ON ABELIANIZATION OF FIRST CLASS CONSTRAINTS

Transcription

1 ON ABELIANIZATION OF FIRST CLASS CONSTRAINTS arxiv:hep-th/ v1 28 Apr 1995 S A Gogilidze, A M Khvedelidze, V N Pervushin Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, , Dubna,Russia The systematic method for the conversion of first class constraints to the equivalent set of Abelian one based on the Dirac equivalence transformation is developed The representation for the corresponding matrix performing this transformation is proposed This representation allows one to lead the problem of abelianization to the solution of a certain system of first order linear differential equations for matrix elements PACS numbers: 0220 Tw, 03, 0365Ge, q JINR Preprint E Permanent address: Tbilisi State University, , Tbilisi, Georgia 2 Permanent address: Tbilisi Mathematical Institute, , Tbilisi, Georgia Electronic address: khved@theorjinrcdubnasu 1

2 I Introduction It is the purpose of this note to describe a practical method for the conversion of non - Abelian constraints into the Abelian form in the theories with first class constraints For explanation of the practical importance of this procedure let us briefly recall the general principles of the description of the standard Dirac theory of Hamiltonian systems with constraints [1] - [4] For the sake of simplicity as usually we will discuss the main ideas using a mechanical system, ie systems with a finite number of degrees of freedom, with having in mind that the direct extension of the results to a field theory in general is possible only in the local sense Suppose that the system with 2n - dimensional phase space Γ acquires the following set of irreducible first class constraints ϕ α (p, q), (α = 1, 2,, m) ϕ α (p, q) = 0, {ϕ α (p, q), ϕ β (p, q)} = f αβγ (p, q)ϕ γ (p, q) (11) This means that the dynamics of our system is constrained on the certain submanifold 1 of the total phase space or, in another words, not all canonical coordinates are responsible for the dynamics The generalized Hamiltonian dynamics is described by the extended Hamiltonian which is a sum of canonical Hamiltonian H C (p, q) and a linear combination of constraints with arbitrary multipliers u α (t) H E (p, q) = H C (p, q) + u α (t)ϕ α (p, q) (12) The arbitrariness of the functions u α reflects the presence in the theory of coordinates whose dynamics is governed in an arbitrary way According to the principle of gauge invariance for physical quantities, these coordinates do not affect them and thus can be treated as ignorable 1 In the next we will symbolize by notation Γ c this 2n m dimensional submanifold of Γ, Γ c Γ 2

3 The main problem that arise is identification of these coordinates If there are in the theory only Abelian constraints {ϕ α (p, q), ϕ β (p, q), (13) one can find these ignorable coordinates as follows In this case, it is always possible [5] - [8] to define a canonical transformation to a new set of canonical coordinates q i Q i = Q i (q i, p i ), p i P i = P i (q i, p i ), (14) so that m of the new P s ( P 1,,P m ) become equal to the abelian constraints (13) P α = ϕ α (q i, p i ) (15) In the new coordinates we have the following system of canonical equations Q = {Q, H Ph }, P = {P, H Ph }, P = 0, with arbitrary functions u(t) and with the physical Hamiltonian Q = u(t), (16) H Ph H C (p, q) ϕα=0 H C(P, Q ) P α =0 The physical Hamiltonian H Ph depends only on the remaining n m pairs of the new canonical coordinates (Q 1, P 1,Q n m, P n m ) wich are a gauge invariant physical variables This means that the coordinates Q α conjugated to the momenta P are ignorable coordinates and the canonical system admits explicit separation of phase space into sectors: physical and nonphysical one q 1 p 1 2n q n p n 2(n m) Q P 2m Q P 3 Physical sector Nonphysical sector (17)

