Symplectic embedding of second class systems

Davi Rodrigues

Symplectic embedding of second class systems

Davi Rodrigues

2004, Nuclear Physics B - Proceedings Supplements

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An alternative approach to embed second class systems using the Wess-Zumino (WZ) variables is proposed[l]. This is developed within the symplectic framework[2,3].

SUPPLEMENTS Nuclear Physics B (Proc. Suppl.) 127 (2004) 170-173 www.elsevierphysics.com Symplectic Embedding of Second Class Systems A.C.R. Mendesa *, C. Nevesbt, W. Oliveirab$, D. C. Rodriguesb c§ “Centro Brasileiro de Pesquisas Fisicas, Rua Xavier Sigaud 150, 22290-180, Rio de Janeiro, RJ, Brasil bDepartamento de Fisica, Instituto de Ciencias Exatas, Universidade Federal de Juiz de Fora, CEP 36036-330, Juiz de Fora, Minas Gerais, Brazil CInstituto de Fisica, Universidade Federal do Rio de Janeiro 21945-970, Rio de Janeiro, Brasil An alternative This is developed 1. General approach to embed second class systems using the Wess-Zumino within the symplectic framework[2,3]. _ j,‘(o), (1) where the symplectic variables are ,pb = ai, pi, with a = 1,2 ,..., N witha=N+l,N+2 variables is proposed[l]. plectic tensor is given by formalism In this section, we describe the alternative embedding technique that changes the second class nature of a constrained system to the first one. This technique follows the Faddeev and Shatashivilli idea[l] and is based on a contemporary framework that handles constrained models, namely, the symplectic formalism[2,3]. In order to systematize the symplectic embedding formalism, we consider a general noninvariant mechanical model whose dynamics is governed by a Lagrangian L(ai,&,t)(with i = 1,2,. . .,N), where ai and I& are the space and velocities variables, respectively. Notice that this model does not result in the loss of generality or physical content. Following the symplectic method the zeroth-iterative first-order Lagrangian is written as L(o) = Af)j(ob (WZ) If this symplectic matrix is singular, it has a zeromode (V(O)) which can generate a new constraint when contracted with the gradient of symplectic potential, This constraint is introduced into the zerothiterative Lagrangian, Eq.( l), through a Lagrange multiplier q, generating the next one c(l) = = A(${(% + $-$“) - V(o), ,~+)j(l)Y _ V(l), (5) with y = 1,2,. . . , (2N + l), v(l) = v@)lQ(0)=o, 5clJ-r = (t(O)“, 7) and A-,(l) = (A&0’,R(O)). As a consequence, the first-iterative symplectic tensor is computed as ,..., 2N. (2) with A?) are the one-form canonical momenta and V(O) is the symplectic potential. The sym*e-mail: albertQfisica.ufjf.br te-mail:cnevesOfisica.ufjf.br te-mail:wilsonOfisica.ufjf.br se-mail:cabralQif.ufrj.br 0920-5632/$ - see front matter 0 2004 Published by Elsevier B.V doi:l0.1016/j.nuclphysbps.2003.12.033 (6) If this tensor is nonsingular, the iterative process stops and the Dirac’s brackets among the phase space variables are obtained from the inverse matrix (j$))-‘. On the contrary, the tensor has a zero-mode and a new constraint arises, indicating A. C.R. Mendes et al. /Nuclear Physics that the iterative process goes on until the symplectic matrix becomes nonsingular or singular. If this matrix is nonsingular, the Dirac’s brackets will be determined. Oppositely, if this matrix is singular and the respective zero-mode does not generate a new constraint, the system has a symmetry. The systematization of the symplectic embedding formalism begins by assuming that the gauge invariant version of the general Lagrangian (L(ui, cii, t)) is given by &i, b, ‘pp, t) = qai, b, t) + LWZ(% hi, (Pp), (7) where (p = 1,2), ‘pp = (e,e) and the extra term (,Cwz) depends on the original (ai,&) and WZ ((pp) configuration variables. Indeed, this WZ Lagrangian can be expressed as an expansion in orders of the WZ variable ((pP) such as .CWZ(% 4, (Pp) = c J”)(%b, (Pp), (8) n=l with vcn) ( (pp) N (p: and satisfies the following boundary condition, &z(cpP = 0) = 0. The reduction of the Lagrangian, Eq.