First class models from linear and nonlinear second class constraints
Mehdi Dehghani, Maryam Mardaani, Majid Monemzadeh, Salman Abarghouei Nejad
aa r X i v : . [ h e p - t h ] S e p First class models from linear and nonlinearsecond class constraints
Mehdi Dehghani ∗ Department of Physics, Faculty of Science,Shahrekord University, Shahrekord, P. O. Box 115, I. R. Iran
Maryam Mardaani
Department of Physics, University of Kashan, Kashan, Iran.
Majid Monemzadeh † Department of Physics, University of Kashan, Kashan, Iran.
Salman Abarghouei Nejad
Department of Physics, University of Kashan, Kashan, Iran.
Abstract
Two models with linear and nonlinear second class constraints areconsidered and gauged by embedding in an extended phase space.These models are studied by considering a free non-relativisticparticle on the hyper plane and hyper sphere in the configura-tion space. The gauged theory of the first model is obtained byconverting the very second class system to the first class one, di-rectly. In contrast, the first class system related to the free particleon the hyper sphere is derived with the help of the infinite BFTembedding procedure. We propose a practical formula, based onthe simplified BFT method, which is practical in gauging linearand some nonlinear second class systems. As a result of gaug-ing these two models, we show that in the conversion of secondclass constraints to the first class ones, the minimum number ofphase space degrees of freedom for both systems is a pair of phasespace coordinates. This pair is made up of a coordinate and itsconjugate momentum for the first model, but the correspondingPoisson structure of the embedded non-relativistic particle on hy-per sphere is a non-trivial one. We derive infinite correction termsfor the Hamiltonian of the nonlinear constraints and an interact-ing gauged Hamiltonian is constructed by summing over them. At ∗ [email protected], [email protected] † [email protected] the end, we find an open algebra for three first class objects of theembedded nonlinear system. It seems that first and second class constrained systems are physicallydifferent. Although a first class constrained system is a gauge system,the second class one must be reduced to a physical system with physicaldegrees of freedoms. Unlike the second class systems, first class ones aremore convenient to handle in the process of canonical quantization.First class constraints, in the quantized version, acts on all statesto select physical ones from many copies in the Hilbert or Fock space.On the other hand, second class constraints, when they are in the lin-ear combinations of original phase space variables, can be removed inorder to find physical phase space and construct Hilbert space in thequantization procedure.In order to quantize both systems, one must obtain the reducedphase space. The reduced phase space of a first class system is theselection of a plenty of similar points in the phase space that all ofthem satisfy first class identities. This selection is done with the helpof other identities, considered by a gauge fixer. Gauge fixing conditionsconvert the set of first class constraints to second class one. In such away, some degrees of the primary model are removed. In contrast, forsecond class systems, the reduced phase space is constructed by removingnon-physical degrees of freedom corresponding to those constraints. Forboth first and second class systems the removing procedure is done bycalculating Dirac brackets.The gauge fixing process in theory of constrained systems is the keypoint to convert a second class model to a gauge model directly [1] orconceptually [2, 3, 4]. One may imagine a set of second class constraintsas a set of first class one and their gauge fixing conditions. The Diracbracket removes additional degrees of freedom. Thus, in the reverseway, by adding some degrees of freedom to the second class functions,i.e. embedding the model in a extended phase space, we can convertthem to a set of first class constraints. Because the decomposition of aset of second class constraints into the first class one and gauge fixingconditions can be done in many ways, extracting gauge theory from asecond class system has more than one solution. In this article, we seethis point in the conversion of two sets of constraints. Although forthe conversion of linear constraints we use a direct gauging process, foranother one the famous method BFT [2, 3, 4] rewritten for Abelian andnon-Abelian systems is used [5].Somehow in a vise versa scheme, say converting first class constraintsto second class ones, people manipulate Hamiltonian constrained sys-tems to enhance their symmetries and do quantization by modern andmore mathematical technique