4 The straight generalization of this method to the non - Abelian case is unattainable; identification momenta with constraints is forbidden due to the non - Abelian character of constraints (11) However, there is a general proof of a possibility of a local replacement of the constraints (11) by the equivalent set of constraints forming an abelian algebra [2], [4], [8], [9], [10] This general observation Abelianization statement reads For a given set of m first class constraints it is always possible to chose locally m new equivalent constraints ϕ α (p, q) = 0 Φ α (p, q) = 0 (18) that define the same constraint surface Γ C so the Poisson brackets between the new constraints strongly vanishes, ie {Φ α (p, q), Φ β (p, q) (19) Thus, one can deal with this equivalent set of Abelian constraints to construct the reduced phase space, the space of physical degrees of freedom To reveal ignorable coordinates, we need an explicit form of the new Abelian constraints Φ α (p, q) We would like to emphasize that in all proofs of the abelianization it has been assumed that the constraints form the functional group [2], [10] In the present paper, we shall point out two alternative schemes of realization of the abelianization procedure: via constraints resolution and via generalized canonical transformation for general non - Abelian constraints of type (11) The generalized canonical transformations [12] are those preserving the form of all constraints of the theory as well as the canonical form of the equations of motion It will be shown that in a constructive fashion it is possible to convert constraints into the Abelian form with the help of the Dirac equivalence linear transformation Φ α (p, q) = D αγ ϕ γ (p, q) (110) with the nonsingular matrix D det D αγ 0 constraints 4

5 The main point of our result is that this matrix D can be determined by linear first order differential equations The remaining part of this note is the proof of this statement and the application to the special example: non - Abelian Christ and Lee model [13] II Abelianization : alternative schemes A Abelianization via constraint resolution The direct way of abelianization of constraints is as follows [10], [11] Under the assumption that ϕ α (p, q) are m independent functions one can resolve the constraints (11) for m of p s p α = F α (p, q) (211) where p denotes the remaining p s Let us pass to a new equivalent to the ϕ α (p, q) constraints Φ α (p, q) = p α F α (p, q) (212) By the explicit computing one can convince that the Poisson brackets of the new constraints {Φ α (p, q), Φ β (p, q)} are independent of p α, but as it is clear that they are again the first class the unique possibility is that their Poisson brackets with each other must vanish identically Thus, after transformations to the new constraints Φ α (p, q) we can realize the above - mentioned canonical transformation (14) such that m of the new P s become equal to the modified constraints Φ α (212) P α = Φ α (q i, p i ) (213) with the corresponding conjugate ignorable coordinates Q α 5

6 B Abelianization of constraints via Dirac s transformation In this section, it will be demonstrated that due to the freedom in the representation of constraint surface Γ c ϕ α (p, q) = 0, {ϕ α (p, q), ϕ β (p, q)} = f αβγ (p, q)ϕ γ (p, q) (214) one can always pass with the help of Dirac s transformation from these first class non - Abelian constraints to the equivalent ones so that the new constraints will be Abelian Φ α (p, q) = D αβ (p, g)ϕ β (p, q), (215) {Φ α (p, q), Φ β (p, q) (216) According to (216) the matrix D αβ must satisfy the set of the nonlinear differential equations {D αγ (p, g)ϕ γ (p, q), D βσ (p, g)ϕ σ (p, q) (217) Such a formulation of the abelianization statement means a possibility to find a particular solution for this very complicated system of nonlinear differential equations Beyond the question eq (217) in this form does not contain any practical value but, as it will be shown here, there is a particular solution to this equation which can be represented as D = D 1 (p, q) D m (p, q) }{{} m (218) where each matrix D k has a form of the product of k s m m matrices 0 D k = R ak+k (p, q) S ak+i (p, q) (219) (a k k(k+1) 2 ) i=k 1 R ak+k = k m k {}}{{}}{ I 0 0 B a k+k (220) 6

