arXiv subjects

Alberto Molgado

Publications and source records attributed to Alberto Molgado.

At least 19 recordsLinked to original sources

arXiv2026

Phase space quantization of anisotropic cosmologies: Taub and Kantowski-Sachs models

We introduce an explicit construction of the non-diagonal and diagonal Wigner distributions for the homogeneous but anisotropic Taub and Kantowski-Sachs cosmological models within the framework of phase space deformation quantization. Conventional canonical quantization of these models via the Wheeler-DeWitt equation is inherently plagued by factor ordering ambiguities. To circumvent these issues, we employ the totally symmetric Weyl quantization map and the Moyal star product. By means of a canonical separation of the Hamiltonian constraint, we are able to resolve the formal convergence problems typically associated with the star product. Furthermore, to establish a rigorous connection with conventional quantum cosmology, we calculate the standard wave functions directly from the diagonal Wigner distributions, recovering the exact physical states in terms of modified Bessel functions in both cases.

gr-qc↗

arXiv2026

Geometric formulation for Palatini-Cartan gravity

Motivated by the increasing efforts to understand the covariant structure of physical models associated with General Relativity using different kinds of geometric frameworks, in this article we analyze the four-dimensional Palatini-Cartan model for gravity, which is a well-known generalization of General Relativity, from the perspective of various geometric-covariant formalisms for classical field theory. At the Lagrangian level, we do not only recover the correct field equations of the theory, which are equivalent to the torsion-free condition and the Einstein equations, but we also study the gauge symmetries of the model in order to construct the Lagrangian momentum map associated with the action of the gauge symmetry group on the configuration space of the system and, consequently, its corresponding Noether currents. Within the multisymplectic approach, we analyze the action of the gauge symmetry group on the multi-momenta phase space of the model, and we also introduce the induced momentum map that allows us to recover the admissible Cauchy data of the system. Further, we also apply the algorithm to treat singular systems within the polysymplectic framework, in which, in order to obtain the correct field equations of the model, we introduce a non-trivial Dirac-Poisson bracket characterized by the generalized Moore-Penrose inverse of the matrix induced by the second class constraints of the system. Finally, using the multisymplectic framework as a starting point, we perform the space plus time decomposition of the system to recover the instantaneous Lagrangian and the extended Hamiltonian of the theory, as well as the gauge structure that characterize the Palatini-Cartan model for gravity within the instantaneous Dirac-Hamiltonian formalism.

gr-qc↗

arXiv2026

Star product for qubit states in phase space and star exponentials

In this paper, we formulate the phase space description of qubit systems using coadjoint orbits of $SU(2)$ and the Stratonovich-Weyl correspondence, yielding a deformation quantization on the sphere. The resulting star product reproduces the operator algebra of complexified quaternions and its antisymmetric part induces the Lie-Poisson structure associated with the Kirillov-Kostant-Souriau symplectic form. We show that quantum dynamics can be expressed entirely in phase space through star exponentials of Hamiltonian symbols, leading to an explicit representation of the propagator. Further, we establish the equivalence between the coherent-state path integral formulation on $S^2$ and the algebraic description in terms of star exponentials. Some examples illustrating the construction of the star-exponential functions and the resulting Poisson structure are included.

quant-ph↗

arXiv2025

The Jaynes-Cummings model in Phase Space Quantum Mechanics

In this paper, we address the phase space formulation of the Jaynes-Cummings model through the explicit construction of the full Wigner function for a hybrid bipartite quantum system composed of a two-level atom and a quantized coherent field. By employing the Stratonovich-Weyl correspondence and the coadjoint orbit method, we derive an informationally complete quasi-probability distribution that captures the full dynamics of light-matter interaction. This approach provides a detailed phase space perspective of fundamental quantum phenomena such as Rabi oscillations, atomic population inversion, and entanglement generation. We further measure the purity of the reduced quantized field state by means of an appropriate Wigner function corresponding to the bosonic field part in order to investigate the entanglement dynamics of the system.

