arXiv subjects

Jose B. Almeida

Publications and source records attributed to Jose B. Almeida.

At least 19 recordsLinked to original sources

arXiv2008

Different algebras for one reality

The most familiar formalism for the description of geometry applicable to physics comprises operations among 4-component vectors and complex real numbers; few people realize that this formalism has indeed 32 degrees of freedom and can thus be called 32-dimensional. We will revise this formalism and we will briefly show that it is best accommodated in the Clifford or geometric algebra G(1,3) x C, the algebra of 4-dimensional spacetime over the complex field. We will then explore other algebras isomorphic to that one, namely G(2,3), G(4,1) and Q x Q x C, all of which have been used in the past by PIRT participants to formulate their respective approaches to physics. G(2,3)is the algebra of 3-space with two time dimensions, which John Carroll used implicitely in his formulation of electromagnetism in 3 + 3 spacetime, G(4,1) was and it still is used by myself in a tentative to unify the formulation of physics and Q x Q x C is the choice of Peter Rowlands for his nilpotent formulation of quantum mechanics. We will show how the equations can be converted among isomorphic algebras and we also examine how the monogenic functions that I use are equivalent in many ways to Peter Rowlands nilpotent entities.

physics.gen-ph↗

arXiv2008

A geometric algebra approach to the hydrogen atom

Monogenic functions in the algebra of 5-dimensional spacetime have been used previously by the author as first principle in different areas of fundamental physics; the paper recovers that principle applying it to the hydrogen atom. The equation that results from the monogenic condition is formally equivalent to Dirac's and so its solutions resemble closely those found in the literature. The use of the monogenic condition as point of departure as not only the advantage of being a unified approach but also provides very strong links with geometry that are completely lost in the usual approach.

physics.atom-ph↗

arXiv2008

How much in the Universe can be explained by geometry?

The paper uses geometrical arguments to derive equations with relevance for cosmology; 5-dimensional spacetime is assumed because it has been shown in other works to provide a setting for significant unification of different areas of physics. Monogenic functions, which zero the vector derivative are shown to effectively model electrodynamics and relativistic dynamics if one allows for space curvature. Applying monogenic functions to flat space, the Hubble relation can be derived straightforwardly as a purely geometrical effect. Consideration of space curvature induced by mass density allows the derivation of flat rotation curves for galaxies without appealing for dark matter. Similarly, a small overall mass density in the Universe is shown to provide a possible explanation for recent supernovae observations, without the need for a cosmological constant.

physics.gen-ph↗

arXiv2007

The hidden geometric character of relativistic quantum mechanics

The presentation makes use of geometric algebra, also known as Clifford algebra, in 5-dimensional spacetime. The choice of this space is given the character of first principle, justified solely by the consequences that can be derived from such choice and their consistency with experimental results. Given a metric space of any dimension, one can define monogenic functions, the natural extension of analytic functions to higher dimensions; such functions have null vector derivative and have previously been shown by other authors to play a decisive role in lower dimensional spaces. All monogenic functions have null Laplacian by consequence; in an hyperbolic space this fact leads inevitably to a wave equation with plane-like solutions. This is also true for 5-dimensional spacetime and we will explore those solutions, establishing a parallel with the solutions of the Dirac equation. For this purpose we will invoke the isomorphism between the complex algebra of 4x4 matrices, also known as Dirac's matrices. There is one problem with this isomorphism, because the solutions to Dirac's equation are usually known as spinors (column matrices) that don't belong to the 4x4 matrix algebra and as such are excluded from the isomorphism. We will show that a solution in terms of Dirac spinors is equivalent to a plane wave solution. Just as one finds in the standard formulation, monogenic functions can be naturally split into positive/negative energy together with left/right ones. This split is provided by geometric projectors and we will show that there is a second set of projectors providing an alternate 4-fold split. The possible implications of this alternate split are not yet fully understood and are presently the subject of profound research.

quant-ph↗

arXiv2006

Can physics laws be derived from monogenic functions?