7 k m k {}}{{}}{ S ak+i = 0 C a k+i k C a k+i k C a k+i m C a k+i m (221) } {{ } k i and the corresponding matrix elements satisfy a set of first order linear differential equations ( see below (225) - (228)) Just the linear character of these equations alows one to speak about a practical usage of the proposed method of abelianization As it will be explained below, the constraints which are obtained as a result of the action of k s matrices (constraints at the a k + k -th step ) Φ a k+k α = ( D k D k 1 D 1) αβ Φ0 β (222) obey the algebra where k constraints have zero Poisson brackets with any one From the algebraic standpoint the proposed method of abelianization is nothing but an iterative procedure of constructing equivalent algebras A a i of constraints Φ a i α In a m steps the m - dimensional non - Abelian algebra is converted into an equivalent Abelian one in such a manner that at the a k - th step the obtained algebra A a k Z k [A] = (Φ a k 1, Φ a k 2,, Φ a k k ) possesses a center with k elements A 0 S1 } {{ } A 2 S3 } {{ } D 1 D 2 A 1 R 2 A 3 S 4 A 4 R 5 A 5 Sak A a k Ra k +k }{{} A ak+k (223) D k The matrix D k converts the algebra A k into the algebra A k+1 in which the center contains one element more than the previous one 7

8 The proof of the validity of the representation (219) and the equations for the matrices S and R are obtained by induction Suppose that Φ a k α - are a set of constraints (obtained as a result of the action of k 1 matrices D i ) with algebra having the center Z k [A] = (Φ a k 1, Φ a k 2, Φ a k k ) than a matrix Dk from (218) perform the transformation to the new constraints Φ a k+1 1 α = D k αβφ a k+1 β (224) which form the algebra with the center Z k+1 [A] = ( Φ a k+1 1, Φ a k+1 2,, Φ a k+1 ) if the matrices S, R are the solutions to the following set of linear differential equations: Φ a k+1 k+1 {Φ a k+i 1 1, S a k+i {Φ a k+i 1 k 1, S a k+i k, = {Φ a k+i 1, S a k+i (225) {Φ a k+i 1 k, S a k+i } = f a k+i 1 k γ k S a k+i γ k f a k+i 1 k i+1 (226) {Φ a k+k 1 1, B a k+k β k {Φ a k+k 1 k 1, B a k+k β k = {Φ a k+k 1, B a k+k β k (227) {Φ a k+k 1 k, B a k+k β k } = f a k+k 1 kγ k β k B a k+k where = k + 1,,m, = 1, 2,, k 1 and f a k+i αγβ constraints algebra A a k+i at the a k + i -th step γ k (228) are the structure functions of Let us begin in a consecutive order the construction of algebras with the center containing 1, 2,, m elements For determination of the new algebra with one central element (let ϕ 1 ) one can act in the following way : a) from the beginning exclude ϕ 1 from the left hand side of eq (214); b) then realize abelianization with all others 8

9 To achieve first, we can perform the transformation with the matrix S 1 Φ 1 α = S1 αβ ϕ β of type (221) S 1 = C C C m (229) In the expanding form it is Φ 1 1 = Φ 0 1 = ϕ 1 The new constraints algebra remains the algebra of first class Φ 1 α 1 = ϕ α1 + C 1 α 1 ϕ 1 (230) {Φ 1 1, Φ 1 α 1 } = f1α 1 1 1Φ f1α 1 1 γ 1 Φ 1 γ 1 {Φ 1 α 1, Φ 1 β 1 } = fα 1 1 β 1 1 Φ1 1 + f1 α 1 β 1 γ 1 Φ 1 γ 1 (231) and the new structure functions fαβγ 1 are determined through the old one f αβγ and the transformation functions Cα 1 1 as follows f1α = f 1α1 1 + f 1α1 γ 1 Cγ {Φ 0 1, C1 α 1 } (232) fα 1 1 β 1 1 = 1 ( ) fα1 β f α1 β 1 γ 1 Cγ {Cα 1 1, Cβ 1 1 }Φ 0 1 f1α C1 β 1 + {Φ 0 α 1, Cβ 1 1 } (α 1 β 1 ) (233) fα 1 1 β 1 γ 1 = f α1 β 1 γ 1 + Cα 1 1 f 1β1 γ 1 Cβ 1 1 f 1α1 γ 1 (234) f1α 1 1 γ 1 = f 1α1 γ 1 (235) Let us now choose the transformation functions Cβ 1 1 so that the Poisson bracket of first constraints Φ 1 1 with all other modified constraints do not contain it {Φ 1 1(p, q), Φ 1 α 1 (p, q)} = f1α 1 1 γ(p, q)φ 1 γ(p, q) γ 1 (236) 9

