Symplectic Projector and Physical Degrees of Freedom of The Classical Particle

Transcription

1 Sympleti Projetor and Physial Degrees of Freedom of The Classial Partile M. A. De Andrade a, M. A. Santos b and I. V. Vanea arxiv:hep-th/ v3 7 Sep 2003 a Grupo de Físia Teória, Universidade Católia de Petrópolis Rua Barão de Amazonas 124, , Petrópolis - RJ, Brasil b Departamento de Físia, Universidade Federal Rural do Rio de Janeiro , Seropédia - RJ, Brasil Departamento de Físia e Matemátia, Fauldade de Filosofia, Ciênias e Letras de Ribeirão Preto USP, Av. Bandeirantes 3900, Ribeirão Preto , SP, Brasil Abstrat The sympleti projetor method is applied to derive the loal physial degrees of freedom of a lassial partile moving freely on an arbitrary surfae. The dependene of the projetor on the oordinates and momenta of the partile is disussed. maro@bpf.br masantos@ufrrj.br ion@dfm.fflrp.usp.br, ivanea@ift.unesp.br. 1

2 1 Introdution The problem of finding the physial degrees of freedom of a onstrained system an be traed bak to the work of Dira [1] and represents the entral issue in the quantization of realisti models. The lassial method to deal with this sort of problems is to enlarge the phase spae by adding non-physial variables to the original ones and to define the physial surfae with the help of a nilpotent operator ating on the all degrees of freedom. This is the well known BRST method (see [2] for a reveiw.) However, in many ases when the struture of the onstraints is simpler, there are other ways of finding the physial degrees of freedom. In partiular, when the onstraints are of seond lass only, a loal sympleti projetor an be written down and the loal physial degrees of freedom an be omputed from it ([3, 4, 5, 6, 7, 8, 9, 10, 11]). A lass of systems that may exhibit simple seond lass onstraints is given by a partile moving on a surfae defined by a holonomi funtion f(x) = 0. The way in whih suh system an arise and the problems related to the quantization of it were addressed in [12]. In this letter we are going to study the dynamis of the lassial partile on a surfae from the point of view of the sympleti projetor method. This study is usefull for understanding the loal degrees of freedom of the partile as well as the range of the appliability of the method to onstrained systems. The paper is organized as follows. In Setion 2 we review the method of the sympleti projetor. In Setion 3 we derive the loal degrees of freedom for a free partile on a surfae embedded in R n and the orresponding Hamiltonian. The last setion is devoted to disussions. 2 Brief Review of The Sympleti Projetor Method The sympleti projetor an be defined for any system subjet to seond lass onstraints φ m ( ξ M) = 0. Here ξ M = (x a,p a ), M = 1,2,...,2N are the oordinates in the phase spae whih is assumed to be isomorphi to R 2N and m = 1,2,...,r = 2k. The sympleti projetor has the following form [4] Λ MN = δ MN J ML δφ m δφ n δξ L 1 mn δξ N, (1) where J MN is the sympleti two-form and 1 mn is the inverse of the matrix mn = {φ m,φ n }. (2) The sympleti projetor given by the relation (1) projets the phase spae variables ξ M onto loal variables on the onstraint surfae ξ ξ M = Λ MN ξ N. (3) Starting with a 2N-dimensional phase-spae and 2k seond lass onstraints, we are led to a vetor with 2(N k) independent omponents. As was noted in [3], the boundary onditions that should be satisfied by the normal oordinates and the orresponding momenta to the onstraint surfae are given by the following relations x m (t) = φ m (x) = 0, (4) p m (t) = ṗ m (t) = 0. (5) 2