named BRST method and de Rham co-homology [6]. In this manner the phase space spreads by new variableswith opposite Grassmannian number with respect to primary variables.See for a more related model to paper[7], which SU (3) linear sigmamodel is considered.In the present article we make a comparison between gauging a the-ory with linear constraints and gauging a theory with nonlinear ones.In this manner, in section (2) we directly calculate the gauged model ofa free non-relativistic particle on a hyper plane. Although this modeland its results are a part of our comparison, it teaches us the con-cept of gauging and embedding in a simple way. Section (3), whichis the main part of current paper, is somehow a revision of gaugingthe Skyrme model or related models such as nonlinear sigma ( O ( N ) in-variant) model, done with the help of the BFT embedding formalism.Most papers about these models are focused on the consistent canonicalquantization and their quantum spectrum. This family of models wereconsidered in several approaches including: the symplectic embedding[8, 9, 10, 11], the BFT formalism [9, 12, 13, 17, 14, 15, 16], Stuckelbergfield shifting [19, 18] or mixed approaches based on first principles of themaking gauge systems [9, 18, 20, 21, 22].The problem stems from second class essence of the models, so peopletry to change it, quantize it and resile it. We want to know what kindof gauge theory can be constructed from such models, specially withthe help of the BFT method, and what is their classical characteristics.To fulfil our goal, we clean the model from mess and limit it to thedefinite degrees of freedom to see the result, clearly. Extensions to morerealistic models are not rigorous. In part (3.1) we make a brief reviewon the BFT formalism. We reduce the formula of the BFT for ourpurposes to apply it on the model of free non-relativistic particle on ahyper sphere. This model and the application of the BFT method on themodel is introduced in (3.2). We find a general form of the embeddedHamiltonian of the free particle on the hyper sphere in (3.2) and makesome comments on it in (3.3). We present the conclusions of our resultsin section (4). Consider a free non-relativistic particle that its location is described bythe coordinate q i , which is displayed by a D -dimensional array ~q . Tohave a second class constrained system we assume the particle is confinedon a hyperplane. The mass of the particle play no rule in our analysis,so we scale the momenta by the mass of particle. The dynamics of sucha system is described by the total Hamiltonian, H T = ~p.~p + λφ ,φ = ~a.~q ~a.~a = 0 . (1)Here ~a is a constant vector independent from phase space variables,which is normal of the hyperplane. with the help of this quantity wecalculate the dynamics of φ and derive secondary constraint, φ = ~a.~p. (2)We see that the consistency of hyperplane in configuration space, as aprimary constraint, leads us to another hyperplane in momenta sub-phase space with the same normal vector. The non-vanishing normalvector condition for the normal vector ~a makes primary and secondaryconstraints as the second class ones and truncates the consistency pro-cess of constraints.Now, we transform our constraints to first class ones by extendingthe phase space. By a direct sum, we paste the auxiliary variables tothe phase space and deduce the new phase space as follows,( q i , p i ) ⊕ ( Q j , P j ) , i = 1 , . . . D j = 1 , . . . d { q i , p i ′ } = δ ii ′ { Q j , P j ′ } = δ jj ′ . (3)In the extended phase space we require that the constraints be correctedby new variables, linearly. f φ = φ + ~b. ~P , f φ = φ + ~c. ~Q, (4)where ~b and ~c are two unknown vectors which are determined by twoconditions. The first one is the first class condition as, { f φ , f φ } ≈ , (5)in which, the weak equality is the equality which is defined on the cor-rected constraints surface. The second condition is deriving f φ from theconsistency of the f φ . The later also required additional corrections tothe Hamiltonian. It means that in the new phase space, the system isaffected by a potential where for simplicity we assume it as a function V = V ( ~Q ) in the new configuration space. So, we arrive to followingequations. ~b.~c = ~a.~a,~c. ~Q + ~b. ∇ ~Q V = 0 , (6)where, the ∇ ~Q is the gradient operator with respect to ~Q . It is worth-while to note that there is a third condition which implies that after theinvestigating the consistency of f φ in the new model, no other constraintmust be appeared. { φ , H c } + e λ { f φ , f φ } ≈ . (7)In above equation both terms vanish identically. Hence, no new equationemerges. The first term comes from the characteristic of the primarymodel. The second one is due to the first classiness of the new model.The set of partial differential equations (6) have many solutions. Acategory of solutions can be derived by considering ~b and ~c as constantvectors. In this way, the primary free second class system (1) convertsto the following interactive gauge system. f H c = ~p.