10 These m 1 requirements f 1 1α 1 1 = 0, according to (232), mean that the transformation function C 1 α 1 must satisfy the following set of linear nonhomogeneous differential equations {Φ 0 1, C 1 α 1 } = f 1α1 1 + f 1α1 γ 1 C 1 γ 1 (237) Note that the problem of existence of solution to such a set of equations is studied very well ( see eg [15] ) Suppose, we find some particular solution Cα 1 1 for (237), then one can determine all structure functions of the modified algebra according to eq(232), f1α = 0 (238) fα 1 1 β 1 1 = f α1 β 1 1 f α1 β 1 γ 1 Cγ {Cα 1 1, Cβ 1 1 }Φ {Φ 0 α 1, Cβ 1 1 } + {Φ 0 β 1, Cα 1 1 } (239) fα 1 1 β 1 γ 1 = f α1 β 1 γ 1 + Cα 1 1 f 1β1 γ 1 Cβ 1 1 f 1α1 γ 1 (240) f1α 1 1 γ 1 = f 1α1 γ 1 (241) Now let us again keep first constraint unchanged and perform the Dirac transformation on the remaining part of constraints Φ α1, α 1 = 2, 3,, m Φ 2 1 = Φ 1 1 = Φ0 1 = ϕ 1 Φ 2 α 1 = B 2 α 1 β 1 Φ 1 β 1 (242) with the requirement that the new constraints have zero Poisson brackets with the first one Φ 1 1 {Φ 2 1, Φ2 α 1 (243) One can verify that this requirement means that the transformation functions B α1 β 1 are the solution to the equation {Φ 1 1, B 2 α 1 β 1 } = f 1γ1 β 1 B 2 α 1 γ 1 (244) With the help of a solution of eq (244) the modified algebra has the following structure functions f1α = 0, (245) 10

11 f 2 α 1 β 1 1 = B 2 α 1 δ 1 B 2 β 1 σ 1 f 1 δ 1 σ 1 1, (246) f 2 α 1 β 1 γ 1 = [ {B 2 α 1 δ 1, B 2 β 1 σ 1 }Φ 1 σ 1 + {B 2 α 1 δ 1, Φ 1 σ 1 }B 2 β 1 σ 1 {B 2 β 1 δ 1, Φ 1 σ 1 }B 2 α 1 σ 1 + B 2 α 1 κ 1 B 2 β 1 σ 1 f 1 κ 1 σ 1 δ 1 ] (B 2 ) 1 δ 1 ρ 1 (247) Thus, as a result of two transformations D 1 = S 1 R 2 we obtain the modified algebra A 2 of constraints Φ 2 α with the central element Φ2 1 The structure functions f 2 α 1 β 1 γ {Φ 2 1, Φ 2 α 1 (248) {Φ 2 α 1, Φ 2 β 1 } = f 2 α 1 β 1 1 Φ2 1 + f2 α 1 β 1 γ 1 Φ 2 γ 1 (249) of algebra (248),(249) possess the significant property : due to the fact that Φ 2 1 is the central element the structure functions obey the following property: {Φ 2 1, f2 α 1 β 1 γ (250) To verify this, it is enough to calculate the Poisson bracket of Φ 2 1 Jacobi identity case with (249) and use the To extend the center of the new algebra A 2 by Φ 2 2, we will act by analogy with the previous a) exclude Φ 2 1 and Φ2 2 from the left hand side of eq (249) b) then achieve the abelianization with all others To carry out the first point of this program we will deal with two consecutive transformations S 3 and S 4 Let us require that the first transformation S 3 of type (221) Φ 3 1 = Φ 2 1 Φ 3 2 = Φ 2 2 Φ 2 α 2 = Φ 2 α 2 + C 3 α 2 Φ 2 1, α 2 = 3,,m (251) lead to the new algebra of constraints so that Φ 3 1 is again the central element {Φ 3 1, Φ 2 α 1 (252) 11

