Scalar QC D 4 on the null-plane
aa r X i v : . [ h e p - t h ] A ug Scalar
QC D on the null-plane R. Casana ∗ , B.M. Pimentel † and G. E. R. Zambrano ‡§ Departamento de F´ısica, Universidade Federal do Maranh˜ao (UFMA),Campus Universit´ario do Bacanga, CEP 65080-040, S˜ao Lu´ıs - MA, Brasil. Instituto de F´ısica Te´orica (IFT/UNESP), UNESP - S˜ao Paulo State UniversityRua Pamplona 145, CEP 01405-900, S˜ao Paulo, SP, Brazil
Abstract
We have studied the null-plane hamiltonian structure of the free Yang-Mills fields and thescalar chromodynamics (
SQCD ). Following the Dirac’s procedure for constrained systems we haveperformed a detailed analysis of the constraint structure of both models and we give the generalizedDirac brackets for the physical variables. In the free Yang-Mills case, using the correspondenceprinciple in the Dirac’s brackets we obtain the same commutators present in the literature. To quantize the theory on the null-plane, initial conditions on the hyperplane x + = cte and equal x + -commutation relations must be given and the hamiltonian must describe the time evolution from aninitial value surface to other parallel surface that intersects the x + -axis at some later time. Althoughthe prescription has a lot of similarities with the conventional approach there are significant differenceswhen we perform the quantization of the theory. Inside the null-plane framework, the lagrangian whichdescribes a given field theory is singular and at least second class constraints appear, these can beeliminated by constructing Dirac’s brackets (DB) and the theory can be quantized, via correspondenceprinciple, in terms of a reduced number of independent fields, the physical ones. Thus, the Dirac’smethod [1] allows built the null-plane hamiltonian and the canonical commutation relations in termsof the independent fields of the theory.The quantization of relativistic field theory at the null plane time, proposed by Dirac [2], has foundimportant applications [3] in both gauge theories and string theory [4]. It is interesting to observethat the null-plane quantization of a non-abelian gauge theory using the null-plane gauge condition, A − , identified the transverse components of the gauge field as the degrees of freedom of the theoryand, therefore, the ghost fields can be eliminated of the quantum action [5].In [6], Tomboulis has quantized the massless Yang-Mills field in the null-plane gauge A a − = 0and derived the Feynman rules. However, in [7], McKeon has shown that the null-plane quantizationof this theory leads a set of second-class constraints in addition to the usual first-class constraints,characteristics of the usual instant form quantization, what implies in the introduction of additional ∗ [email protected] † [email protected] ‡ [email protected] § On leave of absence from Departamento de F´ısica, Universidad de Nari˜no, San Juan de Pasto, Nari˜no, Colombia A a + = 0, such gaugeprovides a generating functional for the renormalized Green’s functions that takes to the Mandelstan-Leibbrandt’s prescription for the free gluon propagator.On the other hand, in [9], Neville and Rohlich have studied the scalar electrodynamics and haveobtained the commutation relations between free fields from the commutations relations of the freefield operators at unequal times but the commutation relation representing the interaction was notcomputed but they affirmed to be derived solving a quantum constraint. This last commutationrelation was determined in [10], Casana, Pimentel and Zambrano have calculated all the commutationrelations following a careful analysis of the constraint structure of