Spin tensor and its role in non-equilibrium thermodynamics

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

Spin

tensor

and

its

role

in

non-equilibrium

thermodynamics

Francesco Becattini

a

,

∗

,

Wojciech Florkowski

b

,

Enrico Speranza

c a_{UniversitàdiFirenzeandINFNSezionediFirenze,ViaG.Sansone1,I-50019,SestoFiorentino,Firenze,Italy} b_{InstituteofNuclearPhysicsPolishAcademyofSciences,PL-31342Kraków,Poland}

c_{InstituteforTheoreticalPhysics,GoetheUniversity,D-60438FrankfurtamMain,Germany}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received21August2018

Receivedinrevisedform6December2018 Accepted9December2018

Availableonline18December2018 Editor:J.-P.Blaizot

Itisshownthatthedescriptionofarelativisticfluidatlocalthermodynamicequilibriumdependsonthe particular quantumstress-energytensoroperator chosen,e.g.,thecanonicalorsymmetrizedBelinfante stress-energytensor.WearguethattheBelinfantetensorisnotappropriatetodescribearelativisticfluid whosemacroscopicpolarizationrelaxesslowlytothermodynamicequilibriumandthataspintensor,like thecanonicalspintensor,isrequired.Asaconsequence,thedescriptionofapolarizedrelativisticfluid involvesanextensionofrelativistichydrodynamics includinganew antisymmetricrank-two tensoras adynamical field.We showthat thecanonicaland Belinfantetensorsleadtodifferentpredictions for measurablequantitiessuchas spectrumand polarizationofparticlesproducedinrelativisticheavy-ion collisions.

©2018TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3_.

1. Introduction

Themeasurementofa finiteglobalpolarization ofparticlesin relativistic heavy-ion collisions [1], in agreement with the pre-dictionsof relativistic hydrodynamics[2,3] (see also [4–11]),has opened a new perspective in the phenomenology of these colli-sionsaswellasinthetheoryofrelativisticmatter,showingforthe firsttimeadirectmanifestationofquantumfeaturesinthisrealm. While a formularelating mean polarization with thermal vortic-ity(see Sec. 3) atlocalthermodynamic equilibriumwasobtained inRef. [12],basedonaneducatedansatz,an exactformulaisstill missingevenatglobalthermodynamic equilibriumwithrotation. Meanwhile,theexperimentshaveproved tobe abletoprobe po-larizationdifferentiallyinmomentumspace[13,14] and,fromthe theorystandpoint, the issuehasbeen raised [15] about the rele-vanceofthespintensorinthedescriptionofarelativisticfluid.

Indeed,theproblemofthephysicalsignificanceofthespin ten-sor — mostly in relativistic gravitationaltheories, notably in the Einstein–Cartantheory—isalong-standingone[16] andhasbeen rediscussed morerecently in Refs. [17,18], whereit was demon-strated that well known thermodynamic formulae for transport coefficientssuchasviscositydodependonthepresenceofaspin tensor.However,uptonow,itsrelevanceforthepolarization mea-suredinrelativisticheavy-ioncollisionshasnot beendiscussedin

*

Correspondingauthor.

E-mailaddress:becattini@fi.infn.it(F. Becattini).

detail,eventhoughitwasusedtoderivethepolarizationformula inRef. [12].Itisoneofthemaingoalsofthispapertopresentand fullyexplorethistheoreticalissueindetail.

Our conclusion is that for a general relativistic fluid the spin tensorissignificant,ifspindensity“slowly”relaxestowards equi-librium,whereslowlymeansonatimescalewhichiscomparable tothefamiliarhydrodynamictimescaleofevolutionofchargeand momentum densities. In this case, we will see that the descrip-tionofthepolarizationofparticlesandthecalculationofitsfinal value requiresa newthermodynamic potential, akinto chemical potential ortemperature, coupledto thespin tensor.Accordingly, relativistichydrodynamicsshouldbe extendedtoincludethespin tensoramongtheevolveddensities.

Thepaperisorganizedasfollows:InSec.2wedefine thespin tensoranditscloserelationshipwiththestress-energytensor. Sec-tion3describesthelocalthermodynamic equilibriumdensity op-erator and its dependence on the pseudo-gauge transformations of the stress-energy and spin tensors. In Sec. 4 we discuss the physicsofdifferentlocalthermodynamicequilibria,whileinSec.5

we consider consequences of usinga spin tensorfor measurable quantities such as spectra and polarization. We summarize and concludeinSec.6.

