
Hojman, S. A., & Asenjo, F. A. (2020). Classical and Quantum Dispersion Relations. Phys. Scr., 95(8), 7 pp.
Abstract: It is showed that, in general, classical and quantum dispersion relations are different due to the presence of the Bohm potential. There are exact particular solutions of the quantum (wave) theory which obey the classical dispersion relation, but they differ in the general case. The dispersion relations may also coincide when additional assumptions are made, such as WKB or eikonal approximations, for instance. This general result also holds for nonquantum wave equations derived from classical counterparts, such as in ray and wave optics, for instance. Explicit examples are given for covariant scalar, vectorial and tensorial fields in flat and curved spacetimes.



Hojman, S. A., & Asenjo, F. A. (2020). Dual wavefunctions in twodimensional quantum mechanics. Phys. Lett. A, 384(13), 5 pp.
Abstract: It is shown that the Schrodinger equation for a large family of pairs of twodimensional quantum potentials possess wavefunctions for which the amplitude and the phase are interchangeable, producing two different solutions which are dual to each other. This is a property of solutions with vanishing Bohm potential. These solutions can be extended to threedimensional systems. We explicitly calculate dual solutions for physical systems, such as the repulsive harmonic oscillator and the twodimensional hydrogen atom. These dual wavefunctions are also solutions of an analogue optical system in the eikonal limit. In this case, the potential is related to the refractive index, allowing the study of this twodimensional dual wavefunction solutions with an optical (analogue) system. (C) 2020 Elsevier B.V. All rights reserved.



Asenjoa, F. A., & Hojman, S. A. (2019). Correspondence between dark energy quantum cosmology and Maxwell equations. Eur. Phys. J. C, 79(9), 5 pp.
Abstract: A FriedmannRobertsonWalker cosmology with dark energy can be modelled using a quintessence field. That system is equivalent to a relativistic particle moving on a twodimensional conformal spacetime. When the quintessence behaves as a free massless scalar field in a Universe with cosmological constant, the quantized version of that theory can lead to a supersymmetric Majorana quantum cosmology. The purpose of this work is to show that such quantum cosmological model corresponds to the Maxwell equations for electromagnetic waves propagating in a medium with specific values for its relative permittivity and relative permeability. The form of those media parameters are calculated, implying that a Majorana quantum cosmology can be studied in an analogue electromagnetic system.



Rubio, C. A., Asenjo, F. A., & Hojman, S. A. (2019). Quantum Cosmologies Under Geometrical Unification of Gravity and Dark Energy. Symmetry, 11(7).
Abstract: A FriedmannRobertsonWalker Universe was studied with a dark energy component represented by a quintessence field. The Lagrangian for this system, hereafter called the FriedmannRobertsonWalkerquintessence (FRWq) system, was presented. It was shown that the classical Lagrangian reproduces the usual two (second order) dynamical equations for the radius of the Universe and for the quintessence scalar field, as well as a (first order) constraint equation. Our approach naturally unified gravity and dark energy, as it was obtained that the Lagrangian and the equations of motion are those of a relativistic particle moving on a twodimensional, conformally flat spacetime. The conformal metric factor was related to the dark energy scalar field potential. We proceeded to quantize the system in three different schemes. First, we assumed the Universe was a spinless particle (as it is common in literature), obtaining a quantum theory for a Universe described by the KleinGordon equation. Second, we pushed the quantization scheme further, assuming the Universe as a Dirac particle, and therefore constructing its corresponding Dirac and Majorana theories. With the different theories, we calculated the expected values for the scale factor of the Universe. They depend on the type of quantization scheme used. The differences between the Dirac and Majorana schemes are highlighted here. The implications of the different quantization procedures are discussed. Finally, the possible consequences for a multiverse theory of the Dirac and Majorana quantized Universe are briefly considered.