3 Then, the physial Hamiltonian is given by the original one written in terms of the oordinates that are obtained after projetion, i. e. the loal oordinates on the physial surfae. These variables are independent, free of onstraints and they obey anonial ommutation relations. The equations of motion follow from the usual Hamilton-Jaobi equations:. ξ = {ξ,h }, (6) where {, } are the Poisson brakets. From the analogy between the Dira matrix D MN = {ξ M, ξ N } D = J MN J ML J KN δφ m δφ n δξ L 1 mn δξ K, (7) and the projetor given by the relation (1) one an see [10] that the following relation holds: Λ = DJ. (8) We note that the trae of the projetor matrix gives the degrees of freedom of the system. In order to quantize the theory, one should start with the physial Hamiltonian obtained above. Then, the observables of the quantum theory should depend only on the oordinates ξ. 3 Free Partile on a Surfae f(x) = 0 in R N Let us onsider a partile moving freely on a smooth arbitrary surfae Σ embedded in R N and defined analytially by the equation Σ : f(x) = 0, (9) where x = {x a }, a = 1,2,,N are the Cartezian oordinates in R N. The movement on the surfae Σ an be obtained by introduing the f (x) in the Hamiltonian through a Lagrange multiplier λ H = 1 2m p ap a λf(x) (10) and then interpreting λ as an independent variable. Thus, the extended phase spae is oordinatised by x µ = (λ,x a ) and p µ = (p λ,p a ) and it is endowed with an Eulidean metri and a sympleti two-form. The onstraints on the dynamis are derived in the usual manner, by the Dira algorithm whih gives the following set of equations φ 1 = p λ (11) φ 2 = f(x) (12) f x a (13) φ 3 = pa m φ 4 = pa p b m 2 a b f + λ m a f a f. (14) The onstraints (11)-(14) are of seond lass, therefore one an apply the Sympleti Projetor Method to determine the loal physial degrees of freedom [3, 10]. To this end, 3

4 one omputes firstly the Dira brakets in the extended phase spae. The non-zero Dira brakets are {λ, x b } D = k b {λ, p b } D = v b where we have used the following notations k b = {x a, p b } D = δ a b n a n b = δ a b {p a, p b } D = ω ab, (15) 2p m f n b (16) v b = p p d m f [ d n b + 3( n a )( d n a )n b ] 2λn n b (17) ω ab = p (n b n a n a n b ) = ω ba (18) The normal vetor to the surfae n a has unit norm and is defined in the usual way n a = 1 a f = af, f mα f a f a f. (20) If we write the omponents of a sympleti vetor in the extended phase spae in the order (λ,x a,p λ,p a ) then the Dira brakets an be disposed into a (2N + 1) (2N + 1) matrix whih is alled the Dira matrix D = 0 k b 0 v b k a 0 0 δ a b v a δ a b 0 ω ab (19). (21) The sympleti projetor is given by the equation (8) and has the following form 0 v b 0 k b 0 Λ = δ a b k a (22) 0 ω ab v δ b a a By ating with the projetor (22) on the degrees of freedom of the extended phase spae we obtain the loal physial degrees of freedom of the partile [3]. Loality, in this ase, has the meaning of loal on the surfae Σ. Note that the trae of the matrix (22) is equal to the number of the degrees of freedom, whih in this ase is 2N 2. The projeted degrees of freedom are given by the following relations λ x a = v b x b k b p b = δ a bx b + k a p λ p λ = 0 p a = ω ab x b + v a p λ + δ a b p b. (23) Now let us disuss the above sytem. We an see that there are some distint ases to deal with. 4

5 3.1 General Λ (ξ) In the ase of an arbitrary funtion f, the entries of the projetor matrix are funtions of the oordinates of the partile in the phase spae. Therefore, there is no general method to separate the linearly independent physial oordinates ξ µ, where µ,ν... = 1,2,... 2(N 1) from the linearly dependent ones ξ α, where α,β,... = 1,2,3,4. However, we an derive some general relations to be satisfied by the linearly dependent oordinates as well as by the projetor. Sine there are onstants α µ suh that ξ α = α µξ µ, (24) then one an use the relation Λ 2 = Λ to find the following relations that should be satisfied by α µ s (δ ν µ Λ µ α (ξ) α ν )Λ ν N (ξ) ξ N = 0, (25) ( ) δβ α Λ α β (ξ) β µλ µ N (ξ) ξn = 0. (26) The above equations should be satisfied simultaneously for all ν and all β. By omparing the number of equations with the number of onstants s, we see that s may be ompletely determined only if N = 1, 2. This ondition, although neessary, is not suffiient. Indeed, if we treat (25) and (26) as a system that solves s, we see that the onstants α µ are expressed as funtions on ξ s. However, by hypothesis, s should be real numbers whih implies that these funtions be onstants. Let us look at the time evolution of an arbitrary projeted oordinate ξ M. On the onstraint surfae, it is given by its Poisson braket with the physial Hamiltonian H ξ M = {ξ M,H } PB, (27) where H = H (ξ ) depends only on the linearly independent variables and in the Poisson brakets means that they should be omputed using the projeted variables. In partiular, the relation (27) guarantees that the mapping by Λ is anonial. On the other hand-side, if we interpret ξ M (ξ) as funtion on the phase spae, then its time evolution should be given by the following relation ξ M (ξ) = {ξ M (ξ),h (ξ)} DB. (28) However, on the onstraint surfae Σ, the relations (27) and (28) should oinide, i. e. the following relation should hold {ξ M,H (ξ )} PB = {ΛM N (ξ) ξn,h (ξ)} DB Σ={φ m (ξ)=0} (29) The above relation represents a onsisteny ondition that should be satisfied by the sympleti projetor. 3.2 Constant Sympleti Projetor There are surfaes for whih the entries of the sympleti projetor are onstant funtions on the oordinates ξ. These surfaes will satisfy the following relations 2p m f b f f = Ca, (30) 5

