General Solution and Canonical Quantization of the Conic Path Constrained Second-Class System

Relativistic conic motion as a second-class dynamical system

“…so that E can actually represent the physical energy. In this way, by expanding in powers of the dimensionless quantity (E − V )/mc 2 , the relativistic solution (36) can be shown to match the non-relativistic approximation displayed in [29]. Depending on the specific form of the potential function, eq.…”

“…It is worth mentioning that, in the well-defined mathematical limit lim c→∞ S ± (x) = 0, (33) we recover the non-relativistic differential equation discussed in ref. [29]. Next, we observe that (34) represents an integration factor for the differential equation (25).…”

“…In similar fashion, closer to our current problem, the non-relativistic motion along a conic path as a Dirac-Bergmann constrained system has been recently investigated in refs. [27][28][29]. In [27], a full BRST formulation for the conic constrained motion has been achieved with its corresponding conserved BRST charge obtained in an operatorial quantization framework, while in [28] the symplectic Faddeev-Jackiw-BarcelosNeto-Wotzasek (FJBW) quantization approach has been employed with a thorough calculation of all Faddeev-Jackiw brackets leading to a gauge-invariant description for that model.…”

“…In [27], a full BRST formulation for the conic constrained motion has been achieved with its corresponding conserved BRST charge obtained in an operatorial quantization framework, while in [28] the symplectic Faddeev-Jackiw-BarcelosNeto-Wotzasek (FJBW) quantization approach has been employed with a thorough calculation of all Faddeev-Jackiw brackets leading to a gauge-invariant description for that model. The general solution and quantum algebra realization in terms of differential operators for the non-relativistic case has been obtained in detail in [29]. In spite of being constructed initially for comparative purposes, the models originally introduced in references [18,27,28] considerably grew up and acquired proper live interest being subsequently used as starting points for further discussions regarding (anti-)(co-)BRST symmetry [30,31], superfields [32], firstorder formalism and gauge invariance [33], Batalin-Fradkin-Fradkina-Tyutin (BFFT) Abelian conversion [34] and gauge-unfixing [35], giving rise to important genuine applications in field theory such as the Lorentz violating bumblebee model [34] and the non-linear sigma model [22,35].…”

“…On the other hand, the very starting from a relativistic version in (1+0) could certainly shed new light to model conception in field theory. Filling that gap, in the current work, we consider the relativistic motion along a general conic path as a Dirac-Bergmann constrained dynamical system, generalizing the classical ideas contained in [27][28][29]. Exploring the emerging algebraic structure, we provide a thorough and detailed analysis of the model concerning both its Lagrangian and Hamiltonian descriptions and perform its canonical quantization.…”