Faddeev-Jackiw analysis for the charged compressible fluid in a higher-derivative electromagnetic field background
Albert C. R. Mendes, Everton M. C. Abreu, Jorge Ananias Neto, Flavio I. Takakura
aa r X i v : . [ phy s i c s . g e n - ph ] M a y Faddeev-Jackiw analysis for the charged compressible fluid in ahigher-derivative electromagnetic field background
Albert C. R. Mendes, ∗ Jorge Ananias Neto, † and Flavio I. Takakura ‡ Departamento de F´ısica, Universidade Federal de Juiz de Fora, 36036-330, Juiz de Fora - MG, Brazil
Everton M. C. Abreu § Grupo de F´ısica Te´orica e Matem´atica F´ısica, Departamento de F´ısica,Universidade Federal Rural do Rio de Janeiro, 23890-971, Serop´edica - RJ, Brazil andDepartamento de F´ısica, Universidade Federal de Juiz de Fora, 36036-330, Juiz de Fora - MG, Brazil (Dated: November 5, 2018)In the present paper we will discuss the Faddeev-Jackiw symplectic approach in the analysis of acharged compressible fluid immersed in a higher-derivative electromagnetic field theory. We haveobtained the full set of constraints directly from the zero-mode eigenvectors. Besides, we havecomputed the Dirac brackets for the dynamic variables of the compressible fluid. Finally, as aresult of the coupling between the charged compressible fluid and the electromagnetic field we havecalculated two Dirac brackets between the fluid and electromagnetic fields, which are both zerowhen there is no coupling between them.
PACS numbers: 03.50.Kk, 11.10.Ef, 47.10.-gKeywords: Faddeev-Jackiw formalism; compressible fluid; electromagnetic background
I. INTRODUCTION
The search for a connection between fluid dynam-ics and electromagnetism is an old concept and ithas played a crucial role in the development of theMaxwell equations [1, 2]. Thomson applied anal-ogous formulations connecting electrostatics, heattransfer and elasticity of solids, that later leadMaxwell to formulate his theory of electricity andmagnetism [1]. This analogy was first applied tothe set of Maxwell equations concerning fluid dy-namics in the early 1962 [3] only to the case ofthe one-dimensional Rayleigh problem. Recently,the generalization of the Maxwell set of fluid equa-tions was introduced in terms of an incompressibleflow, particularly with interest in turbulent flow [4].Even more recently, the fluid Maxwell equationswere generalized to the compressible flow case [5].Besides, other generalizations of fluid dynamics havebeen constructed proposing noncommutative, non-Abelian and supersymmetric formulations, to men-tion a few [8].In [7], some of us have introduced a Lagrangiandescription for the compressible fluid together withthe scenario where a charged fluid was immersed inan electromagnetic field. The interaction betweenthem from the Lagrangian density was discussed. ∗ Electronic address: albert@fisica.ufjf.br † Electronic address: jorge@fisica.ufjf.br ‡ Electronic address: flavio@fisica.ufjf.br § Electronic address: [email protected]
This analogy has been explored in the literature withapplications in quark-gluons plasma (QGP) [8–13]which is a dense liquid that flows with very littleviscosity almost being an ideal fluid.Having said that, we can consider this work aspart of a sequence of other ones from these authorsupon the analysis of this mentioned analogy betweenthe structure of the fluid dynamics and electrody-namics [7, 14, 15]. The purpose of the present paperis to analyze the Lagrangian density which describesthe charged compressible fluid immersed in an elec-tromagnetic field, obtained in [7], from the point ofview of the Faddeev-Jackiw method [16] applied tothis model.The Faddeev-Jackiw (FJ) [16] method is asymplectic description of constrained quantization,where the degrees of freedom are identified by meansof the so-called symplectic variables. The essentialpoint of the FJ method is to make the system intoa first order Lagrangian with some auxiliary fields,but the method does not depend on how the aux-iliary fields are introduced to make the first orderLagrangian. It was applied recently in non-Abeliantheories [17].The work is organized in such a way that in section2 we have reviewed briefly the FJ method. In sec-tion 3 we have analyzed the theory via the Faddeev-Jackiw method and finally in the last section wepresent the conclusions.