12 and the Poisson brackets of the second constraint Φ 3 2 with all other modified constraints does not contain Φ 3 1 {Φ 3 2, Φ2 α 2 } = γ 1 f 3 2α 2 γ Φ3 γ (253) This requirement leads to the following equations {Φ 3 1, C3 α 2 {Φ 2 2, C3 α 2 } = f 2 2α 2 γ 2 C 3 γ 2 f 2 2α 2 1 (254) What about the existence of the solutions to these equations One can verify that the integrability condition for the system of differential equations (254) is nothing else but (250) In full analogy with the previous case one can express the new structure functions f 3 α,βγ through f 2 α,βγ and verify that they obey the following property {Φ 3 1, f3 2α 2 γ (255) Now one can realize the transformation S 4 of type (221) Φ 4 1 = Φ 3 1 Φ 4 2 = Φ 3 2 Φ 4 α 2 = Φ 3 α 2 + C 4 α 2 Φ 2 2 (256) so that Φ 4 1 is again central element {Φ 4 1, Φ2 α 1 (257) and the Poisson brackets of the second constraint Φ 2 4 with all other modified constraints do not contain Φ 4 1 and Φ4 2 {Φ 4 2, Φ4 α 2 } = f 2α2 γφ 3 γ (258) γ 1,2 This requirement leads to the following equations : {Φ 4 1, C4 α 2 {Φ 3 2, C 4 α 2 } = f 3 2α 2 γ 2 C 4 γ 2 f 3 2α 2 2 (259) 12

13 This system is consistent in virtue of equations (255) For the new structure function one can again to verify that {Φ 4 1, f4 2α 2 γ (260) as for the previous step (see eq(255) Now for abelianization of two constraints Φ 4 1, Φ 5 2 it is enough to perform last transformation with matrix R of type (220) if its elements are the solution to the equations {Φ 4 1, B5 α 2 β 2 {Φ 4 2, B5 α 2 β 2 } = f2γ 4 2 β 2 Bα 5 2 γ 2 (261) As a result {Φ 5 1, Φ5 α } = {Φ5 2, Φ5 α Note that (260) provides the existence of the solution to eq(261) As a result, one can easy verify that the new structure functions possess the property {Φ 5 α 2, f 5 α 2 β 2 γ, α 2 = 1, 2 (262) Thus, in five steps (for summary, see Table 1) we obtain an equivalent to initial algebra A 5 with two central elements Φ 5 α 2 Now let us suppose that by acting in such a way we get the algebra A k 1 (see Table 2) with the center composed by k 1 elements {Φ a k 1 α, Φ a k 1 β } = f a k 1 αβγ Φa k 1 γ (263) Z k 1 = ( ) Φ a k 1 1, Φ a k 1 2, Φ a k 1 k 1 {Z k 1, Φ a k 1 α and the structure functions of this algebra have the property {Φ a k 1, f a k 1 β k γ, = k 1,, m (264) 13

14 Now by direct calculations it is easy to verify that via the action of transformation of the matrix D k with elements which are the solutions to eqs (225) and (228), we obtain the algebra A k with the center composed by k elements Z k 1 = ( ) Φ a k 1 1, Φ a k 1 2,, Φ a k 1 k 1 {Z k 1, Φ a k 1 α (265) The conditions (264) play the role of the integrability conditions for the system of eqs(225), (228) For completion we must only to prove the following property of structure functions To verify this one can consider the algebra of constraints {Φ a k+k +1, f a k+k +1 β k+1 γ (266) {Φ a k+k +1, Φ a k+k β k+1 } = f a k+k +1 β k+1 γ k+1 Φ a k+k γ k+1 + f a k+k +1 β k+1 γ k+1 Φ a k+k γ k+1 (267) and compose the Poisson bracket of (267) with Φ a k+k +1 Taking into account that Φ a k+k +1 are the central elements of the algebra A k, we immediately get the desired result (266) with the help of Jacobi identity III Christ and Lee model In this section we will apply the above described procedure to the well known example nonabelian Christ and Lee model described by the Lagrangian L(x,ẋ,y) = 1 2 (ẋ [y,x])2 V (x 2 ) where x and y - are the three- dimensional vectors, (x 1, x 2, ), (y 1, y 2, y 3 ) It is easy to verify that except for three primary constraints there are two independent constraints π = L ẏ = 0, Φ 0 1 = x 2 p 3 p 2 Φ 0 2 = p 1 x 1 p 3 (31) 14

