3D Field Model

Continuous or Discrete? 

A 3D Field Model

Consider several basic questions:

1. Is the nature of space, and time, continuous or discrete?
   

• Is the construction of Space, or Time, analog or digital, or both?
  
  • If analog, do we use the current 2D wave model, or upgrade to 3D?
    
  • If 3D wave analog model is used, do we use Plank's methodology?
    
  • If digital, then is QED methodology needed, and referable and preferable?
    
  • If quantifiable into particles, do they have shapes?
    
• If Space and Time are not separate, but tied together in relationships, then what and how many relationships are there, and then are they each continuous or discrete?
  
• Does Spacetime interact with photons?
  
  • Via particle (discrete point methodology)
    
  • Via wave (2D planar methodology)
    
  • Via field (3D structural methodology)
    

2. Is the metric of the natural co-ordinate system, essential to the field paradigm, 
   continuous or discrete?
   

• Are we in a linearized co-ordinate system, aka x, y, z, … ?
  
• Are we in a rotational co-ordinate system, aka r, Θ, Φ, … ?
  
• Are we in a co-ordinate system that is differential or integral?
  

3. What are the interactions between Spacetime and matter in its different forms?
   

4. How do the collections of ions and massive particles, at wide ranges of density, that are populating space, affect a photon pathway, as in a preferential manner of said transmission as through a continuous, or discrete, 3D field?
   

To really start afresh, in this Physics treatise, meta-anything will not be considered. Discrete methodology implies particles of all kinds, however we will limit ourselves for brevity, to the standard set of reasonably stable known ones of photon, electron, proton, neutron, and neutrino. They have their mirrored forms, but don't want to confuse them here with their so-call anti-forms, as there is disagreement about how the anti- is to be considered and represented. Clarity will be made further in this treatise. They can further be combined into various kinds and forms of matter.

A 3D field of particles will be like being immersed deep in an ocean, with neutral buoyancy (no sense of gravity), and all the air you can breath. Going up, down, left, right, etc. all feels the same, and takes the same effort.  A 3D Field that is described via the Continous methodology implies wave-like functions. Waveforms can be viewed in several different ways. The areas of bright light, that are visible on the dorsal surface of the 'mermaids', that are playing in the '3D field', in the video, are cross-sections of 3D waves of light that were intercepted by a more opaque interface.

The continuous methodology, as is applied in the time domain, can be viewed through the analysis of mathematical functions, physical signals or time series of physical dimensional data, with respect to time. In the time domain, the signal or function's value is known for all real numbers, for the case of continuous time, or at various separate instants in the case of discrete time. An oscilloscope is a tool commonly used to visualize real-world signals in the time domain. In the time-domain graph below, it is being shown how a signal changes with time, whereas a frequency-domain graph shows how much of the signal lies within each given frequency band over a range of frequencies, and a spectrum analyzer is a better tool for showing the harmonics.

As is shown, waveforms can start to be seen in 3D, via Fourier analysis which then starts to show how a 3D continuous field can be imagined.

Axiom

An axiom, or postulate, is a premise or starting point of reasoning. As classically conceived, an axiom is a premise so evident as to be accepted as true without the usual controversy. The word comes from the Greek ἀξίωμα (āxīoma) 'that which is thought worthy or fit' or 'that which commends itself as evident.'  As used in modern logic, an axiom is simply a premise or starting point for reasoning. Axioms define and delimit the realm of analysis; the relative truth of an axiom is usually taken for granted within the particular domain of analysis, and serves as a starting point for deducing and inferring other relative truths. No explicit view regarding the absolute truth of axioms is ever taken in the context of modern mathematics, as such a thing is considered to be an irrelevant and impossible contradiction in terms.