II. FADDEEV-JACKIW FORMALISM
We will begin with a first-order time derivativeLagrangian, which arises from a standard second-order one with auxiliary fields. The first step is toconstruct the symplectic Lagrangian L = a i ( ξ ) ˙ ξ i − V ( ξ ) , (1)where a i are the arbitrary one-form components and i = 1 , ..., N . Since the first-order system is con-structed through a closed two-form, if it is non-degenerated, it defines a symplectic framework onthe phase space, which is described by the coordi-nates ξ i . Besides, if this two-form is singular, withconstant rank, it is defined as a pre-symplectic two-form. Hence, considering the components, the sym-plectic form can be defined by f ij = ∂∂ξ i a j ( ξ ) − ∂∂ξ j a i ( ξ ) , (2)and the equations of motion are f ij ˙ ξ j = ∂∂ξ i V ( ξ ) , (3)where the two-form f ij can be either singular or non-singular. In this last case it has an inverse f ij ˙ ξ i = f ij ∂∂ξ j V ( ξ ) , (4)where we have that (cid:8) ξ i , ξ j (cid:9) = f ij . To consider aconstrained system described by (1), it means thatthe symplectic matrix is singular. And the con-straints of the system have to be determined, ofcourse. Consider that the rank of f ij is 2 n . Inthis case we have N − n = M zero-mode vectors ν α , α = 1 , ..., M . The system is then constrainedthrough M equations with no time-derivatives. Wewill have constraints that reduce the degrees of free-dom’s number. Hence, multiplying (3) by the (left)zero-modes ν α of f ij we have the (symplectic) con-straints with the structure of algebraic relationsΩ α ≡ ν αi ∂∂ξ i V ( ξ ) = 0 . (5)So, we can construct the first-iterated Lagrangianby including the corresponding Lagrange multipliersrelative to the obtained constraints L = a (1) i ( ξ ) ˙ ξ i + Ω α λ α − V (1) ( ξ ) . (6)The Lagrange multipliers λ can be considered as thesymplectic variables which can increase the sym-plectic variables set. This move reduces the num-ber of ξ ’s. After that, the procedure can be en-tirely repeated until all the constraints can be elim-inated and the completely reduced, unconstrained and canonical system remains. But notice that inthe case of gauge theories, we have no new constraintthrough the zero-mode. And the symplectic matrixremains singular. Hence, we can consider mandatoryto introduce gauge condition(s) to highlight the sin-gularity. In this way the procedure can be finishedin terms of the original variables. And the basicbrackets can be determined. III. FADDEEV-JACKIW ANALYSIS FORTHE CHARGED COMPRESSIBLE FLUIDIMMERSED IN AN ELECTROMAGNETICFIELD