(7), into its first order form preceeds the begining of conversion process, thus i(O) = A~O)f”)6 + 7ree - V(O) 7 a where rre is the canonical momentum to the WZ variable, that is, (9) conjugated B (Pnx. Suppl.) 127 (2004) The corresponding 170-I symplectic tensor is (13) which should be a singular matrix. The implementation of the symplectic embedding scheme consists in computing the arbitrary function (G(ui,pi, X,)). To this end, the correction terms in order of X,, within by SC”) (ai, pi, X,), must be computed as well. If the symplectic matrix, Eq.(13), is singular, it has a zero-mode I, and, consequently, we have (14) where we assume that this zero-mode is go) = ( y” 0 0) ) G(ai,pi, A,) = 2 G(n)(ai,~i, n=O A,), (11) with G(n)(ui,pi, A,) - Xg. In this context, the zeroth one-form canonical momenta are given by with & =1,2,.. . ,N, with &= N + 1, with 6= N + 2. (12) (15) where y”, is a generic line matrix. Using the relation given in Eq.(14) together with Eq.(13) and Eq.(15), we get yy!JJ = 0. (16) In this way, a zero-mode is obtained and, in agreement with the symplectic formalism, this zero-mode must be contracted with the gradient of the symplectic potential, namely, As a consequence, a constraint The expanded symplectic variables are $“)a E (ai,pi, yP) and the new symplectic potential becomes V(O) = V(O) + G(ai,pi, A,), (p = 1,2), where X, = (0, ~0). The arbitrary function G(ai , pi, X,) is expressed as an expansion in terms of the WZ fields, namely 171 73 arise as being (18) Due to this, the first-order Lagrangian ten as r(O)6 L(1) = A!$ +n&+R?j-P(l), is rewrit- (19) where v(l) = V(O) Io=s. Note that the symplectic variables are now c(l)& q (ai, pi, r], A,) (with & = 1,2,. . . , N + 3) and the corresponding symplectic matrix becomes A. C.R. Mendes et al. /Nuclear Physics B (Pmt. Suppl.) 127 (2004) 17&l 73 172 where dependent zero-modes, Eq.(22), through canonical transformation (Vet) =(‘J) - T.tS[f,‘,(,,) where bar means transpose matrix. For example, av(0) i@O)” aG(ai,Pi,&) ap)” ’ >I a fme = --anI >I ai-2 fal)= ~apb + + Wai, pi, A,> ap” = + (21) ’ aW 9 apb [ ( wai, Pi,&I) apb . a a$o)a )I Since our goal is to unveil a WZ symmetry, this symplectic tensor must be singular, consequently, it has a zero-mode, namely, fi(‘tiCa, = (l-q/) l a b>, which satisfies the relation fi$%)~!$ = 0. Note that the parameters (a,b) can be 0 or 1 and Y indicates the number of choices for fi(1)a5. As a consequence, there are two independent set of zero-modes, given by While, in the context of the BFFT formalism, different choices for the degenerated matrix X leads to different gauge invariant version of the second class model[4]. Now, it becomes clear that the arbitrariness presents on the BFFT and iterative constraint conversions methods has its origin on the choice of the zero-mode. From relation Vet) -(l)’ I(‘-’ aa = 0, together with Eq.