In mathematics, the term axiom is used in two related but distinguishable senses: "logical axioms" and "non-logical axioms". Logical axioms are usually statements that are taken to be true within the system of logic they define (e.g., (A and B) implies A), while non-logical axioms (e.g., a + b = b + a) are actually defining properties for the domain of a specific mathematical theory (such as arithmetic). When used in the latter sense, "axiom," "postulate", and "presumption" may be used interchangeably. In general, a non-logical axiom is not a self-evident truth, but rather a formal logical expression used in deduction to build a mathematical theory. As modern mathematics admits multiple, equally "true" systems of logic, precisely the same thing must be said for logical axioms - they both define and are specific to the particular system of logic that is being invoked. To axiomatize a system of knowledge is to show that its claims can be derived from a small, well-understood set of sentences (the axioms). There are typically multiple ways to axiomatize a given mathematical domain.

In both senses, an axiom is any mathematical statement that serves as a starting point from which other statements are logically derived. Within the system they define, axioms (unless redundant) cannot be derived by principles of deduction, nor are they demonstrable by mathematical proofs, simply because they are starting points; there is nothing else from which they logically follow otherwise they would be classified as theorems. However, an axiom in one system may be a theorem in another, with the reverse also being true.

Axiomatic Schema

It is understood that the use of axioms in an axiomatic based set theory, as are used in the branches of logic, mathematics, and computer science will be employed. The various axiom schema of specification, axiom schema of separation, subset axiom scheme and axiom schema of restricted comprehension are schema of axioms based in Zermelo–Fraenkel set theory, which essentially says that any definable subclass of a set is a set. The evolution of the implied mathematics will proceed from properly established mathematical axioms. The evolution of the physics proposed, proceeds from here, towards the establishment of more fundamental physical axioms.

The Discrete Methodology:

We will view what we see and experience as a set of particles combined together as the optimum set of discrete 'points' as defined above. We will include in the context of particles the ones mentioned above and again here for reference: photon, electron, proton, neutron, and neutrino, which are classified and referred to in 'the standard model' as such:

Photon

The Photon is an elementary particle, is the quantum of light and all other forms of electromagnetic radiation, and the force carrier for the electromagnetic force, even when static via virtual photons. The effects of this force are easily observable at both the microscopic and macroscopic level, because the photon has zero rest mass; this allows long distance interactions. Like all elementary particles, photons are currently best explained by quantum mechanics and exhibit wave–particle duality, exhibiting properties of both waves and particles (however, for this particular methodology we will stick to the particle definition). For example, a single photon may be refracted by a lens or exhibit wave interference with itself, but also act as a particle giving a definite result when its position is measured (and also as a mechanism for the transfer of momentum).

Electron

The electron (symbol: e−) is a subatomic particle with a negative elementary electric charge, and belongs to the lepton particle family, and is generally thought to be an elementary particle because it has no known components or substructure and have a mass that is approximately 1/1836 that of the proton. Quantum mechanical properties of the electron include an intrinsic angular momentum (spin) of a half-integer value in units of ħ, which means that it is a fermion. Being a fermion, it has been shown that no two electrons can occupy the same quantum state, in accordance with the Pauli exclusion principle. The electron also has properties of both particle and wave, and so can collide with other particles and can be diffracted like light. Experiments with electrons have demonstrated this duality.

Hadron

Hadrons are categorized into two families: baryons (such as protons and neutrons, made of three quarks) and mesons (such as pions, made of one quark and one antiquark). Other types of hadron may exist. Of the hadrons, protons and neutrons bound to atomic nuclei are stable, whereas others are unstable under ordinary conditions; free neutrons decay in 15 minutes. Experimentally, hadron physics is studied by colliding protons or nuclei of heavy elements such as lead, and detecting the debris in the produced particle showers.

Neutrino

The Neutrino (/njuːˈtriːnoʊ/) is an electrically neutral, weakly interacting elementary subatomic particle with half-integer spin. The neutrino (meaning "small neutral one" in Italian) is denoted by the Greek letter ν (nu). All evidence suggests that neutrinos have mass but that their mass is tiny even by the standards of subatomic particles. Their mass has never been measured accurately. Neutrinos do not carry electric charge, which means that they are not affected by the electromagnetic forces that act on charged particles such as electrons and protons. A typical neutrino passes through normal matter unimpeded.