quant-ph↗

arXiv2025

Star exponentials and Wigner functions for time-dependent harmonic oscillators

In this paper, we address the Wigner distribution and the star exponential function for a time-dependent harmonic oscillator for which the mass and the frequency terms are considered explicitly depending on time. To such an end, we explore the connection between the star exponential, naturally emerging within the context of deformation quantization, and the propagators constructed through the path integral formalism. In particular, the Fourier-Dirichlet expansion of the star exponential implies a distinctive quantization of the Lewis-Riesenfeld invariant. Further, by introducing a judicious time variable, we recovered a time-dependent phase function associated with the Lewis-Riesenfeld construction of the standard Schr\"odinger picture. In particular, we applied our results to the cases of the Caldirola-Kanai and the time-dependent frequency harmonic oscillators, recovering relevant results previously reported in the literature.

quant-ph↗

arXiv2025

Geometric analysis for the Pontryagin action and boundary terms

In this article, we analyze the Pontryagin model adopting different geometric-covariant approaches. In particular, we focus on the manner in which boundary conditions must be imposed on the background manifold in order to reproduce an unambiguous theory on the boundary. At a Lagrangian level, we describe the symmetries of the theory and construct the Lagrangian covariant momentum map which allows for an extension of Noether's theorems. Through the multisymplectic analysis we obtain the covariant momentum map associated with the action of the gauge group on the covariant multimomenta phase-space. By performing a space plus time decomposition by means of a foliation of the appropriate bundles, we are able to recover not only the $t$-instantaneous Lagrangian and Hamiltonian of the theory, but also the generator of the gauge transformations. In the polysymplectic framework we perform a Poisson-Hamilton analysis with the help of the De Donder-Weyl Hamiltonian and the Poisson-Gerstenhaber bracket. Remarkably, as long as we consider a background manifold with boundary, in all of these geometric formulations, we are able to recover the so-called differentiability conditions as a straightforward consequence of Noether's theorem.

gr-qc↗

arXiv2025

The Feynman-Kac formula in deformation quantization

We introduce the Feynman-Kac formula within the deformation quantization program. Constructing on previous work it is shown that, upon a Wick rotation, the ground state energy of any prescribed physical system can be obtained from the asymptotic limit of the phase space integration of the star exponential of the Hamiltonian operator. Some examples of this correspondence are provided showing a novel and efficient way of computing the ground state energy for some physical models.

math-ph↗

arXiv2024

Star exponentials from propagators and path integrals

In this paper we address the relation between the star exponentials emerging within the Deformation Quantization formalism and Feynman's path integrals associated with propagators in quantum dynamics. In order to obtain such a relation, we start by visualizing the quantum propagator as an integral transform of the star exponential by means of the symbol corresponding to the time evolution operator and, thus, we introduce Feynman's path integral representation of the propagator as a sum over all the classical histories. The star exponential thus constructed has the advantage that it does not depend on the convergence of formal series, as commonly understood within the context of Deformation Quantization. We include some basic examples to illustrate our findings, recovering standard results reported in the literature. Further, for an arbitrary finite dimensional system, we use the star exponential introduced here in order to find a particular representation of the star product which resembles the one encountered in the context of the quantum field theory for a Poisson sigma model.

math-ph↗

arXiv2021

Tomography in Loop Quantum Cosmology

We analyze the tomographic representation for the Friedmann-Robertson-Walker (FRW) model within the Loop Quantum Cosmology framework. We focus on the Wigner quasi-probability distributions associated with Gaussian and Schrödinger cat states, and then, by applying a Radon integral transform for those Wigner functions, we are able to obtain the symplectic tomograms which define measurable probability distributions that fully characterize the quantum model of our interest. By appropriately introducing the quantum dispersion for a rotated and squeezed quadrature operator in terms of the position and momentum, we efficiently interpret the properties of such tomograms, being consequent with Heisenberg's uncertainty principle. We also obtain, by means of the dual tomographic symbols, the expectation value for the volume operator, which coincides with the values reported in the literature. We expect that our findings result interesting as the introduced tomographic representation may be further benefited from the well-developed measure techniques in the areas of Quantum optics and Quantum information theory.

gr-qc↗

arXiv2021

A review on geometric formulations for classical field theory: the Bonzom-Livine model for gravity