The effective Lagrangian density which describesthe charged compressible fluid immersed in an elec-tromagnetic field is defined, valid for each species( ǫ ), by L = − T µν ( ǫ ) T ( ǫ ) µν − (1 + g )4 F µν F µν − g T µν ( ǫ ) F µν , (7)where T ( ǫ ) µν = ∂ µ U ( ǫ ) ν − ∂ ν U ( ǫ ) µ is the strength tensorof the fluid, the four-vector potential U ( ǫ ) µ ≡ ( U ǫ , ~U ǫ )- U ǫ is the energy function and ~U ǫ is the averagevelocity field [7] - and F µν = ∂ µ A ν − ∂ ν A µ is thestrength tensor for the electromagnetic field. Thespacetime metric elements are η µν = ( − + ++).The coupling constant is g = e ǫ /m ǫ , where e ǫ isthe charge and m ǫ is the mass of the charge. Notethat, when g = 0 we have two uncoupled theories.The Euler-Lagrange equations of motion are(1 + g ) ∂ µ F µν + g∂ µ T µν ( ǫ ) = 0 , (8)and it is easy to see that (7) is invariant under thegauge transformations, A µ → A µ + ∂ µ Λ, for the elec-tromagnetic fields, and U ( ǫ ) µ → U ( ǫ ) µ + ∂ µ Λ, for thecompressible fluid field. In terms of the potentials, U ( ǫ ) α and A α , the above equation reads(1+ g ) [ ✷ A µ − ∂ µ ∂ ν A ν ]+ g h ✷ U ( ǫ ) µ − ∂ µ ∂ ν U ν ( ǫ ) i = 0 . (9)From now on, for simplicity, we will not use thespecies index, and much of what follows is true foreach species. In our model, the symmetric energy-momentum tensor is given byΘ αβ = (1 + g ) (cid:20) η αµ F µλ F λβ + 14 η αβ F µλ F µλ (cid:21) + (cid:20) η αµ T µλ T λβ + 14 η αβ T µλ T µλ (cid:21) (10)+ g (cid:20) η αµ T µγ F γβ + η αµ F µγ T γβ + 12 η αβ T µλ F µλ (cid:21) and it follows directly thatΘ = 12 (cid:16) ~l + ~ω (cid:17) + (1 + g )2 (cid:16) ~E − ~B (cid:17) + g ~l · ~E + g ~ω · ~B (11)which is the energy of the model, where the first termin (11) is the energy of the fluid and the second oneis the energy of the electromagnetic field. The lasttwo terms are the contributions of the interactionbetween the two fields.As we said before, in this paper we want to discussthe Faddeev-Jackiw methodology [16] applied in theanalysis of a higher-derivative theory which, in thiscase, have the higher-derivative in the Maxwell sec-tor. So, rewriting (7) in the form L = − T µν T µν − (1 + g )4 F µν F µν − gU µ ∂ ν F µν , (12)we can introduce another set of canonical pair (Σ µ ≡ ∂ A µ , φ ) in order to have a correct extended phasespace in order to proceed with the canonical analysis.Therefore, we have that L = 12 ( ˙ ~U − ∇ U ) + 12 ( ∇ × ~U ) + 12 (1 + g )( ~ Σ − ∇ A ) + 12 (1 + g )( ∇ × ~A ) − g ~U · ( ∇ Σ − ˙ ~ Σ) − gU ( ∇ .~ Σ − ∇ A ) − g ~U · ( ∇ × ∇ × ~A ) , (13)and to write a first order Lagrangian, we will use anauxiliary field, which is chosen to be the canonicalmomentum due to an algebraic simplification. Inthis case, we have a set of canonical pairs ( U µ , p µ ),( A µ , π µ ) and (Σ µ , φ µ ) and we have directly that p µ = ∂ L ∂ ( ∂ U µ ) , φ µ = ∂ L ∂ ( ∂ Σ µ ) , (14) π µ = ∂ L ∂ (Σ µ ) − ∂ ∂ L ∂ ( ∂ Σ µ ) − ∂ k ∂ L ∂ ( ∂ k Σ µ ) , which results in the following expressions p µ = T µ , φ µ = gη µk U k , (15) π µ = (1 + g ) F µ − gη µk T k + gη µ ∂ k U k . Therefore, making use of the equation of motion forthe canonical momenta associated with the fields U µ , A µ and Σ µ , we have L (0) = − ~p · ˙ ~U + ~φ · ˙ ~ Σ + π µ ˙ A µ − V (0) , (16) where the potential density is V (0) = π µ Σ µ − ~p − ~p · ∇ U −