Measurements of the interaction between energetic photons and hadrons show that the interaction is much more intense than expected by the interaction of merely photons with the hadron's electric charge. Furthermore, the interaction of energetic photons with protons is similar to the interaction of photons with neutrons in spite of the fact that the electric charge structures of protons and neutrons are substantially different. This is because the magnetic aspects of the neutron and proton are greater than the electron, and as will be shown later as being part the polarization of a photon.

If experimentally probed at very short distances, the intrinsic structure of the photon is recognized as a flux of quark and gluon components, quasi-free according to asymptotic freedom in QCD and described by the photon hyper-structure function. In the Standard Model of particle physics, photons are described as a necessary consequence of physical laws having a certain symmetry at every point in space-time. The intrinsic properties of photons, such as charge, mass and spin, are determined by the geometric shape properties of this gauge-type symmetry.

Plasma

Plasma (from Greek πλάσμα, "anything formed") is one of the four fundamental states of matter (the others being solid, liquid, and gas). It comprises the major component of the Sun. Heating a gas may ionize its molecules or atoms (reducing or increasing the number of electrons in them), thus turning it into a plasma, which contains charged particles: positive ions and negative electrons or ions. Ionization can be induced by other means, such as strong electromagnetic field applied with a laser or microwave generator, and is accompanied by the dissociation of molecular bonds, if present. Plasma can also be created by the application of an electric field on a gas such as air. The presence of a non-negligible number of charge carriers makes the plasma electrically conductive so that it responds strongly to electromagnetic fields. Plasma, therefore, has properties quite unlike those of solids, liquids, or gases and is considered a distinct state of matter. Like gas, plasma does not have a definite shape or a definite volume unless enclosed in a container; unlike gas, under the influence of a magnetic field, it may form structures such as filaments, beams and double layers. Some common plasmas are found in stars. In the universe, plasma is the most common state of matter. Most of the matter in the universe, is in the form of the rarefied intergalactic plasma (particularly intracluster medium) and in stars.  Much of the understanding of plasmas has come from the pursuit of controlled nuclear fusion and fusion power, for which plasma physics provides the scientific basis. It is further postulated here, for the discrete methodology, that Spacetime itself, is generally to be thought of as a 'gelatin' of quantum elementary plasma-like particles, with similar qualities. Models like liquid or foam don't leave a history behind them.

If discrete particles can group together and act continuously as a dielectric, gas, or plasma, for the purposes of photon transmission, using Fresnel's system of index of refraction as a function of the indexes, f(n), what are the effects on photon transmissions?

Plasma is found so effective in wave particle interaction it currently leads stealth technology, by using a boundary plasma layer to absorb radar waves. Changes in pressure affect amplitude of the incoming wave and not the frequency. Spectroscopy shows very low flux plasma can still change photon speed to it's own local c/n. Thus the photon speed is controlled by 'n' as within a dielectric medium, and thus bends the incoming photon at the interface of boundary plasma layers between arbitrary pressure regions of the plasma, within an arbitrary unit volume of space.

The Fresnel equations (or Fresnel conditions), deduced by Augustine-Jean Fresnel, describe the behavior of light when moving between media of differing refractive indexes. The reflection of light that the equations predict is known as Fresnel reflection.

When light moves from a medium of a given refractive index n1 into a second medium with refractive index n2, both reflection and refraction of the light may occur. The Fresnel equations describe, as a ratio, what fraction of the light is reflected and what fraction is refracted (i.e., transmitted). They also describe the angle of emission, and the phase shift (color change) of the reflected light. Plasma can also absorb the photons. The act of absorption is where amplitude modulation occurs, creating pressure differentials within the plasma volume.

Most Fresnel equations presume that the interface is flat, planar, and homogeneous, and that the light is a plane wave. The fraction of the incident power that is reflected from the interface is given by the reflectance R and the fraction that is refracted is given by the transmittance T. The media is usually presumed to be non-magnetic.