~p − ~a ~b ~Q. ~Q f φ = ~a.~q + ~b. ~P f φ = ~a.~p + ~a.~a~b.~b ~b. ~Q. (8)We see that the chain structure in the gauged model doesn’t change,i.e. if we consider the f φ as the primary constraint, its consistency givesthe f φ and the consistency of the later vanishes, strongly. Moreover, thealgebra of the embedded constraints is an Abelian one.As the normal vector of the constraint surfaces ( ~a ) is characteristicof the linear second class constrained model, the constant vector ~b whichis incorporate with ~a , is for the gauged system. In addition, we find thesurfaces that are described by the ( f φ , f φ ) are also another hyperplanes.But despite the presence of the constraint surfaces of the primary modelin the coordinate and momentum sub-phase space, in the new model,the ( f φ , f φ ) are hyperplanes on the ~q ⊕ ~P and ~p ⊕ ~Q sub-phase spaces.Also, for primary model, normal vectors of the hyperplanes were thesame, but for the new model they are different.Moreover, There is a comment on the correction term which is addedto the Hamiltonian. A physical interpretation for this term is that inspite of the free model, its gauged model is an interactive one. Thisdescribes an oscillator with incorrect sign in the potential. One mayimagine that oscillator gives its energy from other part, according tothe minus sign. This fact is understood better if one, by a canonicaltransformation, transforms the ~Q coordinates to momenta ~P ′ .Conclusively, we can select a minimal solution by considering only apair of coordinate-momentum conjugate to gauge the free particle on ahyper surface as, f H c = ~p.~p − ~a.~aQ , f φ = ~a.~q + P, f φ = ~a.~p + ~a.~aQ. (9)For other second class constrained models, the extension of phasespace and adding correction terms to the constraints and the Hamilto-nian is not as simple as those models with linear constraints. There hasbeen existed some conversion algorithms to gauge such systems. TheBFT method is one of them which is used in the next section to convertthe simplest model which contains nonlinear second class constraints. In this section we simplify the general form of the BFT algorithm toapply it on a model with nonlinear constraints. The present problem inits general form is related to the problem of the quantization of the freeparticle on sphere that is an introduction to the fundamental problem ofthe quantization in curved space-times. We focus only on the process ofthe conversion a classical second class system to a classical first class one.In other words, we work in the realm of pre-quantization of the systemswith finite degrees of freedom and nonlinear second class constraints.Extension of the BFT formalism to infinite degrees (field) models istrivial but to the models with arbitrary nonlinearity of the constraintsis not.
The BFT algorithm, when it is applied on the systems with bosonicdegrees of freedom, essentially comes from the conditions (3,5). Poissonstructure of the adhered phase space to the primary phase space is notarbitrary but depends on the algebra of second class constraints. Inthis approach, to make a gauge theory, we have two sets of generatorsto generate correction terms for constraints and the Hamiltonian. Theyare denoted by the square matrix B and the vector G for constraints andthe Hamiltonian in the following relations, respectively. The correctionterms are computed by, φ (1) a = χ ab η b ,B (1) ab = { φ (0) a , φ (1) b } − { φ (0) b , φ (1) a } ,B ( n ) ab = n X m =0 { φ ( n − m ) a , φ ( m ) b } + n − X m =0 { φ ( n − m ) a , φ ( m +2) b } ( η ) ,φ ( n +1) a = − n + 2 η d ω − dc χ − cb B ( n ) ba , n ≥ , (10)for embedding the constraints and by G (0) a = { φ (0) a , H (0) } ,G (1) a = { φ (1) a , H (0) } + { φ (0) a , H (1) } + { φ (2) a , H (1) } η ,G ( n ) a = n X m =0 { φ ( n − m ) a , H ( m ) } + n − X m =0 { φ ( n − m ) a , H ( m +2) } η + { φ ( n +1) a , H (1) } η ,H ( n +1) = − n + 1 η a ω − ab χ bc G ( n ) c , n ≥ e H ) and the firstclass constraints ( e φ ) are derived by applying the summation on corre-sponding correction terms. In above equations, the upper indices in theparentheses indicates the order of correction. Also, the η a is a vectorwhich represents the new phase space variables, so the suffix η under thebrackets means the Poisson bracket in the adhered sub-phase space. Thequantities φ (0) a and H (0) are nothing more than uncorrected constraintsand canonical Hamiltonian