This is a paper about geometry and how one can derive several fundamental laws of physics from a simple postulate of geometrical nature. The method uses monogenic functions analysed in the algebra of 5-dimensional spacetime, exploring the 4-dimensional waves that they generate. With this method one is able to arrive at equations of relativistic dynamics, quantum mechanics and electromagnetism. Fields as disparate as cosmology and particle physics will be influenced by this approach in a way that the paper only suggests. The paper provides an introduction to a formalism which shows prospects of one day leading to a theory of everything and suggests several areas of future development.

physics.gen-ph↗

arXiv2006

Monogenic functions in 5-dimensional spacetime used as first principle: gravitational dynamics, electromagnetism and quantum mechanics

Monogenic functions are functions of null vector derivative and are here analysed in the geometric algebra of 5-dimensional spacetime, G(4,1), in order to derive several laws of fundamental physics. The paper introduces the working algebra and the definition of monogenic functions, showing that these generate two 4-dimensional spaces, one with Euclidean signature and the other one with Minkowski signature. The equivalence conditions between the two spaces are studied and relativistic dynamics, not entirely coincident with Einstein's general theory of relativity, is demonstrated. The monogenic condition is then shown to produce Maxwell's equations and electrodynamics both classical and quantized.

physics.gen-ph↗

arXiv2005

Choice of the best geometry to explain physics

Choosing the appropriate geometry in which to express the equations of fundamental physics can have a determinant effect on the simplicity of those equations and on the way they are perceived. The point of departure in this paper is the geometry of 5-dimensional spacetime, where monogenic functions are studied. Monogenic functions verify a very simple first order differential equation and the paper demonstrates how they generate the line interval of special relativity, as well as the Dirac equation of quantum mechanics. Monogenic functions act as a unifying principle between those two areas of physics, which is in itself very significant for the perception one has of them. Another consequence is the possibility of studying the same phenomena in Euclidean 4-dimensional space, providing a different point of view to physics, from which one has an unusual and enriching perspective.

physics.gen-ph↗

arXiv2005

Geometric Drive of the Universe's Expansion

What if physics is just the way we perceive geometry? That is, what if geometry and physics will one day become one and the same discipline? I believe that will mean we will at last really understand physics, without postulates other than those defining the particular space where the physics play is performed. In this paper I use 5-dimensional spacetime as a point of departure and make a very peculiar assignment between coordinates and physical distances and time. I assume there is an hyperspherical symmetry which is made apparent by assigning the hypersphere radius to proper time and distances on the hypersphere to usual 3-dimensional distances. Time, or Compton time to distinguish from cosmic time is the 0th coordinate and I am able to project everything into 4-dimensions by imposing a null displacement condition. Surprisingly nothing else is needed to explain Hubble's expansion law without any appeal to dark matter; an empty Universe will expand naturally at a flat rate in this way. I then discuss the perturbative effects of a small mass density in the expansion rate in a qualitative way; quantitative results call for the solution of equations that sometimes have not even been clearly formulated and so are deferred to later work. A brief outlook of the consequences an hyperspherical symmetry has for galaxy dynamics allows the derivation of constant rotation velocity curves, again without appealing to dark matter. An appendix explains how electromagnetism is made consistent with this geometric approach and justifies the fact that photons must travel on hypersphere circles, to be normal to proper time.

physics.gen-ph↗

arXiv2005

Geometric algebra and particle dynamics

In a recent publication the I showed how the geometric algebra ${G}_{4,1}$, the algebra of 5-dimensional space-time, can generate relativistic dynamics from the simple principle that only null geodesics should be allowed. The same paper showed also that Dirac equation could be derived from the condition that a function should be monogenic in that algebra; this construction of the Dirac equation allows a choice for the imaginary unit and it was suggested that different imaginary units could be assigned to the various elementary particles. An earlier paper had already shown the presence of standard model gauge group symmetry in complexified ${G}_{1,3}$, an algebra isomorphic to ${G}_{4,1}$. In this presentation I explore the possible choices for the imaginary unit in the Dirac equation to show that SU(3) and SU(2) symmetries arise naturally from such choices. The quantum numbers derived from the imaginary units are unusual but a simple conversion allows the derivation of electric charge and isospin, quantum numbers for two families of particles. This association to elementary particles is not final because further understanding of the role played by the imaginary unit is needed.