Notation: In this paper we adopt the natural units, withh

_¯

=

c

=

kB

=

1. The Minkowski metric tensor is diag(1,

−

1,

−

1,

−

1); for the Levi-Civita symbol we use the convention

ε

0123

₌

_{1. We} usethe relativisticnotation withrepeatedindicesassumedto be saturated.OperatorsinHilbertspacearedenotedbyanupperhat,

https://doi.org/10.1016/j.physletb.2018.12.016

0370-2693/©2018TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP3_.

e.g.,



A. Although we work inflat space-time, in manyequations weusecovariantderivative

∇

μ insteadofpartialderivative

∂

μ to emphasizethevalidityoftherelationsingeneralcoordinates.

2. Pseudo-gaugetransformations

Inrelativisticquantumfieldtheoryinflatspace-time,according toNoether’stheorem,foreachcontinuous symmetryoftheaction there is a corresponding conserved current. The currents associ-atedwiththetranslationalsymmetry andtheLorentz symmetry1 are theso-calledcanonical stress-energytensor andthecanonical angularmomentumtensor:



T_Cμν

=



a

∂

L

∂(∂

μ

ψ



a

)

∂

ν

ψ



a

−

gμν

L

,

(1)



J

μ,λν C

=

xλ



T μν C

−

xν



T μλ C

+ 

S

μ,λν C

.

(2)

Here

L

isthelagrangiandensity,while

S



C reads



S

μ,λν C

= −

i



a,b

∂

L

∂(∂

μ

ψ



a

)

D

(

Jλν

)

a_b

ψ



b (3)

withD beingtheirreduciblerepresentationmatrixoftheLorentz grouppertainingtothefield.Theabovetensorsfulfillthefollowing equations:

∇

μ



TCμν

=

0

,

∇

μ

J



μ,λν C

= 

T λν C

− 

TνCλ

+ ∇

μ

S



Cμ,λν

=

0

.

(4)

It turnsout, however, that thestress-energy andangular mo-mentum tensors are not uniquely defined. Different pairs can be generatedeitherbyjustchangingthelagrangiandensityor,more generally,bymeansoftheso-calledpseudo-gaugetransformations [16]:



Tμν

= 

Tμν

+

1 2

∇

λ





λ,μν

_{− }

_

μ,λν

_{− }

_

ν,λμ



_,



S

λ,μν

_{= }

_S

λ,μν

_{− }

_

λ,μν

_,

₍₅₎

where





is a rank-three tensor field antisymmetric in the last twoindices(oftencalledandhenceforthreferredtoas superpoten-tial).InMinkowskispace-time,thenewlydefinedtensorspreserve thetotalenergy,momentum,andangularmomentum(herein ex-pressedinCartesiancoordinates):



Pν

=



 d



μ



Tμν

,



Jλν

=



 d



μ

J



μ,λν

,

(6)

aswellastheconservationequations(4).2_{Intheequation(6)}

_

_is

ageneralspace-likehypersurface.

A special pseudo-gauge transformation is the one where one startswiththecanonicaldefinitionsandthesuperpotentialisthe spintensoritself,thatis,





= 

S

.Inthiscase,thenewspintensor vanishes,

S





=

0,andthenewstress-energytensoristheso-called Belinfantestress-energytensor



TB,



Tμν_B

= 

T_Cμν

+

1 2

∇

λ





S

Cλ,μν

− 

S

μ,λν C

− 

S

ν,λμ C



.

(7)

1 _{ByLorentzsymmetrytransformationsweunderstandhereLorentzboostsand}

rotations.

2 _{ThisstatementonlyappliestoMinkowskispace-time,ingenerallycurved}

space-timesitisnolongertrue[16].

3. Localequilibriumdensityoperator

The densityoperator describing local thermodynamic equilib-rium in quantum field theory was obtained in ref. [19,20] and has been rederived more recently in refs. [21,22]; herein, we briefly summarize thederivation. The localthermodynamic equi-librium densityoperator is obtained by maximizing the entropy S

= −

tr(

ρ



log



ρ

)

with the constraints ofgiven mean densities of conservedcurrentsoversomespace-likehyper-surface

,

a covari-antgeneralizationofahyperplaneinspecialrelativity.The projec-tionsofthemeanstress-energytensorandchargecurrentontothe normalizedvectorperpendicularto



mustbeequaltotheactual ones: nμtr





ρ



Tμν



=

nμTμν

,

nμtr





ρ



jμ



=

nμjμ

,

(8)

where theoperators are in theHeisenberg representation.In ad-ditiontotheenergy,momentum,andchargedensities,oneshould includetheangularmomentumdensityamongsttheconstraintsin Eq. (8),namely:

nμtr





ρ

J



μ,λν



=

nμtr

ρ





xλ



Tμν

−

xν



Tμλ

+ 

S

μ,λν



=

nμ

J

μ,λν

.

