ReviewPath integral treatment for a Coulomb system constrained on D-dimensional sphere and hyperboloid
Introduction
Up to now, the problem of path integral formulation in curved space has not been definitively solved. This is related to the operator-ordering problem in quantum mechanics. In fact, to deduce the good effective potential due to the curvature, one has to refer to the Hamiltonian formulation. As it appears in the Lagrangian, the metric tensor depending on the position, one cannot write the kinetic part at the quantum level clearly, we recourse to the Hamiltonian using the Laplace–Beltrami operator. Then we introduce the momentum operator in the Hamiltonian using the Weyl order from which we deduce the Lagrangian formulation [1]. During the previous decade, a partial Lagrangian solution analogous to this procedure was proposed including a quantum equivalence principle, where all the discretization prescriptions are equivalents [2]. In our opinion, this solution is not complete because during the evolution in curved space the constraints indicating that this evolution alone is not sufficient for a complete description are essential, so we have to supply it with some constraints on the state space to ensure the good interpretability of the theory. According to this program, the quantum corrections are the product of the reduction of the phase space to an effective one using the Dirac brackets method [3], [4] and up to now a concrete bond between these approaches has not been established yet. On the other hand we have an opposition. In fact, in path integral the Dirac brackets method is taken into account by using a delta functional which allows a reduction of phase space. This technique is known as Faddeev–Senjanovic formulation [5]. Furthermore, according to this technique the use of mid-point prescription is privileged [4], [6], [7] contrary to this quantum equivalence principle.
As the problem is still raised, let us poke the discussion by studying the case of simple curved spaces known as homogeneous spaces [8], we particularly take the D-sphere and the D-pseudosphere noted, respectively, as SO (D + 1)/SO (D) and SO (D, 1)/SO (D). These two cases had been treated before using the usual canonical method. We propose to re-examine them within the most natural framework of the constraints, i.e., the Faddeev–Senjanovic formalism. Concretely, we choose the Coulomb potential already treated by [9] with D = 3 and by [10] with D unspecified. In the same way, we will convert the problem to the path integral proper to the dynamic symmetry SU (1, 1) using space–time transformations. However, in our approach the choice of these transformations is carried out so as to avoid the singularity responsible for the instability of the integrals by rejecting them to infinity. Consequently, in a stage of calculations one obtains clearly a stable path integral [11].
In Section 2, we expose the review of general Faddeev–Senjanovic formulation in the case of unspecified variety and interaction. In Section 3, we consider the case of the Coulomb interaction on the D-sphere. We consider the same problem on the D-pseudosphere in Section 4. Section 5 is devoted to concluding remarks.
Section snippets
Review of Faddeev–Senjanovic method
Let us study a particle subjected to the action of scalar and vectorial potential moving on the D-surface immersed in the space of D + 1 dimensions. The Hamiltonian governing the dynamics of this physical system is given bywhere π = (p − eA (x)) and, x, p and A are vectors of D-dimensions. λ is the Lagrange multiplier.
Applying the habitual Dirac procedure, the involved constraints are
The Coulomb problem on SD sphere
For the case of the sphere SD, the function f (x), (x = xi, i = 1, … , D + 1), is given asR being the radius of sphere. The Poisson bracket is then easily evaluated and the propagator (11) is written aswiththe variables χ ∈ [0, π/2], θ1, … , θD−2 ∈ [0, π] and φ ∈ [0, 2π].
Thus, the expression of the propagator (14) becomes
The Coulomb problem on HD pseudosphere
Let the pseudosphere HD immersed in a D + 1 pseudoEuclidian space defined by the equationwhere R being the radius and the scalar product of two vectors is defined byAs previously, the propagator (11) is written asWe introduce the adequate coordinateswith the variables χ ∈ [0, ∞[, θ1, … , θD−2 ∈ [0, π] and φ ∈ [0, 2π].
The quantum
Conclusion
In this paper we achieved a fundamental work concerning the quantification in curved spaces. This problem gives way to an open debate because its final solution has not been established yet. We have tried to deal with this by considering the sphere and the hyperboloid with D dimensions using the constraints method where one is obliged to choose the mid-point prescription contrary to what is stipulated by the quantum principle equivalence. In addition to this, we have treated the case of the
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