arXiv subjects

Jasel Berra-Montiel

Publications and source records attributed to Jasel Berra-Montiel.

27 records · Page 2Linked to original sources

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

Polysymplectic formulation for BF gravity with Immirzi parameter

The polysymplectic formulation of the CMPR action, which is a BF-type formulation of General Relativity that involves an arbitrary Immirzi parameter, is performed. We implement a particular scheme within this covariant Hamiltonian approach to analyze the constraints that characterize the CMPR model. By means of the privileged $(n-1)$-forms and the Poisson-Gerstenhaber bracket, inherent to the polysymplectic framework, the BF field equations associated to the CMPR action are obtained and, in consequence, the Einstein equations naturally emerge by solving the simplicity constraints of the theory. Further, from the polysymplectic analysis of the CMPR action the De Donder-Weyl Hamiltonian formulation of the Holst action is recovered, which is consistent with the Lagrangian analysis of this model as reported in the literature.

gr-qc↗

arXiv2019

Polymeric quantum mechanics and the zeros of the Riemann zeta function

We analize the Berry-Keating model and the Sierra and Rodríguez-Laguna Hamiltonian within the polymeric quantization formalism. By using the polymer representation, we obtain for both models, the associated polymeric quantum Hamiltonians and the corresponding stationary wave functions. The self-adjointness condition provide a proper domain for the Hamiltonian operator and the energy spectrum, which turned out to be dependent on an introduced scale parameter. By performing a counting of semiclassical states, we prove that the polymer representation reproduces the smooth part of the Riemann-von Mangoldt formula, and introduces a correction depending on the energy and the scale parameter, which resembles the fluctuation behavior of the Riemann zeros.

math-ph↗

arXiv2019

De Donder-Weyl Hamiltonian formalism of MacDowell-Mansouri gravity

We analyse the behaviour of the MacDowell-Mansouri action with internal symmetry group $\mathrm{SO}(4,1)$ under the covariant Hamiltonian formulation. The field equations, known in this formalism as the De Donder-Weyl equations, are obtained by means of the graded Poisson-Gerstenhaber bracket structure present within the covariant formulation. The decomposition of the internal algebra $\mathfrak{so}(4,1)\simeq\mathfrak{so}(3,1)\oplus\mathbb{R}^{3,1}$ allows the symmetry breaking $\mathrm{SO}(4,1)\to\mathrm{SO}(3,1)$, which reduces the original action to the Palatini action without the topological term. We demonstrate that, in contrast to the Lagrangian approach, this symmetry breaking can be performed indistinctly in the covariant Hamiltonian formalism either before or after the variation of the De Donder-Weyl Hamiltonian has been done, recovering Einstein's equations via the Poisson-Gerstenhaber bracket.

gr-qc↗

arXiv2019

Polymer Quantum Mechanics as a Deformation Quantization

We analyze the polymer representation of quantum mechanics within the deformation quantization formalism. In particular, we construct the Wigner function and the star-product for the polymer representation as a distributional limit of the Schrödinger representation for the Weyl algebra in a Gaussian weighted measure, and we observe that the quasi-probability distribution limit of this Schrödinger representation agrees with the Wigner function for Loop Quantum Cosmology. Further, the introduced polymer star-product fulfills Bohr's correspondence principle even though not all the operators are well defined in the polymer representation. Finally, within our framework, we also derive a generalized uncertainty principle which is consistent to the ones usually obtained in theories assuming a fundamental minimal length in their formulation.

gr-qc↗

arXiv2017

Polysymplectic formulation for topologically massive Yang-Mills field theory

We analyze the De Donder-Weyl covariant field equations for the topologically massive Yang-Mills theory. These equations are obtained through the Poisson-Gerstenhaber bracket described within the polysymplectic framework. Even though the Lagrangian defining the system of our interest is singular, we show that by appropriately choosing the polymomenta one may obtain an equivalent regular Lagrangian, thus avoiding the standard analysis of constraints. Further, our simple treatment allows us to only consider the privileged $(n-1)$-forms in order to obtain the correct field equations, in opposition to certain examples found in the literature.

hep-th↗

arXiv2017

The Unruh effect for higher derivative field theory

We analyse the emergence of the Unruh effect within the context of a field Lagrangian theory associated to the Pais-Uhlenbeck fourth order oscillator model. To this end, we introduce a transformation that brings the Hamiltonian bounded from below and is consistent with $\mathcal{PT}$-symmetric quantum mechanics. We find that, as far as we consider different frequencies within the Pais-Uhlenbeck model, a particle together with an antiparticle of different masses are created as may be traced back to the Bogoliubov transformation associated to the interaction between the Unruh-DeWitt detector and the higher derivative scalar field. On the contrary, whenever we consider the equal frequencies limit, no particle creation is detected as the pair particle/antiparticle annihilate each other. Further, following Moschella and Schaeffer, we construct a Poincaré invariant two-point function for the Pais-Uhlenbeck model, which in turn allows us to perform the thermal analysis for any of the emanant particles.

gr-qc↗

arXiv2016

Causal Poisson bracket via deformation quantization

Starting with the well-defined product of quantum fields at two spacetime points, we explore an associated Poisson structure for classical field theories within the deformation quantization formalism. We realize that the induced star-product is naturally related to the standard Moyal product through an appropriate causal Green's functions connecting points in the space of classical solutions to the equations of motion. Our results resemble the Peierls-DeWitt bracket analyzed in the multisymplectic context. Once our star-product is defined we are able to apply the Wigner-Weyl map in order to introduce a generalized version of Wick's theorem. Finally, we include some examples to explicitly test our method: the real scalar field, the bosonic string and a physically motivated nonlinear particle model. For the field theoretic models we have encountered causal generalizations of the creation/annihilation relations, and also a causal generalization of the Virasoro algebra for the bosonic string. For the nonlinear particle case, we use the approximate solution in terms of the Green's function in order to construct a well-behaved causal bracket.

hep-th↗

arXiv2015

Deformation quantization of the Pais-Uhlenbeck fourth order oscillator

We analyze the quantization of the Pais-Uhlenbeck fourth order oscillator within the framework of deformation quantization. Our approach exploit the Noether symmetries of the system by proposing integrals of motion as the variables to obtain a solution to the -genvalue equation, namely the Wigner function. We also obtain, by means of a quantum canonical transformation the wave function associated to the Schrödinger equation of the system. We show that unitary evolution of the system is guaranteed by means of the quantum canonical transformation and via the properties of the constructed Wigner function, even in the so called equal frequency limit of the model, in agreement with recent results.

quant-ph↗