of the uncorrected model. The Roman in-dexes a, b, · · · take their values from { , , · · · , ♯ of the second class constraints } and everywhere in this paper the summation convention is consideredfor repeated indices. The square matrices χ and ω determine how theelements of vector η appear in the correction terms. They satisfy themaster equation of the BFT,∆ + χ T ωχ = 0 . (12)By determining ω , also the Poisson structure of the new sub-phasespace will be obtained, because we consider that there is no interactionbetween two parts of the phase space, i.e. it is off-shell.As it has been stated before [23, 24], there are more than one solu-tion for equation(12). Thus, for a given second class system, there areso many corresponding first class systems which divert to it after gaugefixing. In some cases, the elements of the matrix of Poisson bracketsof the second class constraints, say ∆ ab , on the constraint surface areconstant. Therefore, for such cases there is a simple solution for (12)as one could assume the unknown matrices have elements independentfrom the phase space variables, even for the case χ = 1 and ω = − ∆.Such a solution reduces the recursion relations (10,11) to a simple form.In this regime the generating functions B ( n ) is vanished as same as Pois-son brackets on new sub-phase space. Conclusively, the constraints arecorrected by only one term, say f φ a = φ (0) a + η a . The recursion relationfor the n th order of the Hamiltonian reduces to, G ( n ) a = { φ (0) a , H ( n ) } ,H ( n +1) = 1 n + 1 η a ∆ − ab G ( n ) b , n ≥ . (13)The fractional factor can be absorbed in generating vector and the ma-trix ∆ − rearranges the elements of ~η , i.e. we can order the recursionrelations as, G ′ ( n ) a = 1 n + 1 { φ (0) a , H ( n ) } ,η ′ b = η a ∆ − ab ,H ( n +1) = η ′ b G ′ ( n ) b , n ≥ . (14)Although for the problem with which we have encountered, the ∆matrix does not have constant elements, we use the above simplifiedequations in an appropriate manner for our goal. The full dynamics of the free non-relativistic particle confined on a D-dimensional sphere is given by H T = 12 ~p.~p + λ ( ~q.~q − , (15)where we assume the confinement condition as a primary constraint. The λ is Lagrange multiplier, adds the primary constraints φ to the canon-ical Hamiltonian. This is the simplest model with nonlinear constraintin the configuration space which produces its second class partner in thephase space as φ = ~q.~p . Due to the nonlinear nature of the constraints,the extension of phase space in order to convert the constraints as gaugesymmetries of a new model is not trivial. But in the BFT formalismwe have sufficient equations to add a linear term to the constraints inorder to make them first class. This assumption determines the sym-plectic structure of the extended phase space. So, our consideration isthat deformation of constraints by new phase space variables ( η , η ) as f φ = φ + η and f φ = φ + η make them first class constraints. Beforestarting the BFT embedding, the matrix elements of Poisson bracket ofthe constraints off the constraint surfaces is ∆ ab = 2 ~q ǫ ab . During theBFT process, the constraint surface changes, so we can’t compute the0∆ matrix on the constraint surface, unless up to the end of our calcu-lations when it is corrected. This subtle point make the use of the (14)problematic. We eliminate this problem by choosing a suitable ansatz,afterwards. In this way, according to the corrected constraints, we ob-tain the following nontrivial and nonconstant symplectic structure forthe new part of the phase space, { η a , η b } = − − η ) ǫ ab . (16)The two dimensional antisymmetric tensor ǫ ab is characterized by ǫ =1. We define the following objects to employ the relation (14). φ = H c φ = φ + 1 φ = φ . (17)One can shows, these quantities form a closed algebra with followingstructure constants. { φ α , φ β } = f αβγ φ γ , (18) f αβγ = 2( ǫ αβ δ γ + ǫ αβ δ γ − ǫ αβ δ γ ) , where Greek indices take their values from the set { , , } . The δ αβ andthe ǫ αβγ are the conventional, 3-dimensional Kronecker discrete deltafunction and the full antisymmetric tensor with ǫ = 1, respectively.In this version, the three functions are first class in terminology of con-strained systems. But they are not a full chain of a Hamiltonian andsome constraints, specially φ is not a constraint. In other words, theBFT process intends to induce a chain structure of Hamiltonian andconstraints on these objects by adding corrections to them.By running the machinery of the simplified BFT (14) for systemswith constant ∆ matrix (not only weakly but also off the constraintsurface), one can deduce a general formula for n th order correction termof the embedded