Motivated by the study of physical models associated with General Relativity, we review some finite-dimensional, geometric and covariant formulations that allow us to characterize in a simple manner the symmetries for classical field theory by implementing an appropriate fibre-bundle structure, either at the Lagrangian, the multisymplectic or the polysymplectic levels. In particular, we are able to formulate Noether's theorems by means of the covariant momentum maps and to systematically introduce a covariant Poisson-Hamiltonian framework. Also, by focusing on the space plus time decomposition for a generic classical field theory and its relation to these geometric formulations, we are able to successfully recover the gauge content and the true local degrees of freedom for the theory. In order to illustrate the relevance of these geometric frameworks, we center our attention to the analysis of a model for $3$-dimensional theory of General Relativity that involves an arbitrary Immirzi-like parameter. At the Lagrangian level, we reproduce the field equations of the system which for this model turn out to be equivalent to the vanishing torsion condition and the Einstein equations. We also concentrate on the analysis of the gauge symmetries of the system in order to obtain the Lagrangian covariant momentum map associated with the theory and, consequently, its corresponding Noether currents. Next, we aim our attention to describing how the gauge symmetries of the model yield covariant canonical transformations on the covariant multimomenta phase-space, thus giving rise to the existence of a covariant momentum map. Besides, we analyze the physical system under consideration within the De Donder-Weyl canonical theory implemented at the polysymplectic level, thus establishing a relation from the covariant momentum map to the conserved currents of the theory.

gr-qc↗

arXiv2020

Hamiltonian dynamics of gonihedric string theory

We develop in a consistent manner the Ostrogradski-Hamilton framework for gonihedric string theory. The local action describing this model, being invariant under reparametrizations, depends on the modulus of the mean extrinsic curvature of the worldsheet swept out by the string, and thus we are confronted with a genuine second-order in derivatives field theory. In our geometric approach, we consider the embedding functions as the field variables and, even though the highly non-linear dependence of the action on these variables, we are able to complete the classical analysis of the emerging constraints for which, after implementing a Dirac bracket, we are able to identify both the gauge transformations and the proper physical degrees of freedom of the model. The Ostrogradski-Hamilton framework is thus considerable robust as one may recover in a straightforward and consistent manner some existing results reported in the literature. Further, in consequence of our geometrical treatment, we are able to unambiguously recover as a by-product the Hamiltonian approach for a particular relativistic point-particle limit associated with the gonihedric string action, that is, a model linearly depending on the first Frenet-Serret curvature.

hep-th↗

arXiv2020

Star product approach for Loop Quantum Cosmology

Guided by recent developments towards the implementation of the deformation quantization program within the Loop Quantum Cosmology (LQC) formalism, in this paper we address the introduction of both the integral and differential representation of the star product for LQC. To this end, we consider the Weyl quantization map for cylindrical functions defined on the Bohr compactification of the reals. The integral representation contains all of the common properties that characterize a star product which, in the case under study here, stands for a deformation of the usual pointwise product of cylindrical functions. Our construction also admits a direct comparison with the integral representation of the Moyal product which may be reproduced from our formulation by judiciously substituting the appropriate characters that identify such representation. Further, we introduce a suitable star commutator that correctly reproduces both the quantum representation of the holonomy-flux algebra for LQC and, in the proper limit, the holonomy-flux classical Poisson algebra emerging in the cosmological setup. Finally, we propose a natural way to obtain the quantum dynamical evolution in LQC in terms of this star commutator for cylindrical functions as well as a differential representation of the star product using discrete finite differences. We expect that our findings may contribute to a better understanding of certain issues arising within the LQC program, in particular, those related to the semiclassical limit and the dynamical evolution of quantum states.

gr-qc↗

arXiv2020

Quasi-probability distributions in Loop Quantum Cosmology

In this paper, we introduce a complete family of parametrized quasi-probability distributions in phase space and their corresponding Weyl quantization maps with the aim to generalize the recently developed Wigner-Weyl formalism within the Loop Quantum Cosmology program (LQC). In particular, we intend to define those quasi-distributions for states valued on the Bohr compactification of the real line in such a way that they are labeled by a parameter that accounts for the ordering ambiguity corresponding to non-commutative quantum operators. Hence, we notice that the projections of the parametrized quasi-probability distributions result in marginal probability densities which are invariant under any ordering prescription. We also note that, in opposition to the standard Schrödinger representation, for an arbitrary character the quasi-distributions determine a positive function independently of the ordering. Further, by judiciously implementing a parametric-ordered Weyl quantization map for LQG, we are able to recover in a simple manner the relevant cases of the standard, anti-standard, and Weyl symmetric orderings, respectively. We expect that our results may serve to analyze several fundamental aspects within the LQC program, in special those related to coherence, squeezed states, and the convergence of operators, as extensively analyzed in the quantum optics and in the quantum information frameworks.

