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Massless particle in 2d spacetime 9 with constant curvature 9 ∗ 9 1 n George Jorjadze and W lodzimierz Piechocki a † ‡ J 7 †Razmadze Mathematical Institute, 380093 Tbilisi, Georgia 2 v ‡So ltan Institute for Nuclear Studies, 00-681 Warsaw, Poland 9 9 1 February 1, 2008 2 1 8 9 / h Abstract t - p e We consider dynamics of massless particle in 2d spacetimes with con- h : v stant curvature. We analyze different examples of spacetime. Dy- i X namical integrals are constructed from spacetime symmetry related r a to sl(2.R) algebra. Mass-shell condition restricts dynamical integrals to a cone (without vertex) which defines physical-phase space. We parametrize the cone by canonical coordinates. Canonical quantization with definite choice of operator ordering leads to unitary irreducible representations of SO (2.1) group. ↑ PACS: 0460M, 0220S, 0240K ∗ Corresponding author: W lodzimierz Piechocki, Theory Division (Zd.P8), So ltan In- stitute for Nuclear Studies, Hoz˙a 69, 00-681 Warsaw, Poland, E-mail: [email protected], Fax: (48 22) 621 60 85 1 Keywords: 2d spacetime with constant curvature; Dynamical integrals; Canonical quantization; Representations of SO (2.1) group. ↑ 2 1 Introduction We analyze classical and quantum dynamics of a massless particle in two dimensional spacetime with constant curvature R = 0. We consider the 0 6 cases when spacetime is: (i) hyperboloid with R < 0, (ii) half-plane with 0 R < 0 and (iii) stripe with R > 0. Presented results complete our analysis 0 0 of dynamics of a particle with non-zero mass m in 2d curved spacetime 0 [1,2]. Taking formally m 0 leads to some singularities both at classical 0 → and quantum levels. Thus, the massless case needs separate treatment. As the result we get clear picture of the role played by topology and global symmetries of spacetime in the procedure of canonical quantization. Dynamics of a massless particle in gravitational field g (x0,x1) is defined µν by the action [3] 1 S = L(τ) dτ, L(τ) := g (x0(τ),x1(τ)) x˙µ(τ)x˙ν(τ), (1.1) Z −2λ(τ) µν where τ is an evolution parameter, λ plays the role of Lagrange multiplier and x˙µ := dxµ/dτ. It is assumed that λ > 0 and x˙0 > 0. The action (1.1) is invariant under reparametrization τ f(τ),λ(τ) → → ˙ λ(τ)/f(τ). This gauge symmetry leads to the constrained dynamics in the Hamiltonian formulation [4]. The constraint reads Φ := gµνp p = 0, (1.2) µ ν where p := ∂L/∂x˙µ are canonical momenta (we use units with c = 1 = h¯). µ As in [1,2] we use the gauge invariant description. In the case of massive particle the set of spacetime trajectories can be considered as a physical phase-space of the system. This set has natural symplectic structure, which 3 canbeusedforquantization. Thespacetimetrajectoriesofamasslessparticle has no symplectic structure (this set is only one dimensional, since particle velocity is fixed). For the gauge invariant description we use the dynamical integrals constructed from the global symmetries of spacetime. 2 Dynamics on hyperboloid Let (y0,y1,y2) be the standard coordinates on 3d Minkowski space with the metric tensor η = diag(+, , ). A one-sheet hyperboloid H is defined by ab − − (y0)2 +(y1)2 +(y2)2 = m−2, (2.1) − where m > 0 is a fixed parameter. H has a constant curvature R = 2m2 − (see [5]). Any 2-dimensional Lorentzian manifold with constant curvature R can 0 be described (locally) by the conformal metric tensor [5] 1 0 g (X) = expϕ(X)  , X := (x0,x1), (2.2) µν  0 1   −  where the field ϕ(X) satisfies the Liouville equation [6] 2 2 (∂ ∂ )ϕ(X)+R expϕ(X) = 0. (2.3) 0 1 0 − Making use of the parametrization cotmρ cosmθ sinmθ 0 1 2 y = , y = , y = , − m msinmρ msinmρ where ρ ]0,π/m[, θ [0,2π/m[ (2.4) ∈ ∈ we get the conformal form (2.2), with 2 ϕ = logsin mρ, (2.5) − 4 where the spacetime coordinates x0 and x1 areidentified with the parameters ρ and θ, respectively. The function (2.5) satisfies the Liouville equation (2.3) for R = 2m2. 