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We obtain Bargmann–Michel–Telegdi equations of motion of classical spinning particle using Lagrangian variational principle with Grassmann variables.

In this paper, the propagator of a two-dimensional Dirac oscillator in the presence of a uniform electric field is derived by using the path integral technique. The fact that the globally named approach is used in this work redirects, beforehand, our search for the propagator of the Dirac equation to that of the propagator of its quadratic form. The internal motions relative to the spin are represented by two fermionic oscillators, which are described by Grassmannian variables, according to Schwinger’s fermionic model. Once the integration over the anticommuting variables (Grassmannian variables) is accomplished, the problem becomes the one of finding a non-relativistic propagator with only bosonic variables. The energy spectrum of the electron and the corresponding eigenspinors are also obtained in this work.

In noncommutative phase space (NCPS), we investigate the pair creation rate for the problem of scalar and spinorial particles emerging from a vacuum under the influence of uniform external electromagnetic fields using the Schwinger formal technique. It is demonstrated that under specific conditions, where the speed of light c approaches the Fermi velocity ${\upsilon }_F$ and $\bar{\omega }\to {\omega }_c∕2$ (with ${\omega }_c=\frac{|e|\mathcal{ℬ}}{mc}$ representing the cyclotron frequency), the behavior of the Dirac oscillator problem in a uniform electromagnetic field within NCPS is akin to the monolayer graphene problem in NCPS. In addition, the production probability per unit volume per unit time is discussed, along with the limits of parameter deformations.

The Faddeev–Jackiw canonical quantization formalism for constrained systems with Grassmann dynamical variables within the framework of the field theory is reviewed. First, by means of a iterative process, the symplectic supermatrix is constructed and their associated constraints are found. Next, by taking into account the phase space of the system, the constraint structure is considered. It is found that, if there are no auxiliary dynamical field variables, the supermatrix whose elements are the Bose–Fermi brackets between the constraints associated with the independent dynamical field variables coincides with the symplectic supermatrix corresponding to these independent variables. An alternative procedure to obtain the first-class constraints is given. It is shown that for systems with gauge symmetries, by means of suitable gauge-fixing conditions, a nonsingular final symplectic supermatrix can be found. Then, two possible ways of calculating the Faddeev–Jackiw brackets are pointed out. The relation between the Faddeev–Jackiw and Dirac brackets is discussed. Throughout the previous developments, the Faddeev–Jackiw and Dirac algorithms are compared. Finally, the Faddeev–Jackiw canonical quantization method is applied to a simple model and the obtained results are compared with the ones corresponding to the use of the Dirac procedure on this model.

Some time ago, the Faddeev–Jackiw canonical quantization formalism for constrained systems with Grassmann dynamical variables in the field theory context was reviewed. In the present work, the resulting formalism is applied to a classical nonrelativistic U(1) ×U(1) gauge field model that describes the electromagnetic interaction of composite particles in 2+1 dimensions. The model contains a Chern–Simons U(1) field and the electromagnetic field, and it uses either a composite boson system or a composite fermion one. The obtained results are compared with the ones corresponding to the implementation of the Dirac formalism to this model, concluding that the Faddeev–Jackiw and Dirac methods cannot be considered equivalent. A simplified version of the above model is analyzed in the same way, similar to the one used within the framework of condensed matter. In this case, it is observed that when the results obtained by the Faddeev–Jackiw and Dirac methods coincide, the first method is more economical than the second one. For both models, the composite fermion case is explicitly considered.

We have studied the effect of energy-dependent potentials for the relativistic spinning particle using the formalism for supersymmetric path integrals. That leaves behind a new normalization of wave function, which is examined via the Dirac equation and can be confirmed by Feynman’s path integral method. Based on two important examples, Coulomb and Harmonic oscillator potentials, we find that the frequency and the Coulomb’s constant are dependent on spectral parameters. The propagator is calculated and the energy eigenvalues with their corresponding eigenfunctions are deduced.

Starting from the classical nonrelativistic electrodynamics in 1+1 dimensions, a higher-derivative version is proposed. This is made by adding a suitable higher-derivative term for the electromagnetic field to the Lagrangian of the original electrodynamics, preserving its gauge invariance. By following the usual Hamiltonian method for singular higher-derivative systems, the canonical quantization for the higher-derivative model is developed. By extending the Faddeev–Senjanovic algorithm, the path integral quantization is carried out. Hence, the Feynman rules are established and the diagrammatic structure is analyzed. The use of the higher-derivative term eliminates in the Landau gauge the ultraviolet divergence of the primitively divergent Feynman diagrams of the original model, where the electromagnetic field propagator is present. A generalization of the BRST quantization is also considered.

In this paper, we study the creation of particles from the vacuum in monolayer graphene by an electromagnetic field in a noncommutative (NC) phase space coordinate considering the Schwinger method. For two different gauges, the probability of particle creation is calculated using the effective action, and special cases are considered. The essential result is that the noncommutativity of the phase space contributes to the effective action and consequently it has an influence on the process of pair creation from the vacuum.

We perform the Faddeev–Jackiw (FJ) canonical quantization for the Podolsky electrodynamics. To this end, we use an extension of the usual FJ formalism for constrained systems with Grassmann dynamical field variables, proposed by us some time ago. Besides, we compare the obtained results with those corresponding to the implementation of the Dirac formalism to this issue. In this way, we see that the extended FJ and the Dirac formalisms provide the same constraints and generalized brackets, thus suggesting the equivalence between these formalisms, at least for the present case. Furthermore, we find that the extended FJ formalism is more economical than the Dirac one as regards the calculation of both the constraints and the generalized brackets. On the other hand, we also compare the mentioned obtained results with the ones corresponding to the analysis of the issue in study by means of the usual FJ formalism, showing that between the extended and the usual FJ formalisms there are significant differences.

In this study, we propose an extension of the formulation developed by Faddeev and Jackiw to include anticommutative variables in the language of supergeometry, as it could be a cost-effective way to determine the (${\mathbb{Z}}_2$-graded) Poisson structure of theories describing spin-like degrees of freedom. Specifically, we apply the developed approach to pseudoclassical systems to lately use the standard canonical quantization program to check whether their already known quantum description is recovered.

Introducing collective variables, a collective description of nuclear surface oscillations has been developed with the first-quantized language, contrary to the second-quantized one in Sunakawa's approach for a Bose system. It overcomes difficulties remaining in the traditional theories of nuclear collective motions: Collective momenta are not exact canonically conjugate to collective coordinates and are not independent. On the contrary to such a description, Tomonaga first gave the basic idea to approach elementary excitations in a one-dimensional Fermi system. The Sunakawa's approach for a Fermi system is also expected to work well for such a problem. In this paper, on the isospin space, we define a density operator and further following Tomonaga, introduce a collective momentum. We propose an exact canonically momenta approach to a one-dimensional neutron–proton (N–P) system under the use of the Grassmann variables.

In the preceeding paper, introducing $isospin$-dependent density operators and defining exact momenta (collective variables), we could get an exact canonically momenta approach to a one-dimensional (1D) neutron–proton (NP) system. In this paper, we attempt at a velocity operator approach to a 3D NP system. Following Sunakawa, after introducing momentum density operators, we define velocity operators, denoting classical fluid velocities. We derive a collective Hamiltonian in terms of the collective variables.