arXiv subjects
Publications and source records attributed to Sergio A. Hojman.
Two exact different solutions to the Telegraph equation in three-dimensional space are obtained in terms of Airy functions. As a result, these solutions unveil a distinctive propagation pattern along a coordinate that resembles a speed-cone-like coordinate of the system. This unique characteristic leads to effective Schrödinger-like equations, amenable to exact solutions through Airy functions.
It is shown that any mathematical solution for null electromagnetic field knots in flat spacetime is also a null field knotted solution for cosmological electromagnetic fields. This is obtained by replacing the time $t\rightarrow τ=\int dt/a$, where $a=a(t)$ is the scale factor of the Universe described by the Friedman-Lemaître-Robertson-Walker (FLRW) cosmology, and by adequately rewriting the (empty flat spacetimes) electromagnetic fields solutions in a medium defined by the FLRW metric. We found that the dispersion (evolution) of electromagnetic Hopfions is faster on cosmological scenarios. We discuss the implications of these results for different cosmological models.
We investigate the propagation of certain non-plane wave solutions to Maxwell's equations in both flat and curved spacetimes. We find that such solutions (or rather parts of them) exhibit accelerative behaviour, and in particular do not propagate on straight lines. Having established these results, we then turn to their conceptual significance -- which, in brief, we take to be the following: (i) one should not assume that the part of electromagnetic waves from outer space that is subject to detection is localised onto null trajectories; therefore (ii) astrophysicists and cosmologists should at least be wary about making such assumptions in their inferences from obtained data, for to do so may lead to incorrect inferences regarding the nature of our universe.
It is shown that any cosmological anisotropic model produces supersymmetric theories for both massless scalar and electromagnetic fields. This supersymmetric theory is the time-domain analogue of a supersymmetric quantum mechanical theory. In this case, the variations of the anisotropic scale factors of the Universe are responsible for triggering the supersymmetry. For scalar fields, the superpartner fields evolve in two different cosmological scenarios (Universes). On the other hand, for propagating electromagnetic fields, supersymmetry is manifested through its polarization degrees of freedom in one Universe. In this case, polarization degrees of freedom of electromagnetic waves, which are orthogonal to its propagation direction, become superpartners from each other. This behavior can be measured, for example, through the rotation of the plane of polarization of cosmological light.
Harmonic wave functions for integer and half-integer angular momentum are given in terms of the Euler angles $(θ,ϕ,ψ)$ that define a rotation in $SO(3)$, and the Euclidean norm in ${\mathbb R}^3$. Following a classical work by Schwinger, $2$-dimensional harmonic oscillators are used to produce raising and lowering operators that change the total angular momentum eigenvalue of the wave functions in half units. The nature of the representation space $\mathcal H$ is approached from the double covering group homomorphism $SU(2)\to SO(3)$ and the topology involved is taken care of by using the Hurwitz-Hopf map $H:{\mathbb R}^4\to{\mathbb R}^3$. It is shown how to reconsider $H$ as a 2-to-1 group map, $G_0={\mathbb R}^+\times SU(2)\to {\mathbb R}^+\times SO(3)$, translating it into an assignment $(z_1,z_2)\mapsto (r,θ,ϕ,ψ)$ whose domain consists of pairs $(z_1,z_2)$ of complex variables. It is shown how the Lie algebra of $G_0$ is coupled with two Heisenberg Lie algebras of $2$-dimensional (Schwinger's) harmonic oscillators generated by the operators $\{z_1,z_2,\bar{z}_1,\bar{z}_2\}$ and their adjoints. The whole set of operators gets algebraically closed either into a $13$-dimensional Lie algebra or into a $(4|8)$-dimensional Lie superalgebra. The wave functions in $\mathcal H$ can be written in terms of polynomials in the complex coordinates $(z_1,z_2)$ and their complex conjugates, and the representations are explicitly constructed via the various highest weight (or lowest weight) vector representations of $G_0$. A new non-relativistic quantum (Schrödinger-like) equation for the hydrogen atom that takes into account the electron spin is introduced and expressed in terms of $(r,θ,ϕ,ψ)$ and the time $t$. The equation may be solved exactly in terms of the harmonic wave functions hereby introduced.