0 − The Lagrangian (1.1) in this case reads ρ˙2 θ˙2 L = − . (2.6) −2λsin2mρ The hyperboloid (2.1) is invariant under the Lorentz transformations, i.e., SO (2,1) is the symmetry group of our system. The corresponding ↑ infinitesimal transformations (rotation and two boosts) are (ρ,θ) (ρ,θ+α /m), 0 −→ (ρ,θ) (ρ α /m sinmρsinmθ,θ +α /m cosmρcosmθ), 1 1 −→ − (ρ,θ) (ρ+α /m sinmρcosmθ,θ+α /m cosmρsinmθ). (2.7) 2 2 −→ The dynamical integrals for (2.7) read p p p θ ρ θ J = , J = sinmρsinmθ + cosmρcosmθ, 0 1 m −m m p p ρ θ J = sinmρcosmθ + cosmρsinmθ, (2.8) 2 m m ˙ where p := ∂L/∂θ, p := ∂L/∂ρ˙ are canonical momenta. θ ρ Since J isconnected withspace translations(see(2.7)),it defines particle 0 momentum p = mJ . θ 0 It is clear that the dynamical integrals (2.8) satisfy the commutation relations of sl(2.R) algebra J ,J = ε ηcdJ , (2.9) a b abc d { } whereηcd istheMinkowski metrictensorandε istheanti-symmetrictensor abc with ε = 1. 012 5 The mass shell condition (1.2) takes the form p2 p2 = 0 and it leads to ρ− θ the relation 2 2 2 J J J = 0. (2.10) 0 1 2 − − Eq. (2.10) defines two cones. The singular point of the cones J = 0 = J = 0 1 J should be removed, since it corresponds to the massless particle with zero 2 momentum (p = 0 = p ), which does not exist. ρ θ Thus, the dynamical integrals (2.8) define the physical phase-space of the system and it consists of two disconnected cones and , for J > 0 and + − 0 C C J < 0, respectively. 0 According to (2.8) and due to p < 0 (since ρ˙ > 0 and λ > 0) the ρ trajectories satisfy the equations p J J ya = 0, J y J y = ρ = | 0|. (2.11) a 1 2 − 2 1 m2 − m Each point (J ) of the cone or defines uniquely the trajectory on the a + − C C hyperboloid H. The trajectories (2.11) are straight lines in 3d Minkowski space. Hence, the set of trajectories is the set of generatrices of the hyper- ˙ boloid (2.1). For J > 0 we get the ‘right’ moving particle with θ > 0, while 0 ˙ for J < 0 we have the ‘left’ moving one, with θ < 0. Both cones, and , 0 + − C C are invariant under SO (2.1) transformations. ↑ To quantize the system, we consider the cones and separately. We + − C C parametrize as follows + C 1 p q J = (p2 +q2), J = p2 +q2, J = p2 +q2, (2.12) 0 1 2 2 2 q 2 q where (0,0) = (p,q) R2. 6 ∈ Itiseasytoseethat(2.12)defines theone-to-onemapfromtheplanewithout 6 the origin to . The canonical commutation relation p,q = 1 provides + C { } (2.9). For quantization of system we introduce the creation-annihilation op- + C erators a± := (pˆ iqˆ)/√2 and choose the definite operator ordering in (2.12). ± This ordering is defined by the following requirements: a) the operators Jˆ are self-adjoint, a b) they generate global SO (2.1) transformations, ↑ c) the spectrum of Jˆ is positive, 0 d) the Casimir number operator Cˆ := Jˆ2 Jˆ2 Jˆ2 is zero. 0 1 2 − − This leads to the following expressions Jˆ = a−a+, Jˆ = a+√a−a+ +1, Jˆ = √a−a+ +1 a−, (2.13) 0 + − ˆ ˆ ˆ where J = J J . The creation and annihilation operators are defined in ± 1 2 ± the Fock space in the standard way a+ n = √n+1 n+1 , a− n = √n n 1 , | i | i | i | − i where the vectors n , (n 0) form the basis of the corresponding Hilbert | i ≥ space . + H The states n are the eigenstates of Jˆ 0 | i ˆ J n = (n+1) n , (2.14) 0 | i | i and from (2.13) we get Jˆ n = (n+1)(n+2) n+1 , Jˆ n = n(n+1) n 1 . (2.15) + − | i q | i | i q | − i Eqs.(2.13)-(2.15) define the unitary irreducible representation (UIR) of the group SO (2.1). We identify this representation with the representation of ↑ D+ of the discrete series of SL(2.R) group [7]. 