(9)

However, itisclearthatifwehaveBelinfante’sstress-energy ten-sor



TB withassociated vanishing spin tensor, equation (9) is

re-dundant, since the angular momentum densityconstraint is im-pliedin(8).Hence,Eq. (8) remainstheonlyrelevantcondition.The resultingoperatorreads(thesubscript LEstandsforLocal Equilib-rium):



ρ

LE

=

1 ZLE exp

⎡

⎣−



 d



μ



TμνB

β

ν

− ζ

jμ



⎤

_{⎦ ,}

(10)

where

β

and

ζ

are therelevantLagrange multiplierfunctionsfor this problem, whosemeaning isthe four-temperaturevector and the ratio between local chemical potential and temperature, re-spectively[21].The ZLEfactoristhepartitionfunctionwhose def-initionisimpliedbytheconstrainttr



ρ

LE

=

1.

We note that the operator (10) is not the actual density op-erator, because it is not generally stationary as required in the Heisenberg picture.Infact,thetruedensityoperator forasystem in localthermodynamic equilibriumcoincides withthat givenby Eq. (10) attheinitialtime

τ

0,thatis,with



≡ (τ

0),



ρ

=

1 Zexp

⎡

⎢

⎣−



(τ0) d



μ



TμνB

β

ν

− ζ

jμ



⎤

⎥

⎦ .

(11)

Providedthatfluxesatsometimelikeboundaryvanish,itis possi-bletorewritetheactualdensityoperator

ρ



byusingGauss’ theo-rem[21],



ρ

=

1 Zexp

⎡

⎢

⎣−



(τ) d



μ



TμνB

β

ν

− ζ

jμ



+



d



Tμν_B

∇

μ

β

ν

−

jμ

∇

μ

ζ



⎤

_{⎦ ,}

(12)

wherethefirsttermisthelocalthermodynamicequilibriumterm at time

τ

and

denotes thespace-time region encompassedby

thespace-likehypersurfaces

(

τ

),

(

τ

0),andthetime-like bound-aries.Formula(12) istheessenceofZubarev’sformalism[19,20,23] anditnicely separatesthe non-dissipativepart(the first termin theexponent)fromthedissipativeone(thesecondterm).3

The operator (12) becomes stationary, that is independent of thehypersurface

,

whenthefour-temperatureisaKillingvector fieldandtheratiobetweenchemicalpotentialandtemperatureis constant:

∇

μ

β

ν

+ ∇

ν

β

μ

=

0

,

∇

μ

ζ

=

0

.

(13)

The first equation above follows from the fact that Belinfante’s energy-momentumtensorissymmetric.Inaflatspacetime,it im-pliesthat thesecond ordergradients of

β

vanishand

β

itself is givenbytheexpression

β

ν

=

bν

+



νλxλ

,

(14)

whereb is aconstant vector and



isa constant antisymmetric tensor.The conditions (13) defineglobalthermodynamic equilib-rium. One can also check that the (redundant) inclusion of the conservationofangularmomentum(9) inBelinfante’scase(where itisreducedtotheconservationoftheorbitalpartonly)doesnot changetheformofglobalequilibrium,asthisleadstoachangeof thetensor



νλwhichremainsconstant.

Onecannowusethepseudo-gaugetransformationsofSec.2to rewritethelocalthermodynamicequilibriumdensityoperatorasa functionof,e.g.,canonicaltensors.UsingEq. (7),wefind:



ρ

LE

=

1 ZLE exp

⎡

⎣−



 d



μ



TμνB

β

ν

− ζ

jμ



⎤

_⎦

=

1 ZLE exp

⎡

⎣−



 d



μ





T_Cμν

β

ν

+

1 2

β

ν

∇

λ





S

λ,μν C

− 

S

μ,λν C

− 

S

ν,λμ C



− ζ

jμ



.

(15)

Wenowworkouttheintegral involvingthespin tensor. Integrat-ingbypartsweobtain

−

1 2



 d



μ



∇

λ



β

ν

S



Cλ,μν

− β

ν

S



μ,λν C

− β

ν

S



ν,λμ C



+

1 2



 d



μ



∇

λ

β

ν





S

λ,μν C

− 

S

μ,λν C

− 

S

ν,λμ C



.