Hamiltonian, inductively. H ( n ) = 1 n ! φ γ n n Y m =1 η ′ a m f a m γ m − γ m , n ≥ , (19)where summation convention as before is considered for indices for theirdomain which is the set { , } for Roman indices and the set { , , } for Greek indices. Also, the γ takes only the value 0. Besides thecurrent problem, we give an alternative approach to solve the linearproblem in appendix (A) by the manipulation of the above instructionin the presence of central charges. This formula and its twin, that is1given in (A), is one of the main results of this paper for gauging linearand nonlinear second class systems. Those are applicable and practicalwhenever someone tries to perform linear operations on second classfunctions to creates an algebra with the constant structure functionsand central charges.As we see in (19), the H ( n ) can be expanded in terms of three ele-ments of the closed algebra (18). In each level of iteration, due to thepresence of ∆ − that appears in the second equation of the (14), also afactor ( φ ) − is entered in H ( n ) . Thus, the solution is not exactly linearin φ γ as (19). Via afore thoughts, we guess the ansatz, H ( n ) = 1( φ ) n φ µ F µ ( n ; ~η ) , (20)for the n th level of the correction to the Hamiltonian. After some cal-culation which appears in appendix (B) we arrive to the solutions, F ( n + 1; ~η ) = ( − η ) n +1 ,F (1; ~η ) = 0 F ( n + 1; ~η ) = 12 ( η ) ( − η ) n − ,F ( n + 1; ~η ) = η ( − η ) n . (21)Conclusively, the embedded Hamiltonian is a summation on all correctedterms plus the primary canonical Hamiltonian. e H = H c + ∞ X n =1 H ( n ) . (22)According to the equations (21) and the ansatz (20), the output is de-composed into three parts, e H = 1 φ + η ( φ φ + 12 η + η φ ) . (23)The convergence condition for the summations on the series which is ap-peared in the corrected Hamiltonian, forces the value of the new phasespace variable the limitation of η < . Here is worth to noting thatthere is a Lagrangian approach which construct the whole gauged Hamil-tonian without an infinite tower of correction terms [25, 26] In the last stage of the gauging the model (15), we verify whether ourresults are compatible or not. In this way, we check the Abelian or non-Abelian nature of the new first class constraints and their chain structure2between themselves and the Hamiltonian. We purify the Hamiltonianin the form of kinetic and potential terms, by reduction of Hamiltonianon the new constraints surface.In former subsection we see that due the Poisson structure of thenew phase space (16), the corrected constraints are non-Abelian firstclass ones with the following algebra, { f φ , f φ } = 2 f φ . (24)Although the first class constraints become Abelian in the finite orderBFT, here we take out non-Abelian ones because of the non-constancy ofthe ∆ matrix and infinite order of BFT project on this special problem.In the next step, we consider that the f φ is a primary constraint forthe system as like as its uncorrected partner which is described by e H .Afterwards, we simplify the Hamiltonian with the help of that to obtain e H ′ = κH c + 12 ̺ + ̺φ , (25)where we improve new variables by redefinitions, κ = 1 − η , ̺ = η . (26)Then, the consistency of the f φ in the new system gives, { f φ , e H ′ } = 2 ̺ f φ . (27)Which is vanished weakly and no another constraint will be emerged . Consequently the second constraints situates at the primary level,inevitably. So, we project the Hamiltonian on the surface of both con-straints, named it e H on , and then do the consistency. For the f φ itterminates to an identity due to the, { f φ , e H on } = 0 . (28)The curious reality that happen in this stage is that, if we project the e H on the surface of both constraints (assume both of them as primaryconstraints), then the chain structure remains according to the (24), (28)and { f φ , e H on } = 2 κ f φ . (29)The three first class object make an open Lie algebra. Conclusively,after becoming first class, the chain structure of the constraints remains Another possibility is that we encounter to a bifurcation in the process of con-sistency. But, by a direct calculation one can shows that the vanishing of the ̺ eliminates the pair ( κ, ̺ ) as a second class pair, which is not of our favorite. e H on = κ~p − ̺ , κ > { κ, ̺ } = 2 κ, { q i , p j } = δ ij f φ = ~q − κ f φ = ~q.