15 with the algebra {Φ 0 1, Φ0 2 } = x 1 Φ 0 1 x 2 Φ 0 2 (32) The abelianization procedure for this simple case consists in two stages At first step, the transformation S 1 reduces to and equation (225) looks like Φ 1 1 = Φ 0 1 Φ 1 2 = Φ CΦ 0 1 (33) {Φ 0 1, C} = x 2 C + x 1 (34) One can write down a particular solution to this equation C(x) = x 1 arctan So, as a result of first step we get a new algebra ( ) x2 (35) Now let us perform the second transformation R 2 {Φ 1 1, Φ 1 2} = x 2 Φ 1 2 (36) Φ 2 1 = Φ 1 1 with the function B satisfing the equation of type (228) Φ 1 2 = BΦ 1 2 (37) {Φ 1 1, B} = x 2 (38) A particular solution to this equation reads x x 2 3 B(x) = ln (39) Thus, the equivalent to the initial abelian constraints have the form Φ 2 1 = x 2 p 3 p 2 (310) x 2 Φ x 2 [ 3 2 = ln ( p 1 x 1 p 3 ) + x ( ) ] 1 x2 arctan (x 2 p 3 p 2 ) 15

16 IV Concluding remarks We have discussed the iterative procedure of converting first class constraints in an arbitrary singular theory to the Abelian form of constraints The final goal of dealing with a valuable form of abelianization is the construction of the reduced phase space in the complicated non - Abelian theory The application of our procedure to the SU(2) Yang - Mills will be done in separate forthcoming publication V Acknowledgments We would like to thank AT Filippov, GT Gabadadze, AN Kvinikhidze, VV Nesterenko, G Lavrelashvili for helpful and critical discussions This work was supported in part by the Russian Foundation of Fundamental Investigations, Grant No References [1] PAMDirac, Lectures on Quantum Mechanics Belfer Graduate School of Science, (Yeshive University, New York, 1964) [2] K Sundermeyer Constrained Dynamics, Lecture Notes in Physics N 169, (Springer Verlag, Berlin, 1982) [3] AJ Hanson, T Regge, C Teitelboim Constrained Hamiltonian Systems,, (Accademia Nazionale de Lincei, 1976) [4] DM Gitman, IV Tyutin, Quantization of Fields With Constraints (Springer Verlag, Bonn, 1990 ) [5] T Levi-Civita and U Amaldi, Lezioni di Meccanica razionale (Nicola Zanichelli, Bologna, 1927) [6] ET Whittaker, A Treatise on the Analitical Dynamics of Particles and Rigid bodies (Cambridge University Press, Cambridge, 1937) 16

17 [7] LPars, A Treatise on Analitical Dynamics (Heinemann, London, 1964) [8] S Shanmugadhasan, J Math Phys 14, 677 (1973) [9] T Maskawa H Nakajima, ProgTheorPhys 56, 1295 (1976) [10] MHenneux, Phys Rep 126, 1 (1985) [11] JGoldberg, ET Newman, CRovelli, J Math Phys 32, 2739 (1991) [12] PG Bergman and I Goldberg, Phys Rev 98, 531 (1955) ; PG Bergman ibid 98, 544 (1955) [13] NH Christ, TD Lee, Phys Rev 22, 939 (1980) [14] PAM Dirac, Rev Mod Phys 21, 392 (1949) [15] R Courant, Partial Differential Equations (John Willey & Sons, New-York, 1960) 17