(16)

Thefirsttermisadivergenceofanantisymmetrictensor(with re-specttothe

λ

↔

μ

exchange),soitcanbeturnedintoaboundary integral,

−

1 4



∂ dS

˜

μλ



β

ν

S



λ,Cμν

− β

ν

S



μ,λν C

− β

ν

S



ν,λμ C



,

(17)

which vanishes for suitable boundary conditions imposed on

β

and/or

S



. The second term, on the other hand,can be rewritten as

−

1 2



 d



μ



∇

λ

β

ν

S



Cμ,λν

− ∇

λ

β

ν





S

λ,μν C

+ 

S

ν,μλ C



(18)

3 _{Wenoteinpassingthatequation(12) isbasicallytheoneusedtoobtainthe}

firstderivationoftheKuboformulaoftheshearviscosity [24].

by takingadvantage oftheantisymmetryof thelast two indices. Inthisway,wefinallyobtain



ρ

LE

=

1 ZLE exp

⎡

⎣−



 d



μ





Tμν_C

β

ν

−

1 2



λν

S



μ,λν C

−

1 2

ξ

λν





S

λ,μν C

+ 

S

ν,μλ C



− ζ

jμ

 ⎤

⎦ ,

(19) where



λν

=

1 2

(∇

ν

β

λ

− ∇

λ

β

ν

)

(20)

isthethermalvorticity,compareEq. (14),and

ξ

λν

=

1

2

(

∇

ν

β

λ

+ ∇

λ

β

ν

)

(21)

isthesymmetricpartofthegradientofthefour-temperature vec-tor.4

The conclusion we can draw from Eq. (19) is apparent. Ifwe hadusedthecanonicalstress-energytensorinsteadofBelinfante’s form, enforcing only the constraints (8) with



TC replacing



TB to

maximize theentropy,wewouldhaveobtainedaformally analo-gousexpression forthe localthermodynamic equilibriumdensity operator,i.e.,Eq. (10) with



TB replacedby



TC.However,asshown

above, thenew localthermodynamic equilibriumoperator would have not been the same as that given by Eq. (19). Therefore, in general, thereisno equivalenceofdescriptionoflocal thermody-namic equilibriumwith different sets oftensors. This conclusion isnotsurprisingafterallbecausedensities ratherthanintegrated quantities are fixed by the constraints in local thermodynamic equilibriumanddensitiesdo dependon thepseudo-gaugechoice (5):theconceptoflocalthermodynamicequilibriumisnot pseudo-gaugeindependent.

Onemaywonder,however,iftheuseofcanonicaltensors sim-ply requirestheinclusion ofEq. (9) besides Eq. (8). Infact,since Eq. (10) is equivalent to Eq. (19), by inclusion ofthe spin tensor withinthecanonicalapproach,onecanindeedobtainthesame



ρ

LE forthecanonicalandBelinfante’sschemes.Thisissuewillbe dis-cussedingreaterdetailinthenextsection.Nevertheless,atglobal thermodynamicequilibrium, theequivalence isfullyrestored:the tensor(21) vanishesaccordingto(13),thefour-temperaturevector isgivenby(14),



isconstantandthedensityoperatorbecomes:



ρ

=

1 Zexp



−

bμ



Pμ

+

1 2



μν



J μν

_{+ ζ }

_Q



,

(23)

where



P μ and



J μν aregivenbyEq. (6) and



Q isthetotalcharge definedas



Q

=



 d



μ



jμ

.

(24)

Theform(23),dependingonthegenerators,ismanifestlyinvariant underanypseudo-gaugetransformation.

4 _{ItisworthnotingthatforthefreeDiracfield,thecanonicalspintensorhasthe}

form  Sμ,λν C = i 8ψ{γ μ_,[γλ_,γν_{]}ψ , (22)}

whichiscompletelyantisymmetric,thetermproportionaltoξinEq. (19) vanishes. Ithasbeenrecentlypointedoutthatanon-completelyantisymmetricspintensor canbeconnectedtoanon-vanishingenergydipolemoment[25].

4. Localthermodynamicequilibriumwithspintensorandspin hydrodynamics

As it has been mentioned at the end of the previous sec-tion, it seems compelling to include angularmomentum density amongsttheconstraintsdefininglocalthermodynamicequilibrium, seeEq. (9),incaseone startsfromthecanonicalsetoftensorsor anyothersetlinkedtothecanonicalsetbyapseudo-gauge trans-formation. Indeed, asthe orbital part of

J

in Eq. (9) is already takeninto account inthe energy-momentumdensity constraints, theonlyeffectiveadditionalequationis

nμtr





ρ

S



μ,λν



₌

_n

μ

S

μ,λν

.

