Schild Action

An action functional which is polynomial in the partial derivatives of the string coordinates X ^μ(σ^m):

where Σ is the string manifold parametrized by σ^m, m = (0,1), α is a dimensional constant related to the string tension, X ^μν ≡ ɛ ^mn ∂ _m X ^μ ∂ _n X ^ν ≡ {X ^μ, X ^ν} is the worldsheet Poisson bracket , or the tangent bivector to the string worldsheet ; ɛ^mn is the worldsheet totally antisymmetric tensor. This kind of action was originally introduced for the bosonic string in [1] to provide an alternative to the highly nonlinear Nambu-Goto action , and to encode into a single variational principle the dynamics of both timelike and null strings. Action (1) is invariant under area-preserving (symplectic) coordinate transformations: σ^m → ξ^m, det(∂ _m ξⁿ) = 1. This symmetry allows a nonstandard quantization procedure where the worldsheet proper area A ≡ ∫_Σ d ²σ plays the role of evolution parameter [2,3]. The formal analogy between the tangent bivector, written as X ^μν(σ) = ∂₀ X ^μ ∧ ∂₁ X ^ν, and...