Hojman, S. A., & Asenjo, F. A. (2018). Nongeodesic circular motion of massive spinning test bodies around a Schwarzschild field in the Lagrangian theory. Eur. Phys. J. C, 78(10), 7 pp.
Abstract: Recent interest on studying possible violations of the Equivalence Principle has led to the development of space satellite missions testing it for bodies moving on circular orbits around Earth. This experiment establishes that the validity of the equivalence principle is independent of the composition of bodies. However, the internal degrees of freedom of the bodies (such as spin) were not taken into account. In this work, it is shown exactly that the circular orbit motion of test bodies does present a departure from geodesic motion when spin effects are not negligible. Using a Lagrangian theory for spinning massive bodies, an exact solution for their circular motion is found showing that the nongeodesic behavior manifests through different tangential velocities of the test bodies, depending on the orientation of its spin with respect to the total angular momentum of the satellite. Besides, for circular orbits, spinning test bodies present no tangential acceleration. We estimate the difference of the two possible tangential velocities for the case of circular motion of spinning test bodies orbiting Earth.



Asenjo, F. A., & Hojman, S. A. (2017). Do electromagnetic waves always propagate along null geodesics? Class. Quantum Gravity, 34(20), 12 pp.
Abstract: We find exact solutions to Maxwell equations written in terms of fourvector potentials in nonrotating, as well as in Gdel and Kerr spacetimes. We show that Maxwell equations can be reduced to two uncoupled secondorder differential equations for combinations of the components of the fourvector potential. Exact electromagnetic waves solutions are written on given gravitational field backgrounds where they evolve. We find that in nonrotating spherical symmetric spacetimes, electromagnetic waves travel along null geodesics. However, electromagnetic waves on Gdel and Kerr spacetimes do not exhibit that behavior.



Asenjo, F. A., Erices, C., Gomberoff, A., Hojman, S. A., & Montecinos, A. (2017). Differential geometry approach to asymmetric transmission of light. Opt. Express, 25(22), 26405–26416.
Abstract: In the last ten years, the technology of differential geometry, ubiquitous in gravitational physics, has found its place in the field of optics. It has been successfully used in the design of optical metamaterials through a technique now known as “transformation optics.” This method, however, only applies for the particular class of metamaterials known as impedance matched, that is, materials whose electric permittivity is equal to their magnetic permeability. In that case, the material may be described by a spacetime metric. In the present work we will introduce a generalization of the geometric methods of transformation optics to situations in which the material is not impedance matched. In such situations, the material or more precisely, its constitutive tensorwill not be described by a metric only. We bring in a second tensor, with the local symmetries of the Weyl tensor, the “Wtensor.” In the geometric optics approximation we show how the properties of the Wtensor are related to the asymmetric transmission of the material. We apply this feature to the design of a particularly interesting set of asymmetric materials. These materials are birefringent when light rays approach the material in a given direction, but behave just like vacuum when the rays have the opposite direction with the appropriate polarization (or, in some cases, independently of the polarization). (C) 2017 Optical Society of America



Asenjo, F. A., & Hojman, S. A. (2017). New nonlinear modified massless KleinGordon equation. Eur. Phys. J. C, 77(11), 5 pp.
Abstract: The massless KleinGordon equation on arbitrary curved backgrounds allows for solutions which develop “tails” inside the light cone and, therefore, do not strictly follow null geodesics as discovered by DeWitt and Brehme almost 60 years ago. A modification of the massless KleinGordon equation is presented, which always exhibits null geodesic propagation of waves on arbitrary curved spacetimes. This new equation is derived from a Lagrangian which exhibits currentcurrent interaction. Its nonlinearity is due to a selfcoupling term which is related to the quantum mechanical Bohm potential.



Asenjo, F. A., & Hojman, S. A. (2017). Birefringent light propagation on anisotropic cosmological backgrounds. Phys. Rev. D, 96(4), 12 pp.
Abstract: Exact electromagnetic wave solutions to Maxwell equations on anisotropic Bianchi I cosmological spacetime backgrounds are studied. The waves evolving on Bianchi I spacetimes exhibit birefringence (associated with linear polarization) and dispersion. The particular case of a vacuumdominated anisotropic Universe, which reproduces a FriedmannRobertsonWalker Universe (for late times)while, for earlier times, it matches a Kasner Universeis studied. The electromagnetic waves do not, in general, follow null geodesics. This produces a modification of the cosmological redshift, which is then dependent on light polarization, its dispersion, and its nonnull geodesic behavior. New results presented here may help to tackle some issues related to the “horizon” problem.