(25)

Fortheabove tobe actually anindependent constraint,the intro-ductionofanantisymmetrictensorfield



λν isnecessary.Belowit isdubbedaspintensorpotential orshortlyaspinpotential.5_In

anal-ogywiththechemicalpotential,thecomponentsof



playarole ofLagrange multiplierscoupled to the spin tensor. Includingthe furtherconstraint (25) the construction oflocal equilibrium den-sityoperatorbymeansofentropymaximizationyields:



ρ

LE

=

1 ZLE exp

⎡

⎣−



 d



μ





Tμν

β

ν

−

1 2



λν

S



μ,λν

_{− ζ}

_jμ

⎤

⎦ .

(26)

Onecannowseekfortheconditionsrequiredforthedensity op-eratortobestationary,i.e.,forglobalthermodynamicequilibrium. Byrequiringthattheintegrandhaszerodivergence,onealways re-trievesthecondition(13),i.e.,

β

oughttobeaKillingvectorand

ζ

tobe aconstant.Furthermore,if



=



, theformofthedensity operator(23) dependingonintegralgeneratorsisalsorecovered.6

Ifwenow carry outa pseudo-gaugetransformationwith





=



S

inEq. (5) toeliminate thespin tensorandrecoverBelinfante’s stress-energy tensor (by Belinfante’s tensors we now understand those obtained by the pseudo-gauge transformation with





= 

S

ofanyoriginal tensors



T μν and

S



μ,λ_{ν ) by meansof aderivation}

analogoustothatpresentedintheprevioussectiononeobtains



ρ

LE

=

1 ZLE exp

⎡

⎣−



 d



μ





Tμν_B

β

ν

−

1 2

(

λν

−



λν

) 

S

μ,λν

+

1 2

ξ

λν



S

λ,μν

_{+ }

_S

ν,μλ



_{− ζ}

_jμ

 ⎤

⎦ ,

(27)

where



is the thermal vorticity. Consequently, the local ther-modynamicequilibriumoperator(26),hence(27),isequivalentto thatbuiltdirectlyfromBelinfante’stensor,see Eq. (10) ifthe fol-lowingconditionsaremet:

1. thefield

β

isthesameinbothcases;

5 _{InRef. [15],whereahydrodynamicmodelofparticleswithspin1/2was}

pro-posed,thisquantitywasdenotedasωλνanddubbedthespin-polarizationtensor.

6 _{Wenotethattherequirementofvanishingdivergence,thatis,stationarityin}

Eq. (26),mayimplyadditionalformsofglobalequilibriumwith=.The ap-pearanceofthesesolutionsdependsonthesymmetryfeaturesofthestress-energy andspintensors.Inparticular,ifthestress-energytensorissymmetric,Tμν_{= T}νμ_,

andyetthe spintensordoesnotvanish,thespinpotentialcanbeconstantand independentof orevencanbenon-constant,providedthatitfulfillsthe con-traction(∇μλν)Sμ,λν=0.Thereasonfortheappearanceofsuchsolutionsisthat

ifT issymmetricandS=0 then∇μSμ,λν=0,hence,thereexistsanadditional

conservedcharge — theintegralofthespintensor.

2. the tensor



always coincides with thermal vorticity con-structedfromthe

β

field;

3. theterminvolvingthesymmetriccombinationin

λ

and

ν

in-dicesofthespintensorvanishes.

Thefirstconditionislesstrivialthanitmightlookatafirstglance. Whenimposingtheconstraints(8) withdifferenttensors,thefield

β

beingasolutionoftheconstraints,likee.g.

tr

(

ρ

[β, ζ, . . .]

Tμν

)

nμ

=

Tμνnμ

,

depends on the choice of the stress-energy tensor, namely the pseudo-gauge,anditisthusgenerallydifferent.

Thesameconclusioncanbereachedforamoregeneral pseudo-gauge transformation (5). The formal invariance of the operator (26) holdsif

β



= β

,





=  =



, where

β

 and



 are thenew thermodynamic fields for the new set of stress-energy and spin tensors,andifthetermproportionalto

ξ

vanishes.

Finally,wewouldliketopointoutthatadensityoperatorsuch as(27) involvestheappearanceintheentropycurrentexpression of a term depending on the spin potential and the spin tensor. Following the argument givenin Ref. [21] and assuming the ex-tensivityoflog ZLE, log ZLE

=

log tr

⎛

⎝

exp

⎡

⎣−



 d



μ





Tμν

β

ν

−

1 2



λν

S



μ,λν

_{− ζ}

_jμ

⎤

⎦

⎞

⎠

=



 d



μ

φ

μ

,

itispossibletoreadilyobtainanentropycurrentsμ fromthetotal entropy S

= −

tr(

ρ



LElog

ρ



LE),

sμ

= φ

μ

+

T_LEμν

β

ν

− ζ

jμLE

−

1 2



λν

S

μ,λν

LE

,

(28)

where theLE subscript stands forthe meanvalue withthe den-sity operator (26). This vector is, however, not uniquely defined as it is possible to add vectors orthogonal to nμ to obtain the sametotalentropy.Inspiteofthisambiguity,theentropycurrent oughttobeconservedwhenT μν

=

T_LEμν and

S

μ,λν

₌

_S

μ,λν

LE atany time,acondition whichisapparentlythenaturalextensionofthe definitionof an idealfluid.The entropycurrentconservationis a consequenceoftheequation(5.8)inref. [20] whichcanbereadily extendedtothecasewithspintensor.