6 p p d [ b f d m f f + 3( a f f )( a f d f ) ] bf 2λ f f f b f f = D b, (31) p ( bf f a f f af f b f f ) = O ab, (32) where C a, D b and O ab = O ba are onstant numbers. In this ase, (23) is a system of linear equations with numeri oeffiients (from R). Therefore, one an find the independeent degrees of freedom as follows. Sine the projeted momenta p λ vanishes, we should look for three more vanishing or linearly dependent degrees of freedom in (23). They an be obtained by noting that not all the oeffiients in the r.h.s. of (23) are independent sine they are entries of the projetor (22). Therefore, from the ondition we derive the following relations Λ 2 = Λ (33) v a δ a b k a ω ab = v b, (34) k a δ b a = k b, (35) δ a δ b = δ a b, (36) ω a δ b + δ a ω b = ω ab. (37) It is important to note that (33) is not a supplementary ondition imposed by hand. Rather, it is a built-in relation in the formalism beause Λ is onstruted to be the loal projetor on the onstrained surfae. Therefore, the relations (34)-(37) are natural onstraints. n a n b is the projetor on the normal diretion and δ a b is the projetor onto the tangent spae to Σ. We an use these projetors to split the oordinates (23) into normal and tangential. Then it is easy to show that x a t = δ a b x b = δ a bx b + k a p λ, (38) p ta = δ b a (ω b x + v b p λ + p b ). (39) represent the independent degrees of freedom. They are tangential to the surfae Σ as they should be. The dependent and null degrees of freedom are given by the following relations λ = v a x a t k a p ta, (40) x a n = 0, (41) p λ = 0, (42) p na = n a n b ω b x t. (43) The equations above elliminate four degrees of freedom. Therefore, the number of the loal physial degrees of freedom is 2N 2 as expeted. Note that they are expressed in terms of the global indies in the initial R N spae. One an pass to a loal oordinate system on Σ by performing a general oordinate transformation from the origin of R N to the point P on Σ. In order to elliminate the normal diretion, one has to pik-up one of the diretions of the loal oordinate system parallel to the normal. However, the tangential diretions are determined only up a SO(N 1) transformation. 6

7 The physial Hamiltonian of the system should be entirely written in terms of the tangential degrees of freedom and it is given by the folowing relation H = 1 2m p tap a t + 1 2m na n b ω a ω bd x t x d t. (44) The loal quantization of the system should be done by using (44). In order to quantize the system globally on Σ, additional information should be given, as for example, the range of the loal oordinates and the topologial struture of the theory. It is not lear yet if this information ould be inluded in a global projetor of the surfae or if it should appear from gluing the loal operators on neighbour openings. The two ases analysed above, i. e. when Λ ontaints general funtions on ξ and when it ontains just onstants, represent the two extreme ases of the sympleti projetor method. In the first ase one annont provide a general method for finding the independent degrees of freedom, while in the seond ase they are given by solving a system of linear equations with oeffiients from R. Between the two ases, there are partiular surfaes for whih the physial degrees of freedom an be omputed from the (23), but the way of solving this system depends on the speifi model. However, for all ases the physial Hamiltonian is given by (44) and it should be used for loal quantization. In order to take into aount the global properties of the onstraint surfae, one should extend this formalism. However, even if one limitates to loal analysis, there are surfaes for whih the loal physial oordinates obtained from the sympleti projetor vanish. 4 Disussions As was already noted, the sympleti projetor given by the relation (22) depends on the oordinates of the phase spae (x a,p a ). This implies that the splitting of the projeted oordinates into independent and dependent oordinates also depends on the position of the point at whih this analysis is made on the onstrained surfae. Moreover, from the relations (39) and (43) we see that the variables are really independent only if the entries of the sympleti projetor are real numbers. Also, note that the equations (39) and (43) imply that some of the physial oordinates and momenta an vanish on ertain surfaes. This an be seen by taking p λ = 0 on the surfae in (39). For example, the physial oordinates and momenta vanish whenever Also, only momenta an vanish if a f b f = δ b a f f. (45) ( a b f b f a f)x = δ ab. (46) To onlude, we have disussed the possibility of applying the sympleti projetor to determine the degrees of freedom of a partile on a surfae. From the present analysis results that the independent oordinates and momenta an be found exatly only for ertain surfaes. If the omponents of the projetor are onstant, then the solution an always be obtained from a linear system of algebrai equations with onstant oeffiients. We note that finding nontrivial solutions to the equations (30), (31) and (32) represents an interesting problem. Also, there are surfaes for whih these relations do not hold but whih admitt 7