gr-qc↗

arXiv2020

Superfluids in Polymer Quantum Mechanics

In this work, we analyze the corrections obtained on a homogeneous one-dimensional Bose gas within the high densities limit by means of the polymer quantization scheme. Thus, starting from the Bogoliubov formalism, we analyze the ground expectation value of the polymer momentum operator in terms of semiclassical states, in order to obtain an analytic expression for the ground state energy of the N-body system, which allows us to solve the pathological behavior commonly associated with the one-dimensional Bose-Einstein condensation through the introduction of finite size effects characterized by the contribution of the polymer corrections. We also discuss the speed of sound in our polymer version of the Bose gas and the corresponding relative shift induced by the introduction of a minimum length parameter. Finally, by considering the idea that the Bose-Einstein condensation phenomenon is closely related to that of superfluidity, we investigate the emergent superfluid behavior in our polymer model by implementing an appropriate Landau's criterion. In this case, we are able to consequently analyze the changes in the critical velocity which defines the limit between the superfluid-condensate regions, thus deducing that the polymer length acts as a kind of pseudo-potential which induces a dissipationless flow associated with the superfluid phase even in the absence of self-interactions.

gr-qc↗

arXiv2020

Coherent representation of fields and deformation quantization

Motivated by some well-known results in the phase space description of quantum optics and quantum information theory, we aim to describe the formalism of quantum field theory by explicitly considering the holomorphic representation for a scalar field within the deformation quantization program. Notably, the symbol of a symmetric ordered operator in terms of holomorphic variables may be straightforwardly obtained by the quantum field analogue of the Husimi distribution associated with a normal ordered operator. This relation also allows establishing a $c$-equivalence between the Moyal and the normal star-products. In addition, by writing the density operator in terms of coherent states we are able to directly introduce a series representation of the Wigner functional distribution which may be convenient in order to calculate probability distributions of quantum field observables without performing formal phase space integrals at all.

quant-ph↗

arXiv2020

Hamilton-Jacobi approach for Regge-Teitelboim cosmology

The Hamilton-Jacobi formalism for a geodetic brane-like universe described by the Regge-Teitelboim model is developed. We focus on the description of the complete set of Hamiltonians that ensure the integrability of the model in addition to obtaining the Hamilton principal function $S$. In order to do this, we avoid the second-order in derivative nature of the model by appropriately defining a set of auxiliary variables that yields a first-order Lagrangian. Being a linear in acceleration theory, this scheme unavoidably needs an adequate redefinition of the so-called Generalized Poisson Bracket in order to achieve the right evolution in the reduced phase space. Further this Hamilton-Jacobi framework also enables us to explore the quantum behaviour under a semi-classical approximation of the model. A comparison with the Ostrogradski-Hamilton method for constrained systems is also provided in detail.

gr-qc↗

arXiv2019

Deformation quantization of constrained systems: a group averaging approach

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 considering the star-exponential of the constraints and then integrating with respect to a suitable measure. In this manner, the proposed physical Wigner distribution explicitly solves the constraints by including generalized functions on phase space. Finally, in order to illustrate our approach, our proposal is tested in a couple of simple examples which recover standard results found in the literature.

gr-qc↗

arXiv2019

Covariant momentum map for non-Abelian topological BF field theory

We analyze the inherent symmetries associated to the non-Abelian topological BF theory from the geometric and covariant perspectives of the Lagrangian and the multisymplectic formalisms. At the Lagrangian level, we classify the symmetries of the theory as natural and Noether symmetries and construct the associated Noether currents, while at the multisymplectic level the symmetries of the theory arise as covariant canonical transformations. These transformations allowed us to build within the multisymplectic approach, in a complete covariant way, the momentum maps which are analogous to the conserved Noether currents. The covariant momentum maps are fundamental to recover, after the space plus time decomposition of the background manifold, not only the extended Hamiltonian of the BF theory but also the generators of the gauge transformations which arise in the instantaneous Dirac-Hamiltonian analysis of the first-class constraint structure that characterizes the BF model under study. To the best of our knowledge, this is the first non-trivial physical model associated to General Relativity for which both natural and Noether symmetries have been analyzed at the multisymplectic level. Our study shed some light on the understanding of the manner in which the generators of gauge transformations may be recovered from the multisymplectic formalism for field theory.

gr-qc↗