We use the light-front machinery to study the behavior of a relativistic free
particle and obtain the quantum commutation relations from the classical
Poisson brackets. We argue that the usual projection onto the light-front
coordinates for these from the covariant commutation ralations does not
reproduce the expected results.Comment: To appear in the proceedings "IX Hadron Physics and VII Relativistic
  Aspects of Nuclear Physics: A Joint Meeting on QCD and QGP, Hadron
  Physics-RANP,2004,Angra dos Reis, Rio de Janeiro,Brazi

Sales, J. H. O.

Suzuki, A. T.

Zambrano, G. E. R.

English

arXiv

J. H. O. Sales

Crossref

Relativistic free-particle quantization on the light-front: New aspects

We use the light-front machinery to study the behavior of a re

lativistic free particle and

obtain the quantum commutation relations from the classica

l Poisson brackets. We argue that the

usual projection onto the light-front coordinates for thes

e from the covariant commutation ralations

does not reproduce the expected results

Universidad de Nariño (UDENAR)

arXiv:hep-th/0407121v1  14 Jul 2004Relativistic free-particle quantization on thelight-front: New aspectsJ.H.O.Sales∗†, A.T. Suzuki∗∗ and G.E.R. Zambrano‡∗Centro de Ciências-Universidade Federal de Itajuba, 37500-000 - MG - Brazil†henrique@fatecsp.br∗∗Department of Physics, North Carolina State University, Raleigh, NC 27695-8202‡Instituto de Física Teórica - UNESP Rua Pamplona 145, 01405-900 - SP - BrazilAbstract. We use the light-front machinery to study the behavior of a relativistic free particle andobtain the quantum commutation relations from the classical Poisson brackets. We argue that theusual projection onto the light-front coordinates for these from the covariant commutation ralationsdoes not reproduce the expected results.LIGHT-FRONT QUANTIZATIONThe Lagrangian for a free relativistic particle of mass m, is given by [2] L = −m√·x2.The canonical four momentum pµ is obtained directly frompµ =∂L∂ ·xµ=−m·xµ√·x2(1)and the fundamental Poisson brackets are:{xµ , pν}= δ µν (2)and{xµ ,xν}= 0 = {pµ , pν} (3)Canonical quantization can not be obtained directly from these brackets through the rulesof quantization, because this theory has constraints [3]. From (1), we get immediatelythatφ1 = p2−m2 ≈ 0 (4)Using this basic result in{xµ , p2−m2}={xµ , p2}−{xµ ,m2} (5)we have (observing that the second term on the right hand side yields zero straightfor-wardly) {xµ , p2}= 2pν {xµ , pν} ,which with (3) gives, {xµ , p2}= 2pµ . (6)For the component µ = 0 we have{x0, p2}= 2p0 = 2E, (7)that is, for this particular time component the Poisson bracket relates to the total energyof the system. In the case of µ =+ coordinate we have{xµ , p2}l f= p+g+− = 2p+ (8)This result agrees perfectly with the covariant case (6). From Eq.(4) transformed intothe light frontφ1 = p+p−− p⊥2−m2 ≈ 0 (9)we have the primary constraint. The light front Lagrangian isLl f =−m√·xµ ·xµ =−m√·x+ ·x−−·x⊥2. (10)The canonically conjugate momentum components arep− =−m·x−√·x2, p+ =−m·x+√·x2(11)andp⊥ =−m·x⊥√·x2. (12)The Hamiltonian isH l fc = p·x −Ll f = 0 (13)This result gives us an indication that we useH˜ l f = λ(p+p−− p⊥2−m2)which by the condition that the constraint does not evolve in time we have·φ 1={φ1, H˜ l f}≈ 0. (14)This means that there are no new constraints and we are unable to determine themultiplier λ .Let us then, as before impose a new constraint in the light frontφ2 = x+− s ≈ 0 , (15)so that,{φ1,φ2}=−2p+ (16)Comparing with the result (7) we again perceive that there is an inconsistency here.Instead of the energy p− we get the momentum p+.Constructing the Hamiltonian and with the condition of non evolution in time of theconstraint we haveH =12p+(p+p−− p⊥2−m2)(17)In order to obtain the Dirac brackets, let us construc the matrix C, where Ci j ={φi,φ j}and i, j = 1,2. The Dirac brackets then is given by{xµ , pν}l fD = gµν −{xµ ,φ1}C−112 {φ2, pν}−{xµ ,φ2}C−121 {φ1, pν} .Quantization is now achieved by[xµ , pν ]l fD = i[gµν −(p+gµ−+gµ+p−−2p⊥gµ⊥) 12p+g+ν](18)Clearly we have here the problem of zero modes in the light-front.CONCLUSIONWe have (6) for the component µ = 0 we have {x0, p2} = 2p0 = 2E, that is, for thisparticular time component the Poisson bracket relates to the total energy of the system.For the light-front case, µ = + and therefore{x+, p2}= 2p+. Which means that thelight-front “time” variable x+ relates to the momentum p+, an apparent inconsistencybetween canonically conjugate variables. In the covariante case we have a correlationbetween time and energy in the Poisson brackets, while in the light front this correlationis lost. See Eq.(8). This conclusion can be the origin to problem of zero modes in thelight-front. In the paper [4], we propose a treatment for the presented inconsistencyACKNOWLEDGMENTSJ.H.O.Sales thanks the hospitality of Instituto de Física Teórica/UNESP. A.T.Suzukiacknowledges partial support from CNPq (Brasília)and CAPES (Brasília) andG.E.R.Zambrano acknowledges total support from CAPES (Brasília).REFERENCES1. P.A.M.Dirac, Rev.Mod.Phys.21, 392 (1949).2. L.D.Landau and E.M.Lifshitz, Classical Theory of Fields, Ed. Mir Moscou (1972).3. P.A.M. Dirac, Canad. J. Math. 2 (1950) 129. P.A.M. Dirac, Canad. J. Math. 3 (1951) 1.4. A. T. Suzuki, J. H. O. Sales and G. E. R. Zambrano, hep-th/0405095.

We use the light-front machinery to study the behavior of a relativistic free particle and obtain the quantum commutation relations from the classical Poisson brackets. We argue that the usual projection onto the light-front coordinates for these from the covariant commutation relations does not reproduce the expected results

Suzuki, A.T.

http://sired.udenar.edu.co/4403/1/0407121.pdf

Relativistic free-particle quantization on the light-front: New aspects

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