12 ( ∇ × ~U ) (17) −
12 (1 + g )( ~ Σ − ∇ A ) −
12 (1 + g )( ∇ × ~A ) + ~φ · ∇ Σ + gU ( ∇ · ~ Σ − ∇ A ) + ~φ · ( ∇ × ∇ × ~A ) . The initial set of symplectic variables definingthe extended space is given by the set ξ (0) =( U k , p k , U ; A k , π k , A , π ; Σ k , π k , Σ ), and the corre-sponding canonical non-zero one-form is U a (0) k = − p k ; Σ a (0) k = φ k ; A a (0) k = − π k ; A a (0) = π . Using this result in the symplectic two-form matrix f (0) we have that f (0) ij ( ~x, ~y ) = F ij × × × M ij × × × C ij δ ( ~x − ~y ) (18)with F ij = δ ij − δ ij , M ij = δ ij − δ ij −
10 0 1 0 , C ij = − δ ij δ ij , (19)where we can note that the matrix f (0) ij is singular,which means that there are constraint and it hastwo zero-mode ν γ ≡ ( , , ν U , , , , , , ,
0) and ν γ ≡ ( , , , , , , , , , ν Σ ) where ν U and ν Σ are arbitrary function. From this two zero-mode, wehave the following constraints Ω (0) = Z d ~x ν U ( ~x ) δδU ( ~x ) Z d ~y V (0) ( ~y ) (20)= Z d ~x ν U ( ~x ) h ∇ · ~p ( ~x ) + g ( ∇ · ~ Σ( ~x ) − ∇ A ( ~x )) i = 0 (21)and Ω = Z d ~x ν Σ ( ~x ) δδ Σ ( ~x ) Z d ~y V (0) ( ~y )= Z d ~x ν Σ ( ~x ) [ π ( ~x ) − ∇ .φ ( ~x )] = 0 . (22)Since ν U and ν Σ are arbitrary functions, we obtainthe constraints Ω = ∇ · ~p + g ( ∇ · ~ Σ − ∇ A ) = 0 (23)and Ω = π − ∇ · φ = 0 . (24)According to the symplectic algorithm, the con-straints (23) and (24) are introduced in the La-grangian density by using the Lagrangian multipli-ers. Thus, the first iterated Lagrangian density iswritten as L (1) = − ~p · ˙ ~U + ~φ · ˙ ~ Σ + π µ ˙ A µ + ˙ λ Ω + ˙ λ Ω − V (1) , (25)where λ and λ are the Lagrangian multipliers, andthe first iterated symplectic potential density is V (1) = V (0) (cid:12)(cid:12)(cid:12) Ω=0 , Ω=0 = − ~π · ~ Σ − ~p −
12 ( ∇ × ~U ) −
12 (1 + g )( ~ Σ − ∇ A ) −
12 (1 + g )( ∇ × ~A ) + ~φ · ( ∇ × ∇ × ~A ) . (26)It should be noted that when the constraints Ω and Ω are imposed the dependence in U and Σ disap-pears, once the terms in U and Σ were incorpo-rated in the term introduced to the Kinetic part,which was done by redefining the Lagrange multi-pliers.From the above Lagrangian we have the follow-ing set of simplectic variables defined by ξ (1) =( U k , p k ; A k , π k , A , π ; Σ k , φ k ; λ , λ ), with the newcanonical one-form defined by U a (0) k = − p k ; Σ a (0) k = φ k ; A a (0) k = − π k ; A a (0) = π ; λ a (0) = Ω; λ a (0) = Ω . (27)Hence, the first iterated symplectic matrix is writtenas f (1) ij ( ~x, ~y ) = (cid:18) A ij B j,y − B Ti,x G ij (cid:19) δ ( ~x − ~y ) (28)where A ij = δ ij − δ ij − δ ij
00 0 δ ij , B j,y = ∂ yj
