Coordinate–Free Quantization of Second–Class Constraints

The conversion of second-class constraints into first-class constraints is used to extend the coordinate-free path integral quantization, achieved by a flat-space Brownian motion regularization of the coherent-state path integral measure, to systems with second-class constraints 1 Second-class constraints On performing the Legendre transformation for generalized velocities of a Lagrangian of a dynamical system, one very often gets relations between canonical coordinates and momenta that do not involve time derivatives. They are therefore not equations of motion and are called primary constraints [1]. The primary constraints should be satisfied as time proceeds, which leads to further conditions on dynamical variables known as secondary constraints [1]. Let φa = φa(θ) = 0 be all independent constraints (primary and secondary) in the system; here θ, i = 1, 2, ..., 2N , denote canonical variables that span a Euclidean phase space of the system. The canonical symplectic structure is assumed on the phase space {θi, θj} = ◦ ω ij ; one can, for instance, set q = θ and pn = θ, n = 1, 2, ..., N for canonical coordinates and their momenta, then {pn, qm} = δ n and other components of the canonical symplectic structure are zero. Let H(θ) be the canonical Hamiltonian of the system. Since φa is a complete set of constraints, φ̇a = {φa, H} = C a(θ)φb ≈ 0 , (1.1) where the symbol ≈ implies the weak equality [1] that is valid on the constraint surface φa = 0. Systems with constraints admit a more general dynamical description where the Hamiltonian can be replaced by a generalized oneHT = H+λ (θ, t)φa(θ) with λ a being arbitrary functions of θ and time: θ̇ = {θ, HT} ≈ {θ, H}+ λ{θ, φa} . (1.2) Alexander von Humboldt fellow; on leave from Laboratory of Theoretical Physics, JINR, Dubna, Russia.