Remarks on Generalized Uncertainty Principle Induced From Constraint System

Abstract

The extended commutation relations for a generalized uncertainty principle have been based on the assumption of the minimal length in position. Instead of this assumption, we start with a constrained Hamiltonian system described by the conventional Poisson algebra and then impose appropriate second class constraints to this system. Consequently, we can show that the consistent Dirac brackets for this system are nothing but the extended commutation relations describing the generalized uncertainty principle.

Frenkel electron and a spinning body in a curved background

“…Our description of spin on the base of vector variables gives one more example of physically interesting noncommutative relativistic particle, with the "parameter of noncommutativity" proportional to spin-tensor. There are other examples where the noncommutative geometry emerges from second-class constraints, see [35][36][37][38][39][40][41][42][43][44].…”

Implication of the geodesic equation in the generalized uncertainty principle framework

“…dλ is that we have taken x = X through (15). On the other hand, since we have modified the momentum part P (15), in order to get the GUP effect we have to go to the Hamiltonian formalism and derive Hamilton's equations of motion.…”

“…Consistency of the equivalence principle gives some support to construct quantum theory of gravity with the GUP. But with the proposed algebra (15) in hand, it is impossible to get the GUP effect classically in point particle dynamics. More specifically, only by modifying the momentum part P one cannot obtain any GUP effect in single particle motion.…”

“…The last relation of (15) implies . Substituting this into (19) gives the coefficient v(λ) as v(λ) = 1 2mc −g ρσ dx ρ dλ dx σ dλ .…”

“…We have considered the metric to be as usual η µν ≡ (−1, 1, 1, 1). Now it has been shown that [15] the GUP-corrected Poisson brackets are the Dirac brackets of an extended system in which x µ and p ν are initially treated as independent configuration degrees of freedom with momenta Π (x)µ and Π (p)ν , respectively.…”