Dirac Formulation of Free Open String

Abstract

Dirac formulation of open relativistic strings as systems with constraints is made explicitly. Classical theory is given in the standard light-cone and covariant center-of-mass gauges. It is mentioned that the well-known result D = 26 is affected by using the standard quantization of the mutually independent nonphysical boson creation and annihilation operators. It is shown that in the Dirac formulation these operators are not independent in both the gauges. Concepts of Physics, Vol. IV, No. 4 (2007) DOI: 10.2478/v10005-007-0023-x 487 We give some new conditions on these operators and show that the theory is consistent with Poincare algebra in any dimension D. 488 Concepts of Physics, Vol. IV, No. 4 (2007) Dirac Formulation of Free Open String 1 Hamilton description of classical open string We will study the Nambu–Goto [1] free open string in dimension D. We assume the sign convention gμν = diag(−1, 1, . . . , 1), where μ, ν = 0, 1, . . . , D − 1. The string is described by the functions X(τ, σ), where τ ∈ R and σ ∈ 〈0, π〉. The classical string is described by Lagrangian L(X) = −ω ∫ π 0 Ldσ, where ω > 0 is a constant and the Lagrangian density L(τ, σ) is L = √( ẊX ′ )2 − (Ẋ)2(X ′)2 . A dot means partial derivation with respect to τ , a dash with respect to σ, and XY = gμνXY ν = XμY ν . The boundary conditions are X ′ μ(τ, 0) = X ′ μ(τ, π) = 0. In the Hamiltonian formulation we define momenta Pμ(τ, σ) = δL δẊμ(σ) = ω Ẋμ(X ′X ′)−X ′ μ(ẊX ′) √( ẊX ′ )2 − (Ẋ)2(X ′)2 . (1) From (1) we obtain the relations Φ1 = 1 2 ( P 2 + ω ( X ′ )2) = 0 , Φ2 = PX ′ = 0 (2) called constraints. For the Poisson brackets of two functionals F (X,P ) and G(X,P ) we have { F,G } = ∫ π 0 ( δF δXμ(σ) δG δPμ(σ) − δF δPμ(σ) δG δXμ(σ) ) dσ . (3) In particular, the relation { X(σ), P ν(σ′) } = gδ(σ − σ′) is valid. The Hamiltonian of the system with constraints (2) is