However within a plasma cloud there are always stray magnetic fields, induced by random motion causing differentials between plasma particles, causing a random bunch of ionized particles to behave together, to form a 'plasma field' via magnetic interaction. The plasma does consist of both positively-charged and negatively-charged particles whose average kinetic energy is greater, or distance between particles is greater than the 'Debye' length, or greater than the level of kinetic energy of combining into non-ionized particle systems, or formation of neutral particles such as loose unbound neutrons. Loose bound neutrons are unstable and will change into three different particles, those being the Proton, the Electron, and the Neutrino. Thus a cloud of plasma tends to stay plasma-like, unless acted upon by other conditions.

Plasma is neither flat, nor planar, nor homogeneous. Thus calculations for the Reflection, and Refraction of photons depend on polarization of the incident photon, while the Absorption depends on inline changes in local kinetic energy between source particles in the presence of incident photons.

If the incident photon is polarized with its electric field parallel to the interface differential, and perpendicular to said plasma, containing the incident, reflected, and refracted photons, such a photon is described as p-polarized. The incident photon that is polarized with its electric field perpendicular to said plane, is said to be s-polarized, from the German 'senkrecht' (perpendicular). A-polarization is a photon's forward momentum, being affected by the passage through stray magnetic field interface lines which causes harmonics within the amplitudes of the incident photons. As a consequence of the conservation of energy, the transmission coefficients are given by:

P-Polarized

S-Polarized

A-Polarized

 These relationships hold only for power coefficients, not for amplitude coefficients. If the incident light is not polarized (containing an equal mix of a-, s- and p-polarization), the reflection coefficient is:

Photons can bounce back and forth a number of times between the interface differentials within an arbitrary volume of plasma. The combined reflection coefficient for this case is 3R/(1 + R). The discussion given here presumes that the permeability of plasma, is nearly equal to the vacuum permeability of space. This is approximately true for most dielectric materials, but not for all types of material. The completely general Fresnel equations are more complicated.

Amplitude equations for coefficients corresponding to ratios of the electric field complex-valued amplitudes of the s-polar and p-polar 'ballistic Hypervectors', (not necessarily real-valued magnitudes) are styled after "Fresnel equations", while the equations for the a-polar waves take on 3D Sonar like topology, using differentiated velocities integrated into a geometric shape that provides structuralism for the boundaries that form the interface differentials. These equations take several different forms, depending on the choice of formalism and sign convention used. The amplitude coefficients are usually represented by lower case r, t, and a.

For the following, it is presumed that any incident photon is normal to the Differential Interface, DI, and that any change in angle in the original path is the result of refraction, given by 'n' for single dielectric, and is also shown below as the ratio of n2/n1, and that any change in velocity within the path of the photon, is the effect of variable density regions and near relativistic differential Kinetic Energy levels within the structurally defined arbitrary volume of plasma, with all else in equilibrium outside said volume, gives a dynamic process to the redshift, between the photon source and photon receiver.

The coefficient r is the ratio of the reflected photon's complex electric field amplitude to that of the normal of the DI. The coefficient t is the ratio of the transmitted photon's electric field amplitude to that of the normal of the DI. The photon is split into s and p polarizations, whereby s-polarization is perpendicular, and p-polarization is parallel. The coefficient a is the ratio of attenuation via absorption by the plasma, resulting in heating, or or cooling, via a change in kinetic activity of the plasma through interactions with the incident photons, within the region of plasma, to that of the incident photon's original transmission and propagation vector, resulting in no attenuation of the momentum of the photon, just like a photon's passage through a void in the plasma, or the vacuum permeability of space, where we have postulated, using the discrete methodology, that Spacetime itself, is generally to be thought of as a 'gelatin' of quantum elementary plasma-like particles, with similar qualities, where the photon's velocity appears to be the fastest, 'c', and is considered to be 100% transparent to the passage of photons. Yet a photon's passage is not instantaneous, so there must be some kind of resistance, which is a 'time / distance' type relationship, in our '3D*1T' environment.