4.1. Discussion

It is now time to pause and reflect about the physical inter-pretationofthediscussedformalism. Thenotionoflocal thermo-dynamicequilibriumrequirestheexistenceoftwoseparate space-time scales:a microscopicone over whichinformationisnot ac-cessibleandamacroscopiconewhichisusedtoobservesystem’s evolutiontowardsglobalequilibrium.Itisunderstoodinthechoice oftheoperator(26) thatthespindensityappearsamongdensities which“slowly”evolvetowardsglobalequilibrium,justlikea con-served charge densityorenergydensity.The dissipative, entropy-increasing processesmust drivethesystemtoglobalequilibrium, hence(atleastforsystemswithnon-symmetricstress-energy ten-sor) thespinpotential



shouldconvergetothethermalvorticity (see [26] for a similar discussion). Yet,this process maybe slow enough so thatthe spin relaxationtakesplaceon thesametime scaleasthetypicaldissipativehydrodynamic process.Inthiscase, spindensitycanbethoughtashydrodynamicallyrelevant,andthe

Fig. 1. (Coloronline)a)amacroscopicfluidatrestwithpolarizedparticles(black arrows)andantiparticles(redarrows).Thissituation,evenifmetastable,requiresa spintensortobedescribed,b)ifthespintensorvanishes,thefluidmustberotating (non-vanishingthermalvorticity)tohavepolarizedparticlesandantiparticlesinthe samedirection.

spinpotentialwouldbearelevanthydrodynamicfield.Conversely, ifthedensityoperator were chosen tobe (10), thiswouldimply thatthespinrelaxationtimeismicroscopicallysmallandthevalue ofthespinpotentialagreeswiththermalvorticity.

Thetwosituationsdescribedabovecanbeeffectivelyrephrased ina kinetic picture withcolliding particles: in the firstcase, we would say that the spin-orbit coupling of the particles is much weaker than other processes responsible, for example, for local equilibration of their momentum. In the second case, the spin-orbitcoupling wouldbe asstrong asany other coupling so that the spin degreesof freedom locally equilibrate within the same timescaleasmomentum.

Forthepurposeofillustrationofthefactsdescribedabove,one can envisage a fluid temporarily at restwith a constant temper-ature T , hence

β

= (

1/T

)(1,

0

),

wherein both particles and an-tiparticles are polarized in the same direction (see Fig. 1). Such a situation cannot be described as local thermodynamic equilib-riumby the densityoperator (10) because the onlyway tohave particlesandantiparticlespolarizedinthiscaseisthrougha non-vanishing thermalvorticity [12] which vanishes if

β

is constant. Infact,ifweonlyhadBelinfante’sstress-energytensoratour dis-posaland were able to force the systemto be in such an initial configuration,we could describe it onlyafter a microscopic time scale, when a rotating configuration is established, with



=

0. Conversely,withthedensityoperator(26) such ametastable situ-ation,evolvingintime, canbe describedby anon-vanishing spin potentialeventhoughthermalvorticityisvanishing.

ThissituationisreminiscentoftheEinsteinanddeHaaseffect: angular momentum conservation induces a rotation of the body because of the polarization brought about by a magnetic field. Yet,thereisanimportantdifference:eventhoughtherotation oc-curredona“long” timescale,fortheEinsteinandde Haaseffect to be described there is no need of a spin tensor nor ofa spin potential,becauseoftheabsenceofantimatter:themagnetization tensorMμν asamacroscopicdensity(which,fortheDiracfield,is proportional to i

ψ



[γ

μ

,

γ

ν

_]

_ψ)

_{and themagnetic field as} thermo-dynamicconjugatevariablewouldsuffice.Onlywhenantimatteris involved,thatisinaquantumrelativisticfieldtheory,dothespin tensorandthespinpotentialbecomenecessarytodescribeaslow equilibrationprocessof thepolarization.In thelimitingcaseofa completelyneutral fluid atrest, withthe same numberof parti-clesandantiparticles,themagnetizationtensorwouldbezeroand yet a metastable state withparticles and antiparticles can exist. Mathematically,thisisreflectedinthelinearindependenceofspin

tensor(which isthedualofthe axialcurrent)andmagnetization tensorintheDiractheory.