~p + ̺. (30)At first, we see that the embedded hyper sphere in the configurationspace doesn’t transform to another hyper-sphere but it transforms toa hyper surface with sections, at κ = constants , in the form of hyper-sphere. It is a part of a spherical paraboloid. The minimum number ofauxiliary variables in the primary model are imposed by construction ofthe BFT formalism, spontaneously.The minus sign in front of the second term obliges us to interpretboth κ and ̺ as coordinates, unless as same as the first example ofthis paper, we assume the primary system exchanges the energy withan external system in the form of an oscillator. An extra evidence thatguide us to consider the ( κ, ̺ ) as a coordinate pair is a non-canonicaltransformation κ → ln √ κ, ̺ → ̺, (31)which transforms this pair to usual canonical variables in the sense ofusual symplectic structure on phase spaces [27]. So our consideration isreasonable. If we get the ( κ, ̺ ) as a pure coordinate pair, in quantizationprocess, the new part of phase space is a noncommutative plane withposition-dependent noncommutativity parameter. After the quantiza-tion is applied, it can be investigated in the context of Lie algebra non-commutativity [29, 28] or κ -Minkowski noncommutativity [30, 31, 32]. In this paper we embed a general system with linear second class con-straints directly with the help of the BFT embedding. We show thatthe embedded constraints also remain linear but the Hamiltonian be-comes interactive. For a free particle on a hyper sphere, which can beconsidered as a model with nonlinear constraints, we do the embeddingprocedure by the finite order BFT formalism. But, because of the non-constancy of the ∆ matrix, we derive an infinite series for correctionterms for the Hamiltonian. In both systems, we find that correctionterms for primary constraints are linear with respect to the new coordi-nates. It changes hyper sphere to the part of a spherical paraboloid. Inconversion to first class systems we encounter non-unique solutions. Thisobservation was seen by others in gauging another models in different4approaches [33, 34, 35]. We find that minimal solutions add additionaldegrees of freedom in the number of the second class constraints, whichbrings to mind Wess-Zumino variables in gauging a system [18]. Ournovelty of work is a prescription for computing the Hamiltonian correc-tion terms, based on the BFT formalism. Our general formula is derivedfor the fixed ∆ matrix. We generalize it for nonconstant ∆ matrix, i.e.off the constraints surface. We find a nontrivial Poisson structure for thegauged nonlinear system, that in the quantization produces is a noncom-mutative plane. Ultimately, we show that after gauging the nonlinearsystem, the Hamiltonian and the constraints take place in a chain struc-ture with non-Abelian open algebra. A For the linear problem we name the three objects of closed algebra asfollow. φ = H c , φ = φ , φ = φ . (32)All of the conventions for indices are as same as before. This algebrahas an essential difference with respect to the algebra of the nonlinearproblem. In addition to structure constants it has a central charge inthe form, { φ α , φ β } = f αβγ φ γ + c αβ , (33) f αβγ = ǫ αβ δ γ c αβ = ~a ǫ αβ . So, we are in the situation that we can generalize the (19) in the presenceof the central charges. In the same way which arrives us to the (19),we can show that the compact term of the H ( n ) is decomposed into twoparts, H ( n ) = 1 n ! ( φ γ n n Y m =1 η ′ a m f a m γ m − γ m + c a n γ n − η ′ a n n − Y m =1 η ′ a m f a m γ m − γ m ) . (34)For n = 1 the ambiguity in the second term of the parenthesis disap-pears, because according to the conventions and (33), the c a γ = c a vanishes.In conclusion, we can establish the relations (33, 34) to deduce theminimal solution (9) after truncation at n = 2. It means that we havethe finite order BFT which is due to the constant ∆ matrix.5 B We begin with the (14) and the expression for the matrix elements ofPoisson brackets of the second class constraints which can be writtenin the form ∆ ab = − φ ǫ ab . In continue, we find the following recursionrelation between consecutive corrections of the Hamiltonian. H ( n +1) = − φ η a ǫ ab n + 1 { φ b , H ( n ) } . (35)Afterward, we set the ansatz (20) in the above equation to obtain φ µ F µ ( n + 1; ~η ) = − η a ǫ ab n +1 ( F µ ( n ; ~η ) f bµγ φ γ − nF µ ( n ; ~η ) φ µ φ f b γ φ γ ) . (36)The factor φ in the second term is destroyer, but due to the special formof its coefficient f b γ such a factor is removed. It confirms the suggestedansatz and leads to the coupled recursion relations, F ( n + 1; ~η ) = ( − η ) F ( n ; ~η ) ,F ( n + 1; ~η ) = n +1 ( − ( n − η F ( n ; ~η ) + η F ( n ; ~η )) ,F ( n + 1; ~η ) = n +1 ( η F ( n ; ~η ) − nη F ( n ; ~η )) , (37)for three types of unknown functions F µ ( n, ~η ). The first and third equa-tion of the above equations can be solved immediately up to initial con-ditions F (1; ~η ) and F (1; ~η ). Then, with the help of these solutions wesolve the second equation for n ≥
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