For s-polarization, a positive r or t means that the electric fields of the incoming and reflected or transmitted photon are parallel, while negative means perpendicular. The magnetic field of the photon interacts strongly with all particles, while the electric field of the photon interacts only strongly with the electron. For p-polarization, a positive r or t means that the magnetic fields of the photons are parallel, while negative means perpendicular. It is also presumed that the magnetic permeability, µ, of a plasma layer region is equal to the permeability of free space constant µ₀ The physical constant μ₀, commonly called the vacuum permeability, or permeability of free space, or magnetic constant is an ideal, (baseline) physical constant, which is the value of magnetic permeability in a classical vacuum. Vacuum permeability is derived from production of a magnetic field by an electric current or by a moving electric charge and in all other formulas for magnetic-field production in a vacuum. In the reference medium of classical vacuum, µ0 has an exact defined value: µ0 = 4π×10^−7 V·s/(A·m) ≈ 1.2566370614...×10^−6 H⋅m^−1 or N·A^−2 or T·m/A or Wb/(A·m), in the SI system of units. As a constant, it can also be defined as a fundamental invariant quantity, and is also one of three components that defines free space through Maxwell's equations. In classical physics, 'free space' is a concept of electromagnetic theory, corresponding to a theoretically perfect vacuum, and sometimes referred to as the vacuum of free space, or as classical vacuum, and is appropriately viewed as a reference medium.

Let's take a quick look at the multidimensional units of the vacuum permeability: V·s/(A·m), and see how it is that it can be called a 'medium'. If we look at the units the vacuum permeability, Volt second / Ampere meter, and look at 'time/distance' as the inverse of velocity, which is 'Inertial Resistance' which for the vacuum permeability gives (Volt / Ampere) * IR. The units for the Volt are: (mass x area) / (Ampere x second^3); the ampere, whose SI unit symbol is: A, and SI dimension symbol is: I, often shortened to amp, is the SI unit of electric current and is one of the seven SI base units. It is named after André-Marie Ampère (1775–1836), French mathematician and physicist, considered the father of electrodynamics. In practical terms, the ampere is a measure of the amount of electric charge passing a point in an electric circuit per unit time, with 6.241×10^18 electrons (or one coulomb) per second, constituting one ampere.  The Ohm, has units of kg·m²·s^-3·A^-2 but fortunately, it doesn't come into play in this calculation.  So now we have ((mass x area) / ( 6.241×10^18 electrons x second^4)) * (time/distance).  Using the appropriate math, we can rewrite the units as (mass x distance) / ( 6.241×10^18 electrons x second^3), which we can then rewrite as (one unit of Thrust / second) /(6.241×10^18 electrons).  So, now if we are considering empty space-time, the vacuum of free space, where we have postulated, using the discrete methodology, that Spacetime itself, is generally to be thought of as a 'gelatin' of quantum elementary plasma-like particles, such as the electron, positron, proton, neutron, and the neutrino, then here, we now have that basis for a medium. Then considering Gravity as being sourced from empty space-time, just like the vacuum, and whether the force it applies is a 'push' or a 'pull', (and here it is stipulated that all 'force' as applied to any surface, is by the action of 'push', as in to apply pressure which is opposite of vacuum) where the '4π * Thrust/second per 6.241×10^18 plasma particles', can be seen here as a constant spherical 'Thrust Frequency' type force just from the empty space-time, no 'dark matter' or 'dark energy' required. The vacuum would be 'push-in-towards' each space-time plasma particle, while gravity would be 'push-out-from' each space-time plasma particle.  We just need to know the density of plasma particles in free space.

The physical constant ε₀, commonly called the vacuum permittivity, or permittivity of free space, or electric constant, is an ideal, (baseline) physical constant, which is the value of the absolute (not relative) dielectric permittivity of the classical vacuum. Its value is:
ε₀ ≈ 8.854187817620... × 10−12 F·m−1 (or A^2·s^4·kg^−1·m^−3 in SI base units. 
In the vacuum of classical electromagnetism, the polarization is zero, so the relative permittivity ε_r = 1, and ε = ε₀. What we want to work with here, is the SI units: A^2·s^4·kg^−1·m^−3, so we get (( 6.241×10^18 electrons / second)^2 x second^4 )/(mass x volume) = 38.95×10^36 electrons x (second^2/(mass x volume)), not quite as clean as the permeability, but it gives a 'sense' of density.  And we can now also ask, "Is 38.95×10^36 electrons = 1Kg?