4.2. Relativistichydrodynamicswithspintensor

If we are then to describe a relativistic fluid where polarized configurationsarenotgovernedonlybythermalvorticity,onehas tointroduce aspinpotential tensor



amongthe conjugate ther-modynamicfields,besides

β

and

ζ

(thefirststepsinthisdirection havebeen done inRefs. [15,27–29]).In theassociated relativistic hydrodynamics, its 6 independent componentshave to be deter-mined from the solutions of the 6 additional partial differential equations

∂

λ

S

λ,μν

=

Tνμ

−

Tμν

,

(29)

whichappearbesides thefamiliarones, describingthe continuity ofstress-energytensorandcurrent.Ofcourse,oneneedsthe con-stitutiveequationsofthespintensor,thestress-energytensor,and the current in terms of



to be able to solve all hydrodynamic equations.Ingeneral,theyarefunctionalrelations

jμ

=

jμ

[β, ζ, ],

Tμν

=

Tμν

[β, ζ, ],

S

λ,μν

₌

_S

λ,μν

_{[β, ζ, ].}

(30)

At the lowest order of approximations, the constitutive relations areobtainedbycalculatingthemeanvalueswiththelocal equilib-rium densityoperator (26). Dissipative corrections depending on gradients of

β

and



can be calculated with the same method outlinedwhendiscussingEqs. (11) and(12).

5. Polarizationinrelativisticheavy-ioncollisions

Doestheintroductionofaspintensorinafluidhaveany mea-surableconsequence? Thisisanintriguingquestion withpossible far-reachingconsequences.Eventhough the hydrodynamic model hasbecome one of themostuseful toolsformodeling of heavy-ioncollisions,it shouldbe stressedthatneitherthe stress-energy tensor nor any other density in space and time can be directly measuredoraccessed.Theactualmeasurements,likeinany high-energy physics experiment, involve momentum and polarization of asymptoticparticle states;the hydrodynamic model is justan intermediarybetweensomeinitialstateandthefinalparticle spec-tra. In fact, strictly speaking, even in astrophysics one can only measure the radiation emitted by the plasmas and not the den-sitiesthemselves.

Infact,measurablequantitiesintheseexperimentscanbe gen-erallyexpressedasexpectationvaluesofsomenumberofcreation anddestructionoperatorsofasymptoticstates,specifically,offinal hadronsinthecaseofrelativisticheavy-ioncollisions.Forinstance, the single-particlepolarization matrix ofa particlewith momen-tump reads

W

(

p

)

σ,σ

=

tr

(

ρ

a†

(

p

)

σa

(

p

)

σ

),

(31) where

ρ



is the actual density operator. From the above matrix, wellknownquantitiessuchasthemeanspinvector,thealignment, etc.canbecalculated,includingthemomentumspectrumitselfby justtakingthetrace.Equation (31) makesitapparentthat –ifany –thedependenceofmeasuredparticlespinandmomentaon hy-drodynamicswithspintensorisencodedinthedensityoperator.

IthasbeendiscussedinSec.3,thattheactualdensityoperator istheinitiallocalthermodynamicequilibrium obtainedfromEq. (11), whichincludesdissipativeeffects,asitisclearfromEq. (12).The form (11) is hardly fit to calculate quantities like (31) because

fieldoperatorsaretobeevaluatedattheinitialtime

τ

0 whilethe creationanddestructionoperators arethose ofasymptoticstates. In particular,for relativisticheavy-ion collisions, the operators at theinitialtime are thoseofthequark-gluonfieldswhilethe cre-ation anddestruction operators are those offinal hadrons. Thus, it is necessary to make use of the form (12) involving the ef-fective fields at the final time

τ

, where the effective fields are the hadronic ones, to carry out the calculation of Eq. (31). Nev-ertheless,theeffectofpseudo-gaugetransformationsontheactual densityoperator,writteninanyofitsequivalentforms,canbe as-sessedby studyingitseffectontheinitial-timeform(the latteris moreconvenienttousebecauseofitscompactness).

InahydrodynamicapproachbasedonBelinfante’sscheme,the actualdensityoperatorwouldbepreciselygivenby (11),whilein a canonical-based hydrodynamics with relevant spin densities it wouldread



ρ



=

1 Zexp

⎡

⎢

⎣−



(τ0) d



μ





T_Cμν

β

ν

−

1 2



λν

S



μ,λν C

− ζ

jμ



⎤

⎥

⎦ .

