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Abstract

The Infophysical model of spacetime is used to correspond to the belief that an immaterial

ontology is required to resolve the unexplained “mysteries” of the present understanding of

our physical reality1. The mass-energy relationships of classical and quantum mechanics are

derived without the mass property of physical objects, leaving only the concepts of spatial

density2 and temporal density as a consequence of the Existence/Expression relationship of

Infrarealism. Energy and mass are found to be complementary properties, except for unit

conversion factors, which make them directly proportional to the spatial density

property 𝜎 . Planck’s constant 𝑕 is shown to be not a fundamental constant but a unit

conversion factor. Einstein’s mass-energy relationship 𝐸 = 𝑚𝑐2 is shown to be a

restatement of the temporal density 𝑓 to spatial density 𝜎 of light relationship 𝑓 =

𝜎𝑐 . Finally, there is no need for mass-energy conversion because real objects are waves

only (wavicles), the mass and energy properties of solid objects are our sensory perception

of their spatial density and temporal density properties respectively, leaving only units to

convert. Particles are wavicles travelling at a group velocity lower than the speed of light,

photons are wavicles traveling exactly at the speed of light.

Suggested reading In order to better understand the content of this monograph, the reader should be familiar with the following

reports written by the author:

Reality Unveiled (1). The defining essay of Infrarealism, an immaterial ontology.

Spacetime Unveiled (2). An infophysics monograph proposing the Infophysical Spacetime Model (ISM).

Introduction The purpose of this monograph is to derive the mass-energy relationships of classical and quantum mechanics

without the mass property of physical objects, leaving only the concept of spatial density-energy as a

consequence of the existence and expression relationship of the EEC Universes model of Infrarealism. During

the discourse of this monograph some collateral developments were obtained that provide a new definition of

the quantum mechanical wavefunction as well as the Heisenberg uncertainty principle.

First, let’s clarify some mass-energy concepts under the contexts of Infrarealism and Information Physics

(Infophysics).

1 Note: the philosophical aspects of this monograph can be ignored, if so desired, without any serious damage to its infophysical

spacetime simulation content. 2 I use the term “density” instead of “frequency” because cycles per unit space (spatial frequency) can be thought of as “density of

action” per unit space and cycles per unit time (temporal frequency) can be thought of as “density of action” per unit time.

Classical mechanics, quantum mechanics and infophysics

Modern and past advances in the Physical Sciences have based themselves on ideas that fueled them for more

than five centuries of our history. However, with new discoveries and the rise of different views of our

existence and our surroundings, these ideas no longer have the same footing as they did so many years ago.

The teachings and discoveries of Sir Isaac Newton, who single-handedly founded the base for classical

mechanics and modern calculus-based mathematics, while having helped us for so long in humanity’s

technological advancement, are now ironically holding us back.

Without diminishing the reputation of probably the greatest scientific genius of all time, Sir Isaac Newton, I

must say, that although he probably was the most influential human in the development of modern science,

his materialistic views on inertial force, energy and momentum, are also responsible for the present blockage

to the transcendence of human technology to a new magic. Ironically, without him, we would probably not be

as advanced as we are, yet because of his enormous legacy we are being limited to further advancement.

Classical or Newtonian mechanics explained the behavior of physical processes very well, for quite a while.

Because classical mechanics assumes time and space to be continuous it worked fine until we were able to

examine the lower limits of matter and energy through modern technology. Once we developed the tools to

examine the microscopic world, the discreteness of natural phenomena began to manifest itself. It turns out

that spacetime is not continuous, it just seems that way to us because, without using special equipment, we

smooth out its discrete nature. In other words, the world we live in is not smooth; we smooth it out, it has

dimensions and physical processes that display discrete (whole number) properties only. Consequently, at the

beginning of the 20th century, quantum mechanics was developed in order to take into consideration the

discrete nature of spacetime and its physical processes.

Although we gave up the idea of a continuous world, both classical and quantum mechanics still hang on to

the idea —or assume it— that reality has dimensions and physical processes which possess permanent

presence in time and space. But maintaining the point-of-view that spacetime, matter and energy are

immutable, leads to natural phenomena that cannot be rationally explained, such as particle-wave duality,

mass-energy equivalence, the constancy of the speed of light, quantum entanglement, non-locality or action-

at-a-distance, etc, etc.

By devising Infrarealism and integrating wave mechanics with information theory into what I call Infophysics, I

show in this monograph that most of the above mentioned questions and paradoxes can be explained away.

In this monograph, Infrarealism throws away the concept of permanent dimensions and permanent physical

properties; information theory contributes the Nyquist-Shannon sampling theorem and quantum mechanics

the mathematical tools of wave mechanics. The following table lists the principle assumptions made by each

of the models of physical reality described above:

Table 1. Models of Physical Reality

Physical Theory Reality is discrete Reality is permanent Reality is infinite Classical physics No Yes Yes

Quantum physics Yes Yes/No Yes

Infrarealism Yes No No

Within the context of Infrarealism, reality is discrete, constructed and finite. Except for the basic assumptions

stated above, Infrarealism does not conflict with either quantum or classical physics. This is an example of the

correspondence principle: Any new theory, whatever its character—or details—should reduce to the well-

established theory to which it corresponds when the new theory is applied to the circumstances for which the

less general theory is known to hold3.