the theory and the results obtainedare consistent with the specified in the literature [9].In this paper we will discuss firstly the null-plane structure of the pure Yang-Mills fields and afterits interaction with a scalar complex field ( SQCD ) following the Dirac’s formalism for constrainedsystems. The Hamiltonian analysis follows the spirit outlined in [10]. The work is organized as follow:In the section 2, we study the free Yang-Mills field, being its constrained structure analyzed in detail,thus, we classify the constraints and the appropriated equations of motion of the dynamical variablesare determined by using the extended hamiltonian. The null-plane gauge is imposed to transform theset of first class constraints in second class one and, the Diracs’s brackets (DB) among the independentfields are obtained by choosing appropriate boundary conditions on the fields. In the section 3, theconstraint structure of the scalar chromodynamics ( SQCD ) is analyzed, the set of constraints isclassified and the correct equations of motion are checked by using extended hamiltonian as thegenerator of temporal evolution. Next, we invert the second class matrix by imposing appropriatedboundary conditions on the fields and we calculate the DB among the fundamental dynamical variables.Finally, we give our conclusions and remarks. For any semi-simple Lie group with structure constant f abc the Yang-Mill lagrangian density is L = − F µνa F aµν , (1)with F aµν = ∂ µ A aν − ∂ ν A aµ + gf abc A bµ A cν , the gauge index a, b, c runs from 1 to n . Such lagrangian isinvariant under the following infinitesimal gauge transformations δA µa ( x ) = f abc Λ b ( x ) A µc ( x ) + 1 g ∂ µ Λ a ( x ) . (2)with Λ a ( x ) a infinitesimal function.In the present work, we specialize for convenience to the SU (2) gauge group that only has threegenerators and f abc = ε abc , where ε abc is the Levi-Civita totally antisymmetric tensor in three dimen-sions, thus, we can define everything in such way that we can forget about raising and lowering groupindexes. From (1) we find the Euler-Lagrange equations( D ν ) ab F νµb = 0 , (3)where we have defined the covariant derivative ( D ν ) ab ≡ δ ab ∂ ν − gε abc A cν .2 .1 Structure Constraints and Classification In the null-plane dynamics, the canonical conjugate momenta are π µa ≡ ∂ L ∂ (cid:0) ∂ + A aµ (cid:1) = − F + µa , (4)this equation gives the following set of primary constraints φ a ≡ π + a ≈ , φ ka ≡ π ka − ∂ − A ak + ∂ k A a − − gε abc A b − A ck ≈ . (5)and the dynamical relation for A a − π − a = ∂ + A a − − ∂ − A a + − gε abc A b + A c − , (6)At once, the canonical hamiltonian is given by H C = Z d y H C = Z d y (cid:26) (cid:0) π − a (cid:1) + π − a (cid:0) D x − (cid:1) ab A b + + π ia ( D xi ) ab A b + + 14 (cid:0) F aij (cid:1) (cid:27) . (7)Following the Dirac procedure [1], we define the primary hamiltonian adding to the canonical hamil-tonian the primary constraints H P = Z d y (cid:26) (cid:0) π − a (cid:1) + π − a (cid:0) D x − (cid:1) ab A b + + π ia ( D xi ) ab A b + + 14 (cid:0) F aij (cid:1) + u b φ b + λ bl φ lb (cid:27) (8)where u b and λ bl are their respective Lagrange multipliers.The fundamental Poisson brackets ( P B ) among fields are (cid:8) A aµ ( x ) , π νb ( y ) (cid:9) = δ νµ δ ab δ ( x − y ) . (9)Requiring that H P is the generator of temporal evolutions, the consistency condition of the primaryconstraints, i.e. { φ, H P } = 0, give us for φ a { φ a ( x ) , H P } = (cid:0) D x − (cid:1) ab π − b + ( D xi ) ab π ib ≡ G a ( x ) ≈ , (10)a genuine secondary constraint, which is the Gauss’s law. Also, for φ ka we obtain n φ ka ( x ) , H P o = ( D xk ) ab F b + − + ( D xi ) ab F bik − (cid:0) D x − (cid:1) ab λ bk ≈ , (11)a differential equation which allows to compute λ bk after imposition of appropriated boundary condi-tions. The consistency condition of the secondary constraint yields { G a ( x ) , H P } = gε acb A c + ( x ) G b ( x ) ≈ , (12)the Gauss’s law is automatically conserved. Then, there are not more constraints and the equations(5) and (10) give the full set of constraints.The set of first class constraints is { π + a , G a } and the set of second class constraints is (cid:8) φ ka (cid:9) whosePB’s are n φ ka ( x ) , φ lb ( y ) o = − δ lk (cid:0) D x − (cid:1) ab δ ( x − y ) (13)3 .2 Equations of motion Now we check the equations of motion. The time evolution of the fields is determined by computingtheir PB’s with the so called extended hamiltonian H E , which is obtained by adding to the primaryhamiltonian all the first class constraints of the theory:H E = Z d y (cid:26) (cid:0) π − b (cid:1) + π − b (cid:0) D y − (cid:1) bc A c + + π ib ( D yi ) bc A c + + 14 (cid:16) F bij (cid:17) + λ bl φ lb + u b φ b + v b G b (cid:27) , (14)thus, we have the time evolution of the dynamical variables, i.e, ˙ φ = { φ, H E } , gives˙ A a + = u a (15)˙ A a − = π − a + (cid:0) D x − (cid:1) ac A c + − (cid:0) D x − (cid:1) ab v b (16)˙ A ak = ( D xk ) ac A c + + λ ak − ( D xk ) ab v b (17)˙ π + a = G a (18)˙ π − a = − gε abc π − b A c + + ( D xl ) ab λ bl − gε bca v b π − c (19)˙ π ka = − gε bca π kb A c + + (cid:0) D xj (cid:1) ab F bkj − (cid:0) D x − (cid:1) ab λ bk − gε abc π kc v b , (20)if we demand consistence with the Euler-Lagrange equation of motion (3) we must to choose v b = 0,however, the multiplier u a remains indeterminate.The Dirac’s algorithm requires of as many gauge conditions as first class constraints there are,nevertheless these conditions should be compatible with the Euler-Lagrange equations and togetherwith the first class set they should form a second class set, in such way that the Lagrange multipliers,corresponding to the first class set, are determined. Under such considerations, we choose as the firstgauge condition A a − ≈ , (21)whose consistency condition ˙ A a − = (cid:8) A a − , H E (cid:9) ≈ b = 0 in (16) then the Eq.(21) will hold for all times only if π − a + ∂ x − A a + ≈ , (22)therefore, the equations (21) and (22) constitute our gauge conditions on the null-plane and they areknown as the null-plane gauge. The gauge fixing conditions transform the first class set into second class one, the following stage in theDirac’s procedure is to transform the second class constraints in strong identities. This demands analteration of the canonical brackets (PB) to form a new brackets set, the Dirac’s brackets (DB), withwhich the second class constraints are automatically satisfied. Thus, the prescription for determinethe DB implies in calculating the inverse of the second class matrix, for this purpose, we rename thesecond class constraints asΘ ≡ π + a , Θ ≡ (cid:0) D x − (cid:1) ab π − b + ( D xi ) ab π ib Θ ≡ A a − , Θ ≡ π − a + ∂ x − A a + (23)Θ ≡ π ka − ∂ x − A ak + ∂ xk A a − − gε abc A b − A ck , F ab ( x, y ) ≡ { Θ a ( x ) , Θ b ( y ) } . With theseconsiderations, the Dirac’s bracket of two dynamical variables, A a ( x ) and B b ( y ), is then defined as n A a ( x ) , B b ( y ) o D = n A a ( x ) , B b ( y ) o − Z d ud v n A a ( x ) , Θ c ( u ) o (cid:0) F − (cid:1) cd ( u, v ) n Θ d ( v ) , B b ( y ) o , (24)where F − is the inverse of the constraint matrix.The explicit evaluation of F − involve the determination of an arbitrary function of the variables x + and x ⊥ [10] which can