Asenjo, F. A., & Hojman, S. A. (2017). Class of Exact Solutions for a Cosmological Model of Unified Gravitational and Quintessence Fields. Found. Phys., 47(7), 887–896.
Abstract: A new approach to tackle Einstein equations for an isotropic and homogeneous FriedmannRobertsonWalker Universe in the presence of a quintessence scalar field is devised. It provides a way to get a simple exact solution to these equations. This solution determines the quintessence potential uniquely and it differs from solutions which have been used to study inflation previously. It relays on a unification of geometry and dark matter implemented through the definition of a functional relation between the scale factor of the Universe and the quintessence field. For a positive curvature Universe, this solution produces perpetual accelerated expansion rate of the Universe, while the Hubble parameter increases abruptly, attains a maximum value and decreases thereafter. The behavior of this cosmological solution is discussed and its main features are displayed. The formalism is extended to include matter and radiation.



Hojman, S. A., & Asenjo, F. A. (2017). Spinning particles coupled to gravity and the validity of the universality of free fall. Class. Quantum Gravity, 34(11), 8 pp.
Abstract: Recent experimental work has determined that free falling Rb87 atoms on Earth, with vertically aligned spins, follow geodesics, thus apparently ruling out spingravitation interactions. It is showed that while some spinning matter models coupled to gravitation referenced to in that work seem to be ruled out by the experiment, those same experimental results confirm theoretical results derived from a Lagrangian description of spinning particles coupled to gravity constructed over forty years ago. A proposal to carry out (similar but) different experiments which will help to test the validity of the universality of free fall as opposed to the correctness of the aforementioned Lagrangian theory, is presented.



Hojman, S. A., & Asenjo, F. A. (2016). Comment on “Highly relativistic spingravity coupling for fermions”. Phys. Rev. D, 93(2), 4 pp.
Abstract: We exhibit difficulties of different sorts which appear when using the MathissonPapapetrou equations, in particular in the description of highly relativistic particles presented in R. Plyatsko and M. Fenyk [Phys. Rev. D 91, 064033 (2015)]. We compare some results of this theory and of the aforementioned work with the ones obtained using a Lagrangian formulation for massive spinning particles and show that the issues mentioned in the preceding sentence do not appear in the Lagrangian treatment.



Armaza, C., Hojman, S. A., Koch, B., & Zalaquett, N. (2016). On the possibility of nongeodesic motion of massless spinning tops. Class. Quantum Gravity, 33(14), 18 pp.
Abstract: The motion of spinning massless particles in gravitationally curved backgrounds is revisited by considering new types of constraints. Those constraints guarantee zero mass (P μP μ= 0) and they allow for the possibility of trajectories which are not simply null geodesics. To exemplify this previously unknown possibility, the equations of motion are solved for radial motion in Schwarzschild background. It is found that the particle experiences a spininduced energy shift, which is proportional to the Hawking temperature of the black hole background.



Hojman, S. A., & Asenjo, F. A. (2015). Supersymmetric Majorana quantum cosmologies. Phys. Rev. D, 92(8), 7 pp.
Abstract: The Einstein equations for an isotropic and homogeneous FriedmannRobertsonWalker universe in the presence of a quintessence scalar field are shown to be described in a compact way, formally identical to the dynamics of a relativistic particle moving on a twodimensional spacetime. The correct Lagrangian for the system is presented and used to construct a spinor quantum cosmology theory using Breit's prescription. The theory is supersymmetric when written in the Majorana representation. The spinor field components interact through a potential that correlates the spacetime metric and the quintessence. An exact supersymmetric solution for k = 0 case is exhibited. This quantum cosmology model may be interpreted as a theory of interacting universes.



Hojman, S. A. (2015). Construction of Lagrangian and Hamiltonian structures starting from one constant of motion. Acta Mech., 226(3), 735–744.
Abstract: The problem of the construction of Lagrangian and Hamiltonian structures starting from two firstorder equations of motion is presented. This approach requires the knowledge of one (time independent) constant of motion for the dynamical system only. The Hamiltonian and Lagrangian structures are constructed, the HamiltonJacobi equation is then written and solved, and the second (time dependent) constant of the motion for the problem is explicitly exhibited.



Hojman, S. A. (2014). Origin of conical dispersion relations. Rev. Mex. Fis., 60(5), 336–339.
Abstract: A mechanism that produces conical dispersion relations is presented. A Kronig Penney one dimensional array with two different strengths delta function potentials gives rise to both the gap closure and the dispersion relation observed in graphene and other materials. The Schrodinger eigenvalue problem is locally invariant under, the infinite dimensional Virasoro algebra near conical dispersion points in reciprocal space, thus suggesting a possible relation to string theory.