Particle-wave duality

Particles and waves are identical —have identical properties— in every way except in the way we express

them. Both waves and particles are spatial waves that exist in Infrareality as spatial density (spatial frequency,

or simply density) frames which are expressed by us as particles or waves depending on their group velocity

property. Photons are spatial waves expressed with group velocity equals to the velocity of light, particles are

spatial standing waves expressed with group velocities ranging from zero (stationary) to velocities less than

the speed of light. In all cases the term expression refers to the process of transforming value-frames from

the infrareal density domain to the real space domain. These transformations can be interpreted as the

construction of real spatial objects from infrareal density frames —discrete signals— by means of their Inverse

Discrete Fourier Transform (IDFT). The DFT and its inverse are conveniently used in this monograph to derive

the density-energy relationships of Infophysics because wave mechanics uses these particular mathematical

tools and so does information theory, but I suspect the same results can be obtained using other canonical

transformation tools.

To remain compatible with present physical science I refer to those spatial waves expressed with lower

velocity than the velocity of light in a vacuum (𝑐), as particle-waves, matter-waves or simply, wavicles4, and

those at exactly 𝑐, as light, electromagnetic waves, or simply photons. Waves with group velocities higher

than 𝑐, have not been observed and may not be allowed in nature if the construction process used in wave

expression is bandlimited in accordance with the Nyquist-Shannon Sampling Theorem, as seems to be the

case.

In conclusion, there are no material particles in nature, only spatial waves. What we observe as particles is

only a smoothing process of discrete spatial waves. Within this monograph I modify the wave mechanics to

remove the concept of mass completely from wave equations, leaving us with spatial density, density-

momentum and density-energy as the basic infophysical properties of real objects, properties which substitute

the classical concepts of mass, mass-momentum and mass-energy respectively. We find that, mass is a

mathematical device that —although useful for understanding what we observe as weight— confuses our

understanding of physical processes and is unnecessary in the realms of Infrarealism or infophysics. In the rest

of this monograph I will continue using the concept of wavicles in the context of matter-waves or particle-

waves as conceived by de Broglie.

Transverse vs. longitudinal waves

If we think of particles as waves, we can describe them as periodic disturbances propagating in a linear fashion

through space. The question is, are they transverse or longitudinal waves?

3 Webster's Online Dictionary.

4 The term wavicle —first proposed by A.S. Eddington, The Nature of the Physical World— refers to an entity that has both, wave

and particle properties. I use the term wavicle in this and other monographs interchangeably with the infophysical entity defined here as an lv-wave or lv-wave packet.

They could be either type, but transverse waves transfer energy in the direction perpendicular to their

direction of propagation. Particles don’t behave that way, which leads me to think they are longitudinal

waves.

Now, electromagnetic waves are transverse waves, which pose another question, what happens when

particles transition to electromagnetic radiation? Maybe we should reconsider their nature.

Let’s not worry, at this time, about these questions and let the discourse of this monograph try to elucidate on

them.

Mass-energy equivalence

Again, mass and energy are identical. Mass and energy are the measure of the spatial density of a wavicle. In

Quantum Mechanics (QM) the wave function,Ψ(𝑥,𝑦, 𝑧, 𝑡), is interpreted as the probability of finding a

quantum-mechanical particle at any given point in space and time and it is from this interpretation that the

concepts of energy and momentum are derived; both concepts are identical byproducts of the relationship

between position and tangential velocity of the wave function of the particle. In conclusion, there is no mass-

energy duality because they are one and the same.

Existence | Expression According to the EEC Universes model of Infrarealism:

Existence is a set of infrareal properties, which when expressed by our consciousness, is

observed as the set of the Real properties of a given object. We will start with Existence and

explain how by processing existence sets, our collective consciousness manifests reality. In

other words, Existence is infrareal data that is processed —transformed— to become an

object. From another point of view, objects become, when we express their infrareal

existence. Once more, Existence is data, Expression is a process and we are the processors.

The verbs, to become, to manifest, to observe, to transform or to express, are

interchangeable terms in the context of Infrarealism.

The transformation process from existence data to expressed objects is postulated to be an inverse Fourier

transform or some other canonical transformation of a similar type. Let’s start by borrowing from wave

mechanics the basic concepts of momentum and position spaces. The single-dimension version of the

stationary Gaussian wave function in momentum space is obtained from the spatial wave function Ψ 𝑥 via its

Fourier transform

1) Φ 𝑝 =1

2𝜋ℏ Ψ(𝑥)𝑒−𝑖𝑝