be fixed by considering appropriate boundary conditions [11] on the fields A aµ eliminating the ambiguity in the definition of the inverse of the operator ∂ − related to theirzero modes that give origin to hidden subset of first class constraints which generate improper gaugetransformations [12] what is characteristic of the null-plane constraint structure [13]. Thus, from (24)we obtain the DB among the independent variables of the theory n A ak ( x ) , A bl ( y ) o D = − δ ab δ lk ǫ ( x − y ) δ ( x ⊺ − y ⊺ ) (25) n A ak ( x ) , A b + ( y ) o D = 14 | x − y | ( D xk ) ab δ ( x ⊺ − y ⊺ ) . At once, via the correspondence principle we obtain the commutators among the fields h A ak ( x ) , A bl ( y ) i = − i δ ab δ lk ǫ ( x − y ) δ ( x ⊺ − y ⊺ ) , (26) h A ak ( x ) , A b + ( y ) i = i | x − y | ( D xk ) ab δ ( x ⊺ − y ⊺ ) . (27)The first relationship is exactly that obtained by Tomboulis [6]and starting from it is possible tocalculate the other two expressions determined by him, meanwhile the equation (27) is our contributionto the commutation relations. SQC D The model describing the interaction of Yang-Mills and complex scalar field is given the followinglagrangian density L = η µν ( D µ ) ab Φ † b ( D ν ) ac Φ c − m Φ † a Φ a − F µνa F aµν , (28)where the field strength F aµν and the covariant derivative ( D µ ) ab are defined in the SU (2) adjointrepresentation by F aµν = ∂ µ A aν − ∂ ν A aµ + gε abc A bµ A cν , ( D µ ) ab ≡ δ ba ∂ µ − gε abc A cµ , (29)respectively. Φ c is the complex scalar field which has three components in an internal space and thegauge transformation are rotations in this space what gives a conserved vector quantity named isospin .The field equations are given for ( D ν ) ab F νµb = J µa ( D µ ) ab ( D µ ) bc Φ c + m Φ a = 0 (30)( D µ ) ab ( D µ ) bc Φ † c + m Φ † a = 0 , where J βh is the current density defined by J µa ≡ gε abc nh ( D µ ) bd Φ † d i Φ c + Φ † c h ( D µ ) bd Φ d io . (31)5 .1 Structure Constraints and Classification The canonical conjugate momenta of the gauge field is π µa ≡ ∂ L ∂ (cid:0) ∂ + A aµ (cid:1) = − F + µa , (32)and for the fields Φ a , Φ † a areΠ † a ≡ ∂ L ∂ ( ∂ + Φ a ) = ( D − ) ab Φ † b , Π a ≡ ∂ L ∂ (cid:16) ∂ + Φ † a (cid:17) = ( D − ) ab Φ b (33)respectively.From (32) we get one dynamical relation for A a − π − a = ∂ + A a − − ∂ − A a + − gε abc A b + A c − , (34)and the following set of primary constraints for the gauge sector φ a ≡ π + a ≈ , φ ka ≡ π ka − ∂ − A ak + ∂ k A a − − gε abc A b − A ck ≈ , (35)and from (33) we obtain a set of primary constraints of the scalar sectorΘ a ≡ Π a − ( D − ) ab Φ b ≈ , Θ † a ≡ Π † a − ( D − ) ab Φ † b ≈ . (36)The canonical hamiltonian is H C = Z d y (cid:26) (cid:0) π − b (cid:1) + π − b ( D − ) bc A c + + π kb ( D k ) bc A c + + J + c A c + (cid:27) (37)+ Z d y (cid:26)h ( D xk ) ab Φ † b i h ( D xk ) ad Φ d i + m Φ b Φ b + 14 (cid:16) F bjk (cid:17) (cid:27) and the primary hamiltonian is H P = H C + Z d y n u b φ b + λ bk φ kb + U † b Θ b + Θ † b U b o , (38)where u b and λ bk are the Lagrange multipliers associated to the vector constraints and U † b and U b arethe multipliers associated with the scalar ones.The fundamental Poisson brackets are (cid:8) A aµ ( x ) , π νb ( y ) (cid:9) = δ νµ δ ab δ ( x − y ) (39) n Φ a ( x ) , Π † b ( y ) o = δ ba δ ( x − y ) , n Φ † a ( x ) , Π b ( y ) o = δ ba δ ( x − y ) , (40)and the non null PB’s among the primary constraints n φ ka ( x ) , φ lb ( y ) o = − δ lk (cid:0) D x − (cid:1) ab δ ( x − y ) , (41) n Θ a ( x ) , Θ † b ( y ) o = − (cid:0) D x − (cid:1) ab δ ( x − y ) . Following the Dirac’s procedure, we compute the consistence condition