The Hamilton-Jacobi method of constrained systems is discussed. The equations of motion for three singular systems are obtained as total differential equations in many variables. The integrability conditions for these syatems lead us to the canonical reduced phase-space coordinates with out using any gauge fixing conditions. The operator and the path integral quantization of these systems is discussed.
The derivation of the brackets among coordinates and momenta for classical constrained systems is a necessary step toward their quantization. Here we present a new approach for the determination of the classical brackets which does neither require Dirac's formalism nor the symplectic method of Faddeev and Jackiw. This approach is based on the computation of the brackets between the constants of integration of the exact solutions of the equations of motion. From them all brackets of the dynamical variables of the system can be deduced in a straightforward way.
For a particle moves on a 2D surface f(x)=0 embedded in 3D Euclidean space, the geometric momentum and potential are simultaneously admissible within the Dirac canonical quantization scheme for constrained motion. In our approach, not the full scheme but the symmetries indicated by classical brackets [x,H]_{D} and [p,H]_{D} in addition to the fundamental ones [x,x]_{D}, [x,p]_{D} and [p,p]_{D} are utilized, where the subscript D stands for the Dirac bracket. The generalized centripetal force law p=[p,H]_{D} for particle on the 2D surface play the key role, and there is no simple relationship b…
We consider the problem of constrained motion along a conic path under a given external potential function. The model is described as a second-class system capturing the behavior of a certain class of specific quantum field theories. By exhibiting a suitable integration factor, we obtain the general solution for the associated non-linear differential equations. We perform the canonical quantization in a consistent way in terms of the corresponding Dirac brackets. We apply the Dirac-Bergmann algorithm to unravel and classify the whole internal constraints structure inherent to its dynamical Ham…
Motivated by certain concepts introduced by the Refined Algebraic Quantization formalism for constrained systems which has been successfully applied within the context of Loop Quantum Gravity, in this paper we propose a phase space implementation of the Dirac quantization formalism to appropriately include systems with constraints. In particular, we propose a physically prescribed Wigner distribution which allows the definition of a well-defined inner product by judiciously introducing a star version of the group averaging of the constraints. This star group averaging procedure is obtained by …
So far, it is not well known how to deal with dissipative systems. There are many paths of investigation in the literature and none of them present a systematic and general procedure to tackle the problem. On the other hand, it is well known that the fractional formalism is a powerful alternative when treating dissipative problems. In this paper we propose a detailed way of attacking the issue using fractional calculus to construct an extension of the Dirac brackets in order to carry out the quantization of nonconservative theories through the standard canonical way. We believe that using the …
An operator-valued quantum phase space formula is constructed. The phase space formula of Quantum Mechanics provides a natural link between first and second quantization, thus contributing to the understanding of quantization problem. By the combination of quantization and hamiltonization of dynamics, a quantization hierarchy is introduced, beyond the framework of first and second quantization and generalizing the standard quantum theory. We apply our quantization method to quantum many-body system and propose an eclectic model, in which the dimension of Hilbert space does not scale exponentia…
Generalized Weyl quantization formalism for the cylindrical phase space $S^1 \times \mathbb{R}^1$ is developed. It is shown that the quantum observables relevant to the phase of linear harmonic oscillator or electromagnetic field can be represented within this formalism by the self-adjoint operators on the Hilbert space $L^2(S^1)$.
Kinematics and dynamics of a particle moving on a torus knot poses an interesting problem as a constrained system. In the first part of the paper we have derived the modified symplectic structure or Dirac brackets of the above model in Dirac's Hamiltonian framework, both in toroidal and Cartesian coordinate systems. This algebra has been used to study the dynamics, in particular small fluctuations in motion around a specific torus. The spatial symmetries of the system have also been studied. In the second part of the paper we have considered the quantum theory of a charge moving in a torus kno…
We implement the so-called Weyl-Heisenberg covariant integral quantization in the case of a classical system constrained by a bounded or semi-bounded geometry. The procedure, which is free of the ordering problem of operators, is illustrated with the basic example of the one-dimensional motion of a free particle in an interval, and yields a fuzzy boundary, a position-dependent mass (PDM), and an extra potential on the quantum level. The consistency of our quantization is discussed by analyzing the semi-classical phase space portrait of the derived quantum dynamics, which is obtained as a regul…
Quantization of $R^2$ and $S^1 \times S^1$ phase spaces are explicitly carried out tweaking the techniques of geometric quantization. Crucial is a combined use of left and right invariant vector fields. Canonical bases, operators and their actions are explicitly presented. Arguments of Dirac and also Wu and Yang for monopoles are applied for obtaining the quantization of the phase space area. Equivalence of states in the infinite dimensional prequantum Hilbert space resulting in a physical Hilbert space of dimension equal to the phase space area in units of the Planck constant is demonstrated.…
Mechanical systems (i.e., one-dimensional field theories) with constraints are the focus of this paper. In the classical theory, systems with infinite-dimensional targets are considered as well (this then encompasses also higher-dimensional field theories in the hamiltonian formalism). The properties of the Hamilton-Jacobi (HJ) action are described in details and several examples are explicitly computed (including nonabelian Chern-Simons theory, where the HJ action turns out to be the gauged Wess-Zumino-Witten action). Perturbative quantization, limited in this note to finite-dimensional targe…
A generalized Weyl quantization formalism for a particle on the circle investigated in \cite{1} is developed. A Wigner function for the state $\hat{\varrho}$ and the kernel $\mathcal{K}$ for a particle on the circle is defined and its properties are analyzed. Then it is shown how this Wigner function can be easily modified to give the number-phase Wigner function in quantum optics. Some examples of such number-phase Wigner function are considered.
The Weyl quantization of classical observables on the torus (as phase space) without regularity assumptions is explicitly computed. The equivalence class of symbols yielding the same Weyl operator is characterized. The Heisenberg equation for the dynamics of general quantum observables is written through the Moyal brackets on the torus and the support of the Wigner transform is characterized. Finally, a dequantization procedure is introduced that applies, for instance, to the Pauli matrices. As a result we obtain the corresponding classical symbols.
A conditional symmetry is defined, in the phase-space of a quadratic in velocities constrained action, as a simultaneous conformal symmetry of the supermetric and the superpotential. It is proven that such a symmetry corresponds to a variational (Noether) symmetry.The use of these symmetries as quantum conditions on the wave-function entails a kind of selection rule. As an example, the minisuperspace model ensuing from a reduction of the Einstein - Hilbert action by considering static, spherically symmetric configurations and r as the independent dynamical variable, is canonically quantized. T…