(32)

Equation (32) canbetransformedinthesamewayasinthe pre-vioussectionandturnedintoanexpressionlikeEq. (27),namely



ρ



=

1 Zexp

⎡

⎢

⎣−



(τ0) d



μ





T_Bμν

β

ν

−

1 2

(

λν

−



λν

) 

S

μ,λν C

+

1 2

ξ

λν





S

Cλ,μν

+ 

S

ν,μλ C



− ζ

jμ



⎤

⎥

⎦ .

(33)

Therefore,foranymeasurablequantity



X ,therewouldbea differ-encebetweenthemeanvaluescalculatedwiththedensity opera-tors(11) and(33).Defining:



A

=



(τ0) d



μ



TBμν

β

ν

− ζ

jμ



(34) and



B

=



(τ0) d



μ



−

1 2

(

λν

−



λν

) 

S

μ,λν C

+

1 2

ξ

λν





S

λ,μν C

+ 

S

ν,μλ C



,

(35) onehas



X

=

tr

(

ρ





X

)

−

tr

(

ρ



X

)

=

1 Ztr

(

e A+B

_

_X

₎

₋

1 Ztr

(

e A

_

_X

₎



1



0 dz tr



ρ





XezA





B

−

tr

(

ρ



B

)

e−zA



,

(36)

where the right-hand side is the leading term in the linear re-sponsetheory.If

S



C isthecanonicalspintensoroftheDiracfield,

thetermin

ξ

vanishesbecause

S



C isacompletelyantisymmetric

tensorand,hence,thedifferenceinthetheoreticalvalue depends onthecorrelatorbetween



X and



B,thatistheintegralofthespin tensorweightedbythedifferencebetweenspinpotentialand ther-malvorticity.Theevaluationof(36) isnotaneasy taskbutitcan be envisagedthat thisdifference will be a quantumeffect anda tinyoneformostobservableswhichmostlydependonthe

β

field,

namely velocity and temperature field, such as the momentum spectra.Infact,thedifferencecould besignificantforobservables which dependlinearlyon thethermal vorticity,such as polariza-tion.

6. Conclusions

Insummary,wehaveshownthatthenon-equilibriumor local-equilibrium thermodynamics is sensitive to the pseudo-gauge transformation ofthe stress-energy andspin tensors in quantum field theory.This conclusionis aconsiderable extension of previ-ousarguments[17,18],aswehaveshownherethatpseudo-gauge transformations quantitatively affect theoretical values of mea-surable quantities in high-energy physics experiments, especially polarizationoffinalparticlescreatedinrelativisticheavy-ion colli-sions.

Theinequivalenceofdifferentstress-energytensorsarisesmost clearly in the description of a completely neutral fluid at rest with finitepolarization ofboth particles and antiparticles— this corresponds to ametastable local equilibriumstate thathas a fi-nite valueofsome spintensorandzero(thermal)vorticity.Ifthe stress-energy tensor used is Belinfante’s symmetrizedone, finite polarizationshouldalwaysbeaccompaniedbyamacroscopic rota-tion.

This pseudo-gauge inequivalence has a hydrodynamic coun-terpart. Hydrodynamics ofmetastable polarizedneutral fluids re-quiresaranktwospinpotentialtensor



μν asconjugate thermo-dynamic field tothe spin tensor. Thisaddition ofcourse compli-catesthestandardhydrodynamicequationstobesolved.

Finally,itshouldbepointedoutthatthequestionremainshow to determinethe “right”spin tensor amongall possibleones. In-deed,the canonicalspin tensorobtainedfromEq. (3) isone par-ticular option; other spin tensors can be obtained by applying a pseudo-gauge transformation and they will not yield the same local thermodynamic equilibrium operator, according to the dis-cussion inSect.4following Eq. (27).Apparently,the issuecannot be settled theoretically, but experimentally comparing measured quantitiestothedifferentpredictions.Wenotethat similarissues have been very intensively studied in the context of the proton spindecomposition,forexampleseeRefs. [30,31].

Acknowledgements

Very usefuldiscussions with B.Friman, M.Stephanov, L.Tinti are gratefully acknowledged. W.F. was supported in part by the PolishNationalScienceCentre GrantNo.2016/23/B/ST2/00717.E.S. was supported byBMBFVerbundprojekt 05P2015–Alice atHigh Rate andby theDeutscheForschungsgemeinschaft(DFG) through the grant CRC-TR 211 “Strong-interaction matter under extreme conditions”andbytheCOSTActionCA15213“Theoryofhot mat-terandrelativisticheavy-ioncollisions”(THOR).

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