18 Table 1 Abelianization stages for the algebra with two central elements CONSTRAINTS ALGEBRA CONDITIONS Φ 0 α Φ 0 α = ϕ α {Φ 0 α, Φ0 β } = f0 αβγ Φ0 γ Φ 1 α Φ 1 1 = Φ0 1 {Φ 1 1, Φ1 α } = f1 1αγ 1 Φ 1 γ 1 {Φ 0 1, C1 α 1 } = f 1α1 γ 1 C 1 γ 1 f 1α1 1 Φ 1 α 1 = Φ 0 α 1 + C 1 α 1 Φ 0 1 Φ 2 α Φ 2 1 = Φ1 1 = Φ0 1 {Φ 2 1, Φ1 α {Φ1 1, B2 α 1 β 1 } = f 1 1γ 1 β B2 α 1 γ 1 Φ 2 α 1 = B 2 α 1 β 1 Φ 1 β 1 1 {Φ 2 1, Φ2 α Φ 3 α Φ 3 1 = Φ 2 1 = Φ 1 1 {Φ 3 1, Φ 3 α {Φ 3 1, C 3 α 2 Φ 3 2 = Φ 2 2 {Φ 3 2, Φ 3 α} = f 3 1αγ 1 Φ 2 γ 1 {Φ 2 2, C 3 α 2 } = f 2 2α 2 γ 2 C 2 γ 2 f 2 2α 2 1 Φ 3 α 2 = Φ 2 α 2 + C 3 α 2 Φ 1 1 Φ 4 α Φ 4 1 = Φ 3 1 = Φ 1 1 {Φ 4 1, Φ 4 α {Φ 4 1, C 4 α 2 Φ 4 2 = Φ3 2 = Φ2 2 {Φ 4 2, Φ4 α } = f4 2αγ 2 Φ 4 γ 2 {Φ 3 2, C4 α 2 } = f 3 2α 2 γ 2 C 4 γ 2 f 3 2α 2 2 Φ 4 α 2 = Φ 3 α 2 + C 4 α 2 Φ 3 2 Φ 5 α Φ 5 1 = Φ4 1 Φ1 1 {Φ 5 1, Φ5 α {Φ4 1, B5 α 2 β 2 Φ 5 2 = Φ4 2 Φ2 2 {Φ 5 2, Φ5 α {Φ4 2, B5 α 2 β 2 } = f 4 2γ 2 β 2 B 5 α 2 γ 2 Φ 5 α 2 = B 5 α 2 β 2 Φ 4 β 2 2 {Φ 5 1, Φ5 α } = {Φ5 2, Φ5 α 20

19 Table 2 Abelianization stages for the algebra with k central elements CONSTRAINTS ALGEBRA CONDITIONS k-1 {Φ a k 1 1, Φ a k 1} = {Φ a k 1 2, Φ a k 1} = = {Φ a k 1 α α k 1, Φa k 1 α Φ a k 1 = Φ a k 1 1 = Φ 1 1 {Φ a k 1, Φ a k α Φ a k 2 = Φ a k 1 2 = Φ 2 2 {Φ a k 2, Φ a k α {Φa k 1, C a k {Φa k 2, C a k Φ a k k = Φa k 1 k {Φ a k k, Φa k α } = f a k kαγ 1 Φ a k γ 1 {Φ a k 1 k, C a k } = f a k 1 k γ k C a k γ k f a k 1 k 1 Φ a k = Φ a k 1 α + C a k Φ 1 1 Φ a k+k 1 = = Φ 1 1 {Φ a k+k 1, Φ a k+k α {Φ a k+k 1, B a k+k β k Φ a k+k 2 = = Φ 2 2 {Φ a k+k 2, Φ a k+k α Φ a k+k k = Φ a k 1 k {Φ a k+k k, Φ a k+k α {B a k+k β k, Φ a k+k k } = f a k+k 1 kγ k β k B a k+k γ k Φ a k = B a k+k β k Φ a k+k 1 β k k {Φ a k+1 1 1, Φ a k+1 1 α } = {Φ a k+1 1 2, Φ a k+1 1 α } = = {Φ a k+1 1 k, Φ a k+1 1 α 21