1 7 Quantization of the case can be done in the same way. The corre- − C sponding Hilbert space is again the Fock space, but we have to make − H the following replacements: Jˆ Jˆ and Jˆ Jˆ. As a result we get the ± ∓ 0 0 → → − representation D−. 1 The Hilbert space of the whole system is = and the corre- + − H H ⊕H sponding representation D+ D− describes the SO (2.1) symmetry of the 1 1 ↑ ⊕ quantum system. It is interesting to mention the following: At the classical level our system can be obtained from the massive case in the limit m 0 (where m is a particle mass) and by removing the singular 0 0 → point of the physical phase-space. The quantum theory of the massive case was considered recently in [1]. The corresponding representation is defined by the operators Jˆψ = nψ , Jˆ ψ = √n2 +n+a2 ψ , 0 n n + n n+1 Jˆ ψ = √n2 n+a2 ψ , (2.16) − n n−1 − where ψ := exp inφ (n Z) form the basis of the Hilbert space L (S1) and n 2 ∈ a = m /m. For a = 0 (m = 0) this representation terns into D+ A D−, 0 0 1 1 ⊕ ⊕ where A is a one dimensional trivial representation on the vector ψ . By 0 removing A we get the quantum theory of the massless case. The momentum of a quantum particle pˆ = mJˆ can take only discrete θ 0 values P = mn, where n is a nonzero integer. n 8 3 Dynamics on half-plane Let us consider the Liouville field ϕ = 2logm x0 , given on a plane (x0,x1). − | | This field defines constant spacetime curvature R = 2m2 (see (2.3)). The 0 − Lagrangian (1.1) in this case reads 2x˙+x˙− L = , (3.1) −λm2(x+ +x−)2 where x± := x0 x1. It is assumed that x˙0 > 0, which leads to p +p < 0. + − ± Formally, (3.1) is invariant under the fractional-linear transformations ax+ +b ax− b x+ , x− − , ad bc = 1. (3.2) → cx+ +d → cx− +d − − Thus, formally, SL(2.R)/Z2 (which is isomorphic to SO↑(2.1)) is the symmetry of our system. The transformations (3.2) are well defined on the plane only forc = 0. The corresponding transformations with c = 0 formthe group of dilatations and translations (along x1), which is a global symmetry of the considered spacetime. The infinitesimal transformations for (3.2) are x± x± α , x± x± +α x±, x± x± α (x±)2 (3.3) 0 1 2 → ± → → ± and the corresponding dynamical integrals read P = p p , K = p x+ +p x−, M = p (x+)2 p (x−)2, (3.4) + − + − + − − − where p = ∂L/∂x˙±. ± The dynamical integrals (3.4) satisfy again the commutation relations (2.9) with 1 1 J = (P +M), J = (P M), J = K. (3.5) 0 1 2 2 2 − 9 The mass-shell condition (1.2) leads to p = 0 for P > 0 and p = 0 for + − P < 0. Due to (3.4) and the mass-shell condition, we have 2 K PM = 0 (3.6) − and the trajectories read K P 1 0 x ǫ(P)x = , where ǫ(P) = . (3.7) − P P | | It is clear that x0 = 0 is the singularity line in the spacetime. In the massive case this singularity leads to the dynamical ambiguities [2,8]. However, the dynamics of the massless particle is defined uniquely due to (3.7). The physical phase-space is defined by two cones (3.6) without the line corresponding to P = 0. Thus, we have two disconnected parts: with + P P > 0 and with P < 0. Both cones ( and ) are invariant under − + − P P P dilatations and translations generated by the dynamical integrals K and P. But, the transformations generated by M are not defined globally. Thus, the physical phase-space has the same symmetry as the spacetime. Let us quantize the system corresponding to case (the case can be + − P P done in the same way). We parametrize as follows [9] + P 2 P = p, K = pq, M = pq , (3.8) where(p,q)arethecoordinatesonhalf-planewithp > 0. The canonicalcom- mutation relation p,q = 1 provides the commutation relations of sl(2.R) { } algebra P,K = P, P,M = 2K, K,M = M . (3.9) { } { } { } 10