(20) and fitiiCQ, = (@,, ferential equations involving tained, namely, 51t is important to notice that v is not a fixed parameter b), some dif- cl = Pft)f$ + fG?7 Solving terms, the relations Cz=o G(“)(uij also including (G(O)(% Note that the matrix elements @,, present some arbitrariness which can be fixed in order to disclose a desired WZ gauge symmetry. In addi-(1)6 tion, in our formalism the zero-mode ~(~1~~)is the gauge symmetry generator, which allows to display the symmetry from the geometrical point of view. At this point, we call attention upon the fact that this is an important characteristic since it opens up the possibility to disclose the desired hidden gauge symmetry from the noninvariant model. Different choices of the zero-mode generates different gauge invariant versions of the second class system, however, these gauge invariant descriptions are dynamicaly equivalent, i.e., there is the possibility to relate this set of in- 1 a G(ui,pi, A,) are ob- Pi, above, some correction within can be determined, X ) t”hb boundary conditions Pi, A, = 0)). In order to compute the remaining corrections terms of G(oi, pi, X,), we impose that no more constraints arise from the contraction of the zero-mode ($ii%, ) with the gradient of potential v(l) (ui, pi, X,). This condition generates a general differential equation, which reads as 0 = +;;ta, = I$, aV(l)(ui,pi,X,) a,$)~ dV(‘) (ui, pi) [ O" +‘Cn=O O” +c a,pff W”)(ai,pi, A,) apa m=O 1 @")(Ui,pi,X,) de (25) A.C.R. Mendes et al. /Nuclear Physics B (Pmt. Suppl.) 12 7 (2004) 170-I 73 The last relation allows us to compute all correction terms in order of X,, within Gcn)(ai,pi,X,). Note that this polynomial expansion in terms of X, is equal to zero, subsequently, all the coefficients for each order in this WZ variables must be identically null. In view of this, each correction term in orders of X, can be determined as well. For a linear correction term, we have and rrrgtogether with the canonical momentum relation conjugated to 8, given in Eq.(lO). Due to this, the gauge invariant Hamiltonian is obtained explicitly and the zero-mode $ii;h, is identified as being the generator of the infinitesimal gauge transformation, given by -(l)& “t$)(a) = E”(z+l)’ where e is an infinitesimal 2. Final (26) where the relation V(l) = V(o) was used. For a quadratic correction term, we get aG2) (% Pi, A,) a?@ . From these equations, a recursive equation n 1 2 is proposed as + b 173 (27) for (30) parameter. discussion In summary, we reformulate noninvariant systems as gauge invariant theories using an alternative WZ embedding formalism, based on a geometrical treatment, which allows us to reveal a desired hidden symmetry on second class models. Indeed, these symmetries belong to an independent set of zero-modes, which is a pleasant feature of this formalism. Due to this, it was possible to shed some light on the understanding of the origin of arbitrariness of the usual WZ constraint conversion methods. 3. Acknowledgments This work is supported in part by FAPEMIG and CNPq, Brazilian Research Agencies. (28) which allows us to compute the remaining correction terms in order of 0 and ~0. This iterative process is successively repeated up to Eq.(25) when it becomes identically null(case i) or when an extra term @n) (ai, pi, X,) can not be computed( case ii). Then, the new symplectic potential is written as ~(l)(ai,pi,X,) = V(‘)(ai,pi) + G(ai,pi, A,). (29) For the case i, the new symplectic potential is gauge invariant. For the second case ii, due to some corrections terms within G(ai, pi, X,) that are not yet determined, this new symplectic potential is not gauge invariant. As a consequence, there are some WZ counter-terms in the new symplectic potentiaLwhich can be fixed using Hamilton’s equation of motion for the WZ variables 8 REFERENCES L.D. Faddeev , Phys.Lett.Bl45, 81 (1984). L.Faddeev and R.Jackiw, Phys.Rev.Lett.60, (1988) 1692; N.M.J.Woodhouse, Geometric Qua&i&ion (Clarendon Press, Oxford, 1980). J.Barcelos-Neto and C.Wotzasek, M0d.Phys.Lett.A 7, (1992) 1172; Int.J.Mod.Phys.A7 (1992) 4981. J.Barcelos-Neto, Phys.Rev.D55, (1997) 2265.

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