, like the non-negative diagonal elements andnormalization of the trace (after dividing by ( π ) and integrating over momentum,like for the pure states) does not change the average number of particles. eferences
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