8 solutions to the system (23). In both ases, the loal quantization of the system should be done by using (44). In the general ase, there is no systemati method to split the projeted oordinates into a set of independent variables and a linearly dependent one. The dynamis of the partile on a plane in R 3 an be onsidered as a simple example of a surfae for whih the projetor is onstant. This ase is somewhat degenerate sine the movement is free and the definition of the plane is a matter of hosing a oordinate system in R 3. However, as one an easily hek out, the system (23) gives the orret Hamiltonian and the degrees of freedom. As an example of sympleti projetor that depends on the oordinates but whose Hamiltonian an be obtained by applying the sympleti projetor method one an take the irle S 1. Then the sympleti projetor gives the Hamiltonian equal to the kineti energy in agreement with the Dira algorithm. This result is also true for S n and represents one of the intermediate ases disussed previously. These examples illustrate the above disussion. In order to quantize the system globally on Σ, additional information should be given, as for example, the range of the loal oordinates and the topologial struture of the theory. It is not lear yet if this information ould be inluded in a global projetor of the surfae or if it should appear from gluing the loal operators on neighbour openings. The globality problem invites to extending the loal sympleti operator method to global theories. We hope to report on these topis in a future paper [13]. Aknowledgments M. A. S. and I. V. V. would like to thank to J. A. Helayel-Neto for hospitality at CBPF during the preparation of this work. M. A. de Andrade aknowledeges to J. A. Helayel- Neto and M. Botta for useful disussions. I. V. V. was supported by the FAPESP Grant 02/ Referenes [1] P. A. M. Dira, Letures on Quantum Mehanis (Yeshiva University, N. Y., 1964) [2] Henneaux, Teitelboim, Quantization of Gauge Systems, (Prineton University Press, 1992) [3] C. M. Amaral, Configuration Spae Constraints As Projetors In The Many Body System, Nuovo Cim. B 25, 817 (1975). [4] P. Pitanga and C. M. Amaral, Sympleti Projetor In Constrained Systems, Nuovo Cim. A 103 (1990) [5] M. A. Santos, J. C. de Mello and F. R. Simao, On The Constrution Of The U Matrix From Dira Brakets In QCD, J. Phys. A 21, L193 (1988). [6] M. A. Santos, J. C. de Mello and P. Pitanga, Physial Variables In Gauge Theories, Z. Phys. C 55, 271 (1992). [7] M. A. Santos and J. A. Helayel-Neto, Physial variables for the Chern-Simons- Maxwell theory without matter, hep-th/ [8] L. R. Manssur, A. L. Nogueira and M. A. Santos, An extended Abelian Chern-Simons model and the sympleti projetor method, Int. J. Mod. Phys. A 17, 1919 (2002) hep-th/

9 [9] M. A. De Andrade, M. A. Santos and I. V. Vanea, Unonstrained variables of nonommutative open strings, JHEP 0106, 026 (2001) hep-th/ [10] M. A. De Andrade, M. A. Santos and I. V. Vanea, Loal physial oordinates from sympleti projetor method, Mod. Phys. Lett. A 16, 1907 (2001) hep-th/ [11] M. A. Dos Santos, Consisteny of the sympleti projetors method on multidimensional gauge models. (PhD Thesis, In Portugese), arxiv:hep-th/ [12] P. C. Shuster and R. L. Jaffe, Quantum mehanis on manifolds embedded in Eulidean spae, Annals Phys. 307, 132 (2003), arxiv:hep-th/ [13] M. A. de Andrade, M. A. Santos, I. V. Vanea, in preparation. 9