domingo, 4 de junio de 2017

Classic 4D Model Of the Discrete Universe (C2). I:Quantum

En esta entrada os dejo el capitulo 2 del libro

2. Discrete space-time

The current theories, based on a space-time continuum, sometimes give rise to the appearance of infinites masked with renormalization, and the denomination of singularity, as in the case of black holes. Instead, in the discrete 4D model, the infinities disappear because at no time can the space-time be zero.

2.1 Introduction

General relativity implies that space-time is a continuum. However, there is no experimental evidence for this. Are space and time a continuum or are they composed of indivisible discrete units? We're probably convinced of continuity as a result of education. In recent years however, both physicists and mathematicians have asked if it is possible that space and time are discrete? Smolin states that space is formed from “irreducible pieces of volume that cannot be broken into anything smaller” that he calls “Atoms of Space and Time” [3].
Minimum volume, length or area are measured in units of Planck [3]. Planck's constant, h, which represents the elementary quantum of action, has an important role in quantum mechanics. There are several theories that predict the existence of a minimum length [4,5]. These theories are related to quantum gravity, such as string theory and double special relativity, as well as black hole physics [6–8].
Quantification of space-time maintains relativistic invariance [9] and causation, and allows us to distinguish elementary particles from each other in a simple and natural way [10].
There is evidence of discrete structures on the largest scales, for example superclusters and the redshift [11]. Cowan already said in 1969 that the redshift can only occur with discrete values [12]. This was subsequently confirmed by Karlsson [13].
Heisenberg said that physics must have a fundamental length scale, and with Planck's constant, h, and the speed of light, c, allow derivation of the masses of the particles [14, 15]. Planck’s length can be considered as the shortest distance having any physical meaning. To Sprenger, “a fundamental (minimal) length scale naturally emerges in any quantum theory in the presence of gravitational effects that accounts for a limited resolution of space-time.” The Planck scale appears to combine gravity (G), quantum mechanics (h), and special relativity (c) [16]. Padmanabhan shows that the Planck length provides a lower limit of length in any suitable physical space-time [17, 18]. Also, Messen starts from a minimum length he calls a, and a four-dimensional space, which allows him to characterize the different types of particles by quantum numbers. Then, different states of the particles correspond to different excitations of space-time [19].
From Planck units Planck force is derived, which is associated with the gravitational potential energy and electromagnetic energy. Planck force can be expressed as
where G is the gravitational constant, mp the Planck mass, c the speed of light, ħ the reduced Planck constant (h/2π), and rp the radius of Planck, which can be expressed in terms of G, ħ and c as
Substituting all of these into Equation (2.1), results in

2.2 Definition of time

What relates time to speed? The answer is obvious, space. Here space is not the distance between two points, but the space that fills the universe. We can thus define time as: the variation of the space-universe with speed. Planck’s particle expands in four spatial dimensions at the speed of light, giving rise to the appearance of time as a result of the variation of space with the speed of expansion, c
Let us now consider the gravitational force between the sun and the Earth. To simplify matters, let us suppose that the earth describes a flat circular orbit. To calculate this path we use the coordinate x-y axes. From the solar perspective, the only important thing for the sun to exert a force on the Earth is the distance r between them. It does not matter whether the Earth is on the right, left, behind or in front, as this force does not vary with position. The same happens with the expansion of the universe as only one of the three spatial dimensions is important: the direction between the two points or objects of reference considered. This direction is at all times perpendicular to the fourth dimension (radial direction). The increase of space is two dimensional and for an observer at rest time, will be given by:
If the surface increase, ΔS, is positive, time will be real and positive. This is proper time, pertaining to relativity and to the time we can observe and measure. However, if the surface increase is negative, the time is imaginary. This is the time that must be considered, for example, when we measure particles with energy whose wavelength is less than that of the particle, or when we apply Schrödinger’s equation.

2.3 Real and imaginary time
When we measure electron mass precisely, we need a lot of time, or conversely, little energy, which is equivalent to being at a distance greater than its wavelength. However, when we want to measure its position accurately, we use a lot of energy or short wavelength, which is equivalent to being at distances smaller than the electron’s wavelength. According to the given definition of time, the spherical surface will be formed by the fourth dimension (radial direction), and the direction of the observation (figure 2.1), 
so that
will be real and positive for distances greater than the wavelength.
When we get closer to the particle (a distance shorter than its wavelength) it turns out that the surface variation in the interior of the surface of radius l=cto  is negative, so that:
Here, ti, is the time in the interior of the surface of radius l and te is the time on the exterior of the surface. t0 is the time on the surface, or time at rest, or ct0, which is the wavelength of the quantum mechanics.

2.4 Conclusion

Time is a two-dimensional physical quantity. The fourth dimension we measure is time due to the expansion of the universe. There is also another time, which is the distance between two points in space. The expansion of the universe generates real and positive time. In addition, when we measure particles with long wavelengths, we have a real and positive time. But when we measure particles with short wavelengths, smaller than the wavelength of the particle, then the time is imaginary.


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