math.GM↗

arXiv2004

A General Method for the Determination of Matrix Coefficients for High Order Optical System Modelling

The non-linear transformations incurred by the rays in an optical system can be suitably described by matrices to any desired order of approximation. In systems composed of uniform refractive index elements, each individual ray refraction or translation has an associated matrix and a succession of transformations correspond to the product of the respective matrices. This paper describes a general method to find the matrix coefficients for translation and surface refraction irrespective of the surface shape or the order of approximation. The choice of coordinates is unusual as the orientation of the ray is characterised by the direction cosines, rather than slopes; this is shown to greatly simplify and generalise coefficient calculation. Two examples are shown in order to demonstrate the power of the method: The first is the determination of seventh order coefficients for spherical surfaces and the second is the determination of third order coefficients for a toroidal surface.

physics.optics↗

arXiv2004

Programming matrix optics into Mathematica

The various non-linear transformations incurred by the rays in an optical system can be modelled by matrix products up to any desired order of approximation. Mathematica software has been used to find the appropriate matrix coefficients for the straight path transformation and for the transformations induced by conical surfaces, both direction change and position offset. The same software package was programmed to model optical systems in seventh-order. A Petzval lens was used to exemplify the modelling power of the program.

physics.optics↗

arXiv2004

Wavefront and ray-density plots using seventh-order matrices

The optimization of an optical system benefits greatly from a study of its aberrations and an identification of each of its elements' contribution to the overall aberration figures. The matrix formalism developed by one of the authors was the object of a previous paper and allows the expression of image-space coordinates as high-order polynomials of object-space coordinates. In this paper we approach the question of aberrations, both through the evaluation of the wavefront evolution along the system and its departure from the ideal spherical shape and the use of ray density plots. Using seventh-order matrix modeling, we can calculate the optical path between any two points of a ray as it travels along the optical system and we define the wavefront as the locus of the points with any given optical path; the results are presented on the form of traces of the wavefront on the tangential plane, although the formalism would also permit sagital plane plots. Ray density plots are obtained by actual derivation of the seventh-order polynomials.

physics.gen-ph↗

arXiv2004

The null subspace of G(4,1) as source of the main physical theories

It is the author's belief that a perfect theory will eventually be formulated, where geometry and physics become indistinguishable, so that the complete understanding of space properties, together with proper assignments between geometric and physical entities, will provide all necessary predictions. The author intends to show that GR and Quantum Mechanics (QM) can be seen as originating from properties of the null subspace of 5-dimensional space with signature (-++++), together with its associated geometric algebra G(4,1). Besides generating GR and QM, the same space generates also 4-dimensional Euclidean space where dynamics can be formulated and is quite often equivalent to the relativistic counterpart. Euclidean relativistic dynamics resembles Fermat's principle extended to 4 dimensions and is thus designated as 4-Dimensional Optics (4DO). In this presentation the author uses G(4,1) with imposition of the null displacement length condition and derives the method to transpose between the metrics of GR and 4DO; this transition is proven viable for stationary metrics. It is hopeless to apply Einstein type equations in 4DO, for the simple reason that a null Ricci tensor always leads to a metric diverging to infinity. The author uses geometric arguments to establish alternative equations which are solved for the case of a stationary mass and produce a solution equivalent to Schwarzschild's metric in terms of PPN parameters. As a further development, the author analyses the case of a monogenic function in G(4,1). The monogenic condition produces an equation that can be conveniently converted into Dirac's, with the added advantage that it has built in standard model gauge group symmetry.