Zalaquett, N., Hojman, S. A., & Asenjo, F. A. (2014). Spinning massive test particles in cosmological and general static spherically symmetric spacetimes. Class. Quantum Gravity, 31(8), 21 pp.
Abstract: A Lagrangian formalism is used to study the motion of a spinning massive particle in FriedmannRobertsonWalker and Godel spacetimes, as well as in a general Schwarzschildlike spacetime and in static spherically symmetric conformally flat spacetimes. Exact solutions for the motion of the particle and general exact expressions for the momenta and velocities are displayed for different cases. In particular, the solution for the motion in spherically symmetric metrics is presented in the equatorial plane. The exact solutions are found using constants of motion of the particle, namely its mass, its spin, its angular momentum, and a fourth constant, which is its energy when the metric is timeindependent, and a different constant otherwise. These constants are associated to Killing vectors. In the case of the motion on the FriedmannRobertsonWalker metric, a new constant of motion is found. This is the fourth constant which generalizes previously known results obtained for spinless particles. In the case of general Schwarzschildlike spacetimes, our results allow for the exploration of the case of the ReissnerNordstrom(Anti) de Sitter metric. Finally, for the case of the conformally flat spacetimes, the solution is explicitly evaluated for different metric tensors associated to a universe filled with static perfect fluids and electromagnetic radiation. For some combination of the values of the constants of motion the particle trajectories may exhibit spacelike velocity vectors in portions of the trajectories.



Hojman, S. A., & Koch, B. (2013). Closing a Window for Massive Photons. Adv. High. Energy Phys., , 5 pp.
Abstract: Working with the assumption of nonzero photon mass and a trajectory that is described by the nongeodesic world line of a spinning top we find, by deriving new astrophysical bounds, that this assumption is in contradiction with current experimental results. This yields the conclusion that such photons have to be exactly massless.



Contreras, M., & Hojman, S. A. (2014). Option pricing, stochastic volatility, singular dynamics and constrained path integrals. Physica A, 393, 391–403.
Abstract: Stochastic volatility models have been widely studied and used in the financial world. The Heston model (Heston, 1993) [7] is one of the best known models to deal with this issue. These stochastic volatility models are characterized by the fact that they explicitly depend on a correlation parameter p which relates the two Brownian motions that drive the stochastic dynamics associated to the volatility and the underlying asset. Solutions to the Heston model in the context of option pricing, using a path integral approach, are found in Lemmens et al. (2008) [21] while in Baaquie (2007,1997) [12,13] propagators for different stochastic volatility models are constructed. In all previous cases, the propagator is not defined for extreme cases rho = +/ 1. It is therefore necessary to obtain a solution for these extreme cases and also to understand the origin of the divergence of the propagator. In this paper we study in detail a general class of stochastic volatility models for extreme values rho = +/ 1 and show that in these two cases, the associated classical dynamics corresponds to a system with second class constraints, which must be dealt with using Dirac's method for constrained systems (Dirac, 1958,1967) [22,23] in order to properly obtain the propagator in the form of a Euclidean Hamiltonian path integral (Henneaux and Teitelboim, 1992) [25]. After integrating over momenta, one gets an Euclidean Lagrangian path integral without constraints, which in the case of the Heston model corresponds to a path integral of a repulsive radial harmonic oscillator. In all the cases studied, the price of the underlying asset is completely determined by one of the second class constraints in terms of volatility and plays no active role in the path integral. (C) 2013 Elsevier B.V. All rights reserved.



Hojman, S. A., & Asenjo, F. A. (2013). Can gravitation accelerate neutrinos? Class. Quantum Gravity, 30(2), 10 pp.
Abstract: The Lagrangian equations of motion for massive spinning test particles (tops) moving on a gravitational background using general relativity are presented. The paths followed by tops are nongeodesic. An exact solution for the motion of tops on a Schwarzschild background which allows for superluminal propagation of tops is studied. It is shown that the solution becomes relevant for particles with small masses, such as neutrinos. This general result is used to calculate the necessary condition to produce superluminal motion in part of the trajectory of a small mass particle in a weak gravitational field. The condition for superluminal motion establishes a relation between the mass, energy and total angular momentum of the particle.