The reflected and incident waves propagate in the same medium and make the same angle with the normal to the interface, the amplitude reflection coefficient is related to the reflectance R = |r|². The transmittance T is generally not equal to |t|^2, since the light travels with different direction and speed in the two media. The transmittance is related to t by:

The factor of 'cos θt/cos θi' represents the change in area, resulting in magnification, 'm', of the cross-section of the photon stream, is needed since T, the ratio of powers, is equal to the ratio of (intensity × area).  


In terms of the ratio of refractive indices, and of the magnification 'm' of the incident photon at cross section occurring at surface interface, the Transmittance can be shown to be:

Thereby we can infer as an establishing axiom: The velocity of a photon is controlled by 'n' within a 'single' dielectric medium, and controlled by the ratio of 'n2/n1' within a variable dielectric medium. It is then possible to say, if space itself is discrete, with each point, within any arbitrary volume of space-time, in a state of change, going from a virtual photon, to a virtual electron-positron pair, and possibly other structural functions, that it can also act as a dielectric medium.  

When light makes multiple reflections, instead of refracting, between two or more surfaces, the multiple beams of light generally interfere with one another, resulting in net transmission and reflection amplitudes that depend on the light's wavelength. Typically in plasma conditions, the energy of the individual photons of light may see the way of entropy, and diminish in energy. The interference, however, is seen only when the surfaces are at distances comparable to or smaller than the light's coherence length, which for ordinary white light is few micrometers, using coherent light it can be much larger. An example of interference between reflections is the iridescent colors seen in a soap bubble or in thin oil films on water. A quantitative analysis of these effects is based on the Fresnel equations, but with additional calculations to account for interference.  The scattering of the photons, via passage through plasma clouds, can be readily compared to the iridescent colors seen in a soap bubble as mentioned above and is seen below.

2. Interaction
By observing the current natural conditions of space and matter, and applying the concept of discrete, or 'distinct localities', as defined by real and dynamic physical boundaries, one can expose a hard physical and logical reality underlying all we observe, when considering scattering and it's broader micro and macro implications, via photonic interactions that modulate particle actions, with the reverse being true as well. Photons propagate through dielectric media such as glass or air by interacting with the particles.  As a photon interacts with a particle, the photon can be said to be changing or be in the act of being 'polarized', as the photon's orientation, spin, and direction of travel get modified in the process, think billiards. Then depending on Heisenberg, and the random roll of the dice, the photon may be fully absorbed, be re-emitted at a lower or higher wavelength, bounce off, or just pass through, which will show up as the 'scatter' of each photon.

The standard QED analogy is that of electrons absorbing photons and re-emitting them at 'c' with respect to the electron. In reality, the electron will accept the momentum supplied by the photon, and the velocity of the electron increases. When the electron feels frame resistance from space due to its new increase of velocity, it emits a new photon, that is frequency-dependent upon the spin rate of the electron at emission time.

If the electron is moving near light speed, this means that whatever relative speed an photon arrives at they'll be re-emitted, or scattered, at the new speed-dependent ratio of the different indexes of refraction, as dictated by the pressure differential conditions within the local cloud of plasma. The newly-emitted photons, then will travel at 'c' again when moving through the voids in the plasma. This process changes photon speed and direction, as required, equivalent to photons entering or leaving moving water (n = 1.33) which, as Fizeau first showed, is with respect to the waters relative motion. Even after meeting the fine structure surface boundary of ions, the photon reaches the lens of an instrument, then our eye. As the photon passes through each medium, its speed changes at each interface. As there is more than one interface with a plasma like medium, there is more than one refractive index, thus a system of functional refractive index ratios is needed to better describe plasma interaction within this 3D Field Model of Spacetime.