physics.gen-ph↗

arXiv2004

Euclidean formulation of general relativity

A variational principle is applied to 4D Euclidean space provided with a tensor refractive index, defining what can be seen as 4-dimensional optics (4DO). The geometry of such space is analysed, making no physical assumptions of any kind. However, by assigning geometric entities to physical quantities the paper allows physical predictions to be made. A mechanism is proposed for translation between 4DO and GR, which involves the null subspace of 5D space with signature $(-++++)$. A tensor equation relating the refractive index to sources is established geometrically and the sources tensor is shown to have close relationship to the stress tensor of GR. This equation is solved for the special case of zero sources but the solution that is found is only applicable to Newton mechanics and is inadequate for such predictions as light bending and perihelium advance. It is then argued that testing gravity in the physical world involves the use of a test charge which is itself a source. Solving the new equation, with consideration of the test particle's inertial mass, produces an exponential refractive index where the Newtonian potential appears in exponent and provides accurate predictions. Resorting to hyperspherical coordinates it becomes possible to show that the Universe's expansion has a purely geometric explanation without appeal to dark matter.

physics.gen-ph↗

arXiv2004

Maxwell's equations in 4-dimensional Euclidean space

The paper formulates Maxwell's equations in 4-dimensional Euclidean space by embedding the electromagnetic vector potential in the frame vector $g_0$. Relativistic electrodynamics is the first problem tackled; in spite of using a geometry radically different from that of special relativity, the paper derives relativistic electrodynamics from space curvature. Maxwell's equations are then formulated and solved for free space providing solutions which rotate the vector potential on a plane; these solutions are shown equivalent to the usual spacetime formulation and are then discussed in terms of the hypersphere model of the Universe recently proposed by the author.

physics.gen-ph↗

arXiv2004

An hypersphere model of the Universe - The dismissal of dark matter

One can make the very simple hypothesis that the Universe is the inside of an hypersphere in 4 dimensions, where our 3-dimensional world consists of hypersurfaces at different radii. Based on this assumption it is possible to show that Universe expansion at a rate corresponding to flat comes as a direct geometrical consequence without intervening critical density; any mass density is responsible for opening the Universe and introduces a cosmological constant. Another consequence is the appearance of inertia swirls of expanding matter, which can explain observed velocities around galaxies, again without the intervention of dark matter. When restricted to more everyday situations the model degenerates in what has been called 4-dimensional optics; in the paper this is shown to be equivalent to general relativity in all static isotropic metric situations. In the conclusion some considerations bring the discussion to the realm of 4D wave optics.

physics.gen-ph↗

arXiv2003

3-Dimensional Mapping of Corneal Topography and Thickness

Optical sections of the cornea are obtained by illumination with a collimated beam expanded in a fan shape by a small rotary cylindrical lens. The light diffused from the cornea is observed by two cameras and processed in order to yield the surfaces' profiles. The optical system used to project a thin rotating line on the cornea consists of a white light source provided with optical fiber bundle output which is first conditioned by a set of lenses so that it would produce a spot on the cornea. A small cylinder lens is used to expand the beam in one direction so that a thin line illuminates the cornea, rather than a spot. The cylinder lens is provided with motor driven rotation about an axis normal to its own in order to rotate the line on the cornea such that the projected line scans the whole cornea; the illuminator is completed with a slit aperture. The cornea is not perfectly transparent, scattering some of the light that traverses it; this fact is used for its observation by two cameras. These are placed at pre-defined angles with the illumination axis, so that optical sections of the cornea can be seen; the use of two cameras avoids the need for camera rotation in synchronism with the cylinder. The two cameras' images can be combined in order to simulate a single virtual rotating camera. Image processing is used to extract information about the corneal surfaces profiles and thickness from the optical sections. Several peculiar aspects of processing are discussed, namely the corneal edge detection algorithm, the correction for angle of view and deformation due to observation of the inner surface through the cornea.

physics.med-ph↗

arXiv2003

Standard-model symmetry in complexified spacetime algebra

Complexified spacetime algebra is defined as the geometric (Clifford) algebra of spacetime with complex coefficients, isomorphic $\mathcal{G}_{1,4}$. By resorting to matrix representation by means of Dirac-Pauli gamma matrices, the paper demonstrates isomorphism between subgroups of CSTA and SU(3). It is shown that the symmetry group of those subgroups is indeed $U(1) \otimes SU(2) \otimes SU(3)$ and that there are 4 distinct copies of this group within CSTA.

math.GM↗