arXiv subjects

M. F. Araujo de Resende

Publications and source records attributed to M. F. Araujo de Resende.

8 recordsLinked to original sources

arXiv2026

A tutorial on Dirac quantisation by analysing the problem of a ball on an inclined plane as a Hamiltonian system with constraints

In this paper, we present a detailed review/analysis of the Dirac quantisation of Hamiltonian systems with constraints. To this end, we use, as a guide, the physical example provided by the dynamics of a solid ball rolling, without slipping, down an inclined plane under the action of gravity. After all, however simple this physical system may be, it provides a rich framework for this analysis since, in addition to allowing us to discuss scenarios involving holonomic and non-holonomic constraints, it is also a gauge system. Indeed, due to this latter fact, we have carefully detailed how the transition, from classical to quantum mechanics, must be guided by the Dirac-Bergmann algorithm and by the consequent replacement of Dirac brackets with commutators. As a central result, we demonstrate that the restriction of the Hamiltonian operator of this system with constraints to the physical Hilbert subspace (which is identified with the quantisation of these constraints) reproduces the same Schrödinger equation that can be originally obtained in intrinsic terms, a fact that only reinforces the consistency of the Dirac quantisation method.

quant-ph↗

arXiv2026

Quantum aspects of the classical Maxwell's equations in free space from the perspective of the correspondence principle

Due to the advent of Quantum Mechanics' 100th anniversary in 2025, we wrote this review paper in order to present a discussion that addresses the foundations of this theory. And since the creation of this Mechanics and other quantum theories was guided, for instance, by correspondence principles that needed to be identified between them and other well-established physical theories, this paper will be devoted to discussing the correspondence between these quantum theories and Maxwell's theory of electromagnetism. More precisely, what we will do throughout this paper is discuss how the Maxwell's electromagnetic theory in free space and the stronger formulation of the correspondence principle already pointed, together, to the basis of a Quantum Mechanics that was only be formulated half a century later. And, in order to make this very clear, we will show that the quantum-mechanical description of a photon can already be identified, simply, by manipulating Maxwell's equations in free space with mathematical resources that, for instance, were already well known before the advent of Quantum Mechanics.

quant-ph↗

arXiv2025

Non-Abelian fusion rules from Abelian systems with SPT phases and graph topological order

Since Ref. [1] shows the emergence of non-Abelian fusion rules in some examples of a class of Abelian models, but does not prove whether these rules also exist in other cases, the purpose of this paper is to present such proof emphasizing the importance of the existence of these rules. By the way, as the ground state of these models can be degenerate as a function of their algebra and, hence, they can support some symmetry-protected topological (SPT) phases, we prove that these non-Abelian fusion rules are always necessary for these SPT phase transitions to occur via a condensation mechanism or/and some global symmetry breaking.

quant-ph↗

arXiv2025

Classical gauge theories as systems with constraints: a geometric point of view

In this paper, we briefly review the Hamiltonian formulation of classical systems that are constrained to submanifolds so that, within this context, the true meaning of classical gauge theories becomes clear. Please note that this paper is nothing more than a near-literal translation of Ref. [1], which we originally published in Brazilian Portuguese in 2018. Therefore, if you, the reader, find this paper useful enough to cite it in any of your works, we kindly ask that you (also) cite Ref. [1].

physics.gen-ph↗

arXiv2023

Finite-group gauge theories on lattices as Hamiltonian systems with constraints

In this work, we present a brief but insightful overview of the gauge theories, which are defined on $ n $-dimensional lattices by using finite gauge groups, in order to show how they can be interpreted as a Hamiltonian system with constraints, analogous to what happens with the classical (continuous) gauge (field) theories. As this interpretation is not usually explored in the literature that discusses/introduces the concept of lattice gauge theory, but some recent works have been exploring Hamiltonian models in order to support some kind of quantum computation, we use this interpretation to, for example, present a brief geometric view of one class of these models: the Kitaev Quantum Double Models.

hep-lat↗

arXiv2023

Quantum Double Models coupled to matter fields: a detailed review for a dualization procedure

In this paper, we investigate how it is possible to define a new class of lattice gauge models based on a dualization procedure of a previous generalization of the Kitaev Quantum Double Models. In the case of this previous generalization that will be used as a basis, it was defined by adding new qudits (which were denoted as matter fields in reference to some works) to the lattice vertices with the intention of, for instance, interpreting its models as Kitaev Quantum Double Models coupled with Potts ones. Now, with regard to the generalization that we investigate here, which we want to define as the dual of this previous one, these new qudits were added to the lattice faces. And as the coupling between gauge and matter qudits of the previous generalization was performed by a gauge group action, we show that the dual behaviour of these two generalizations was achieved by coupling these same qudits in the second one through a gauge group co-action homomorphism. One of the most striking dual aspects of these two generalizations is that, in both, part of the quasiparticles that were inherited from the Kitaev Quantum Double Models become confined when these action and co-action are non-trivial. But the big news here is that, in addition to the group homomorphism (that defines this gauge group co-action) allows us to classify all the different models of this second generalization, this same group homomorphism also suggests that all these models can be interpreted as two-dimensional restrictions of the $ 2 $-lattice gauge theories.

quant-ph↗

arXiv2021

Uma história e o uso de espelhos no aprendizado das regras de sinais envolvendo números negativos

The purpose of this paper (which, in many passages, I take the liberty of writing in the first person singular for bringing a personal experience) is to show how teachers of Physics and Mathematics in Basic Education can teach, through a simple and playful activity, the rules of signs that are related to basic arithmetic operations involving positive and negative numbers. How? By using a ruler, or any other object that can be interpreted as a number line, and a flat mirror. By the way, as the justification of why this method works is physical and mathematical, this paper provides a succinct review on the formation of images by flat and spherical mirrors since some teachers of Mathematics may (just as a possibility) not know how it happens. Lastly, given the interdisciplinary nature of this activity, some additional considerations are also presented regarding how these teachers can use it to instigate the students to observe how Physics and Mathematics, together, are part of everyday life for all of us.

math.HO↗

arXiv2019

A pedagogical overview on 2D and 3D Toric Codes and the origin of their topological orders

In this work, we will show how the topological order of the Toric Code appears when the lattice on which it is defined discretizes a three-dimensional torus. In order to do this, we will present a pedagogical review of the traditional two-dimensional Toric Code, with an emphasis on how its quasiparticles are conceived and transported. With that, we want to make clear not only how all these same quasiparticle conception and transportation fit into this three-dimensional model, but to make it clear how topology controls the degeneracy of ground state in this new situation.

quant-ph↗