of every primary constraint.Thus, the consistence condition of the scalar constraints yields:˙Θ a = − gε abc π − b Φ c − gε bcd ( D − ) ab (cid:2) Φ d A c + (cid:3) + ( D k ) ab ( D k ) bc Φ c − m Φ a − D − ) ab U b (42)˙Θ † a = − gε abc π − b Φ † c − gε bcd ( D − ) ab h Φ † d A c + i + ( D k ) ab ( D k ) bc Φ † c − m Φ † a − D − ) ab U † b , U b and U † b , respectively. In this way, there are notmore constraints associated with the scalar sector.In the gauge sector, the consistency condition of φ ka provides˙ φ ka = ( D k ) ab π − b + ( D j ) ab F bjk − J ka − D − ) ab λ bk ≈ λ bk . Finally, the consistence condition of π + a contributes with a secondary constraint˙ φ a = ( D − ) ab π − b + ( D i ) ab π ib − J + a ≡ G a ≈ , (44)which is the Gauss’s law for the scalar chromodynamics. After a laborious work, it is possible toverify that no more further constraints are generated from the consistence condition of the Gauss’ lawbecause it is automatically conserved˙ G a = gε abc h Φ † c ˙Θ b + Φ c ˙Θ † b i ≈ . (45)Therefore, the equation (35), (36) and (44) constitute the full set of constraints of the theory.The non null PB’s among the constraints of the theory are n φ ka ( x ) , φ lb ( y ) o = − δ lk (cid:0) D x − (cid:1) ab δ ( x − y ) , n Θ † a ( x ) , Θ b ( y ) o = − (cid:0) D x − (cid:1) ab δ ( x − y ) , (46) n G a ( x ) , Θ † b ( y ) o = − gε acf Φ † f ( x ) (cid:0) D x − (cid:1) cb δ ( x − y ) , { G a ( x ) , Θ b ( y ) } = − gε acf Φ f ( x ) (cid:0) D x − (cid:1) cb δ ( x − y ) , thus, it is easy to note that π + a is vanishing PB with all the other constraints, therefore, it is a firstclass constraint. The remaining set, n φ ka , Θ a , Θ † a , G a o , is apparently a second class set, however, itis possible to show that their constraint matrix is singular and its zero mode eigenvector provides alinear combination of constraints which is a first class constraint [10]. Such second first class constraintis Σ a ≡ G a − gε abc h Φ † c Θ b + Φ c Θ † b i . (47)Then, the first class constraints set is φ a = π + a , Σ a = G a − gε abc h Φ † c Θ b + Φ c Θ † b i . (48)it is the maximal number of first class constraints and, the second class set is φ ka = π ka − ∂ − A ak + ∂ k A a − − gε abc A b − A ck , Θ a = Π a − ( D − ) ab Φ b , (49)Θ † a = Π † a − ( D − ) ab Φ † b . At this point we need to check that we have the correct equation of motion. The time evolution ofthe fields is determined by computing their PB with the extended hamiltonian which is defined as H E = H C + Z d y n u b φ b + λ bl φ lb + U † b Θ b + Θ † b U b + w b Σ b o (50)7hus, the time evolution of the gauge field yields˙ A a + = u a ˙ A a − = π − a + ( D − ) ac A c + − ( D − ) ab w b ˙ A ak = ( D k ) ab A b + + λ ak − ( D k ) ab w b ˙ π + a = G a ˙ π − a = gε abc π − b A c + + 2 g A a + Φ † b Φ b − g A b + Φ † a Φ b − g A b + Φ † b Φ a (51)+ ( D k ) ab λ bk − gε abc U † b Φ c − gε abc Φ † c U b + gε bca w b π − c ˙ π ka = gε abc π kb A c + − J ka + ( D j ) ab F bjk − ( D − ) ab λ bk + gε abc w b π kc and for the scalar fields the dynamics is given for˙Φ a = U a + gε abc w b Φ c , ˙Φ † a = U † a + gε abc w b Φ † c , ˙Π a = gε cde ( D − ) ad (cid:2) A c + Φ e (cid:3) − gε abc A b + h ( D − ) cd Φ d i + ( D k ) ab h ( D k ) bc Φ c i (52) − m Φ a − ( D − ) ab U b − gε abc w b Φ † c , ˙Π † a = − gε abc A b + h ( D − ) cd Φ † d i + gε bcd ( D − ) ac h A b + Φ † d i + ( D k ) ab h ( D k ) bc Φ † c i − m Φ † a − ( D − ) ab U † b + gε abd w b Π † d , We can note that the set of equation (51) and (52) only will be compatible with the Euler-Lagrangeequations (30) if we set w b = 0 however the multiplier u a still remains