A medium with specific anisotropic refractive indices can induce a supersymmetric behavior in the propagation of polarized electromagnetic waves, in an analogue fashion to a quantum mechanical system. The polarizations of the wave are the ones which behave as superpartners from each other. For this to happen, the anisotropy of the medium must be transverse to the direction of propagation of the wave, with different refractive indices along the direction of each polarization. These refractive indices must follow a very specific relation in order to trigger the supersymetric response of the electromagnetic wave, each of them with spatial dependence on the longitudinal (propagation) direction of the wave. In this form, in these materials, different polarized light can be used to test supersymmetry in an optical fashion.
We show that several Hamiltonians that are $\mathcal{PT}$ symmetric may be taken to Hermitian Hamiltonians via a non-unitary transformation and vice versa. We also show that for some specific Hamiltonians such non-unitary transformations may be associated, via a fractional-Wick rotation, to complex time.
A supersymmetric relativistic quantum theory in the temporal domain is developed for bi-spinor fields satisfying the Dirac equation. The simplest time-domain supersymmetric theory can be postulated for fields with time-dependent mass, showing an equivalence with the bosonic supersymmetric theory in time-domain. Solutions are presented and they are used to produce probability oscillations between mass states. As an application of this idea, we study the two-neutrino oscillation problem, showing that flavour state oscillations may emerge from the supersymmetry originated by the time-dependence of the unique mass of the neutrino.
By adapting the Madelung-Bohm formalism to paraxial wave propagation we show, by using Ermakov-Lewis techniques, that the Gouy phase is related to the form of the phase chosen in order to produce a Gaussian function as a propagated field. For this, we introduce a quantum mechanical invariant, that it is explicitly time dependent despite the fact that the Hamiltonian is itself time-independent. We finally show that the effective Bohm {\it index of refraction} generates a GRIN medium that produces the focusing needed for the Gouy phase.
New localized structured solutions for the three-dimensional linear diffusion (heat) equation are presented. These new solutions are written in terms of Airy functions and either Gaussian or Bessel functions. They accelerate along their propagation direction, while in the plane orthogonal to it, they retain their either Gaussian or Bessel structure. These diffusion (heat) densities retain a localized structure in space as they propagate, and may be considered the heat analogue of Airy light bullets.
Bohr atomic model is based on the assumption that electrons on allowed quantized orbits do not radiate. Its main results include the values of the radii of circular quantized orbits and of the hydrogen atom energy levels. Quantum mechanical justifications of both his hypothesis and of the well known radii relation, which is used to compute the energy levels, are presented.
We report accelerating diffusive solutions to the diffusion equation with a constant diffusion tensor. The maximum values of the diffusion density evolve in an accelerating fashion described by Airy functions. We show the diffusive accelerating behavior for one--dimensional systems, as well as for a general three--dimensional case. We also construct a modulated modified form of the diffusion solution that retains the accelerating features.
It is showed that complex scalar fields with a self-interaction potential may propagate along null geodesics on flat Friedmann--Lemaître--Robertson--Walker universes with different time-dependent scale factors. This occurs provided they self interact adequately, for different forms of potentials, and even for the massive case.
Symmetries of the field equations are used to construct infinitely many nontrivial linearly independent new solutions to different partial differential equations such as the Schroedinger, the diffusion, and the paraxial equations, among many others, including Klein Gordon, Dirac, Maxwell, Rarita Schwinger, linear Einstein field equations and even some especial seed solutions of fully nonlinear general relativity. The construction is done by applying one symmetry operator of the differential system to one nonseparable seed solution of the same system.
We show that two-mode squeezed vacuum-like states may be engineered in the Bohm-Madelung formalism by adequately choosing the phase of the wavefunction. The difference between our wavefunction and the one of the squeezed vacuum states is given precisely by the phase we choose.
A new approach to find exact solutions to one--dimensional quantum mechanical systems is devised. The scheme is based on the introduction of a potential function for the wavefunction, and the equation it satisfies. We recover known solutions as well as to get new ones for both free and interacting particles with wavefunctions having vanishing and non--vanishing Bohm potentials. For most of the potentials, no solutions to the Schrödinger equation produce a vanishing Bohm potential. A (large but) restricted family of potentials allows the existence of particular solutions for which the Bohm potential vanishes. This family of potentials is determined, and several examples are presented. It is shown that some quantum, such as accelerated Airy wavefunctions, are due to the presence of non--vanishing Bohm potentials. New examples of this kind are found and discussed.
It is proved that accelerating nondiffracting gravitational Airy wave--packets are solutions of linearized gravity. It is also showed that Airy functions are exact solutions to Einstein equations for non--accelerating nondiffracting gravitational wave--packets.
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 non--quantum 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.