00 0 0 0 00 0 0 0 0 − − g∂ y , G ij = − δ ij g∂ yj δ ij − ∂ yj − g∂ xi − ∂ xi , (29) and we can see that f (1) ij is a singular matrix. Fromthis result, we can determine its zero-mode as being¯ ν α = (¯ ν Ui , , , , ¯ ν A , , ¯ ν Σ i , ¯ ν φi , ¯ ν λ , ¯ ν λ ) , (30)where ¯ ν Ui = ∂ i ¯ ν λ ; ¯ ν Σ i = ∂ i ¯ ν λ ; ¯ ν φi = − g∂ i ¯ ν λ ; ¯ ν A = − ¯ ν λ and ¯ ν λ , ¯ ν λ are arbitraryfunctions. Thus, from this zero-mode in Eq. (30)we have that Ω = Z d ~x h ¯ ν Ui ( ~x ) δδU i ( ~x ) + ¯ ν Σ i ( ~x ) δδ Σ i ( ~x )+ ¯ ν A ( ~x ) δδA ( ~x ) + ¯ ν φi ( ~x ) δδφ i ( ~x ) i Z d ~y V (1) ( ~y )= Z d ~x ¯ ν λ i ( ~x )[ ∇ .~π ( ~x )] = 0 . (31)Once again, as ¯ ν λ is an arbitrary function, weobtain a new set of constraint relations given by Ω = ∇ · ~π = 0 . (32)Now, following the FJ method, the second-iterated Lagrangian can be written as L (2) = − ~p. ˙ ~U + ~φ. ˙ ~ Σ+ π µ ˙ A µ + ˙ λ Ω+ ˙ λ Ω+ ˙ λ Ω − V (2) , (33)where V (2) = V (1) (cid:12)(cid:12)(cid:12) Ω=0 = V (1) . (34)From the above Lagrangian we can find the followingcanonical non-zero one-form U a (0) k = − p k ; Σ a (0) k = φ k ; A a (0) k = − π k ; (35) A a (0) = π ; λ a (0) = Ω; λ a (0) = Ω; λ a (0) = Ω , which leads to the corresponding third-iterated sym-plectic matrix, f (2) ij ( ~x, ~y ) = (cid:18) A ij ¯B j,y − ¯B Ti,x ¯G ij (cid:19) δ ( ~x − ~y ) (36)where A ij has the same expression given in (29), and ¯B j,y = ∂ yj ∂ yj − − g∂ y , ¯G ij = − δ ij g∂ yj δ ij − ∂ yj − g∂ xi − ∂ xi , (37)and once again, we can see that f (2) is singular andthe zero-mode associated with this matrix is¯¯ ν α = (¯ ν α , ¯¯ ν λ ) , (38)where ¯ ν α has the same expression given by Eq. (30).However, the zero-mode ¯¯ ν α generates the constraint Ω again, the zero-mode does not generate any newconstraints and, consequently, the symplectic matrixremains singular. It characterizes the theory as agauge theory.In order to obtain a regular symplectic matrix agauge fixing term must be added to the theory. Thechoice of this condition can be suggested by manyreasons, the most important being the simplificationthat it may introduce in the theory. In the Maxwelltheory, the condition usually employed to gauge fix-ing is the Coulomb gauge A = 0 , ∇ · ~A = 0 . (39)However, concerning the theory described by the La-grangian in (7), where a charged compressible fluidis immersed in an electromagnetic field, the condi-tion (39) is not sufficient to promote a gauge fixing,to do that we need an extra condition. In this case,an appropriate choice is the “Lorentz Gauge” to acompressible fluid [7], where ∂ α U α = 0 or ∇ · ~U + Γ = 0 , (40)which is directly related to the condition relative tothe compressibility of the fluid [7]. Thus, consideringEqs. (39) and (40) with gauge fixing conditions ¯Ω = ∇ · ~A, and ¯Ω = ∇ · ~U + Γ , (41)we them obtain a new Lagrangian density L (3) = − ~p · ˙ ~U + ~φ · ˙ ~ Σ − ~π · ˙ ~A + ˙ λ ( ∇ · ~p + g ∇ · ~ Σ) + ˙ λ ( π − ∇ · ~φ ) + ˙ λ ( ∇ · ~π ) + ˙ λ ( ∇ · ~A )+ ˙ λ ( ∇ · ~U + Γ ) − V (3) , (42)where V (3) = − ~π · ~ Σ − ~p −
12 ( ∇ × ~U ) −