undetermined in this way theDirac’s formalism tell us to impose one set of gauge conditions, one for every first class constraint.The gauge conditions are chosen in such a way that they are compatible with the Euler-Lagrangeequations of motion, thus one such set is the null-plane gauge conditions is given by relations (21) and(22). We have the following set of second class constraints:Ψ ≡ π + a , Ψ ≡ G a − gε abc (cid:16) Φ † c Θ b + Φ c Θ † b (cid:17) Ψ ≡ A a − , Ψ ≡ π − a + ∂ − A a + (53)Ψ ≡ π ka − ∂ − A ak + ∂ k A a − − gε abc A ck A b − Ψ ≡ Π a − ( D − ) ab Φ b , Ψ ≡ Π † a − ( D − ) ab Φ † b , with these, we define the following constraint matrix M ab ( x, y ) ≡ { Ψ a ( x ) , Ψ b ( y ) } , from where theDB for the dynamical variables are determined via evaluation of the inverse of this matrix.Then, byconsidering appropriate boundary conditions on the fields, [10, 11], a unique inverse of the constraintmatrix is obtained and after a laborious work we obtain the DB for the independent dynamical8ariables of scalar chromodynamics: n A ak ( x ) , A bl ( y ) o D = − δ ab δ lk ǫ ( x − y ) δ ( x ⊺ − y ⊺ ) n Φ a ( x ) , Φ † b ( y ) o D = − δ ba ǫ ( x − y ) δ ( x ⊺ − y ⊺ ) (54) n Φ a ( x ) , A b + ( y ) o D = g ε abc δ ( x ⊺ − y ⊺ ) (cid:26) Φ c ( x ) | x − y | − Z dv Φ c ( v ) ǫ ( x − v ) ǫ ( v − y ) (cid:27)n Φ † a ( x ) , A b + ( y ) o D = g ε abc δ ( x ⊺ − y ⊺ ) (cid:26) Φ † c ( x ) | x − y | − Z dv Φ † c ( v ) ǫ ( x − v ) ǫ ( v − y ) (cid:27) In this work we have studied the null plane Hamiltonian structure of the free Yang-Mills field and itsinteraction with a complex scalar that we named as scalar chromodynamics (
SQCD ).Performing a careful analysis of the constraint structure of Yang-Mills field, we have determinedin addition of the usual first class constraints set a second class ones set, which is a characteristic ofthe null-plane dynamics [10]. The imposition of appropriated boundary conditions on the fields fixesthe hidden subset of first class constraints [13] and eliminates the ambiguity on the operator ∂ − , thatallows to get a unique inverse for the second class constraint matrix [10]. The Dirac’s brackets of thetheory are quantized via correspondence principle; the commutators obtained are the same derived byTomboulis [6].The scalar chromodynamics SQCD hamiltonian analysis has shown further of the free Yang-Mill structure, a contribution of the scalar sector with an additional constraints set. However, as aconsequence of the constraint associated with the scalar part, one of the first class constraints is a linearcombination of the Gauss’ law with the scalar constraints, in a similar way to the scalar electrodynamicscase [10], such first class constraint is given by the zero mode eigenvector of the constraint matrix.Finally, choosing the null-plane gauge condition, which transforms first class constraints in secondclass ones and imposing appropriated boundary conditions on the fields to get a unique inverse of thesecond class constraints matrix and following the Dirac’ procedure we obtain the Dirac’ brackets ofthe canonical variables of the theory. Our results are consistent with those reported in the literature[9, 10] when the abelian case is considered.As the null-plane hamiltonian structure is well-defined, the null-plane quantization, of the modelsreported here and [10], via the path-integral formalism are now in advanced and whose result will bereported elsewhere. Acknowledgements
RC thanks to CNPq for partial support, BMP thanks CNPq for partial support and GERZ thanksCNPq (grant 142695/2005-0) for full support.
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