12 (1 + g ) ~ Σ + 12 (1 + g ) ~A · ( ∇ ~A ) − ~φ · ( ∇ ~A ) (43) is associated with the symplectic variables ξ (3) =( U k , p k ; A k , π k , π ; Σ k , φ k ; λ , λ , λ , λ , λ ). Fromthe expression for the potential V (3) we can see whattheory’s dynamical variables are. They are partof the canonical set ( p k , U k ), ( π k , A k ) and ( φ k , Σ k ).The new canonical one-form is defined by U a (0) k = − p k ; Σ a (0) k = φ k ; A a (0) k = − π k ; (44) λ a (0) = ∂ i p i + g∂ i Σ i ; λ a (0) = π − ∂ i φ i ; λ a (0) = ∂ i π i ; λ a (0) = ∂ i A i ; λ a (0) = ∂ i U i + Γ . These relations can lead us to the correspondingthird-iterated symplectic matrix f (3) ij ( ~x, ~y ) = ˜A ij ˜B ij,y − ˜B Tji,x ˜G ij ! δ ( ~x − ~y ) (45)where ˜A ij = δ ij − δ ij − δ ij δ ij , ˜B ij,y = ∂ yj ∂ yj ∂ yj
00 0 0 ∂ yj − δ ij g∂ yj , ˜G ij = − ∂ yj ∂ xi . (46)We can observe that f (3) ij is not singular, thereforewe can construct its inverse. The inverse of f (3) ij iscalled the symplectic tensor( f (3) ij ) − = δ ij − ∂ i ∂ j ∇ ∂ i ∇ − δ ij + ∂ i ∂ j ∇ g∂ i gδ ij ∂ i ∇ − δ ij + ∂ i ∂ j ∇ ∂ i ∇
00 0 δ ij − ∂ i ∂ j ∇ ∂ i ∇ − g∂ j ∂ i − − g − ∂ j δ ij ∂ i ∇ − gδ ij − δ ij g ∂ i ∇ − ∂ j ∇ − ∇ − ∂ j ∇ ∇
00 0 − ∂ j ∇ − ∇ − ∂ j ∇ g − g ∂ j ∇ ∇ δ (3) ( ~x − ~y ) . (47)Moreover, we can relate λ = U , λ = Σ and λ = A . In this way, from (47) it is possible to identifythe following FJ’s generalized brackets given by { A i ( ~x ) , π j ( ~y ) } = (cid:18) − δ ij + ∂ i ∂ j ∇ (cid:19) δ ( ~x − ~y ) { Σ i ( ~x ) , φ j ( ~y ) } = δ ij δ ( ~x − ~y ) , { U i ( ~x ) , p j ( ~y ) } = (cid:18) δ ij − ∂ i ∂ j ∇ (cid:19) δ ( ~x − ~y ) (48)and { p i ( ~x ) , φ j ( ~y ) } = gδ ij δ ( ~x − ~y ) , { p i ( ~x ) , π ( ~y ) } = g∂ i δ ( ~x − ~y ) , (49)where the Dirac brackets for the electromagneticfields correspond to the one we have found in previewworks [7, 14], as well as the Dirac brackets for thehigher-derivative terms in the electromagnetic fields,Σ and φ . Besides, we have found the Dirac bracketsfor the dynamic variables of the compressible fluid p and U , the last of Eqs.(48). Finally, as a result of thecoupling between the charged compressible fluid andthe electromagnetic field we found two Dirac brack-ets between the fluid and electromagnetic fields, Eqs. (49), which are both zero when there is no couplingbetween them. IV. CONCLUSIONS.
In this paper we have analyzed the gauge invari-ance of the theory which describes the charged com-pressible fluid interacting with an electromagneticfield (7) by using the Faddeev-Jackiw method. Wehave found the constraints, the gauge transforma-tions and we have obtained the generalized FJ brack-ets.
Acknowledgments
The authors thank CNPq (Conselho Nacional deDesenvolvimento Cient´ıfico e Tecnol´ogico), Brazil-ian scientific support federal agency, for partial fi-nancial support, Grants numbers 302155/2015-5,302156/2015-1 and 442369/2014-0 and E.M.C.A.thanks the hospitality of Theoretical Physics De-partment at Federal University of Rio de Janeiro(UFRJ), where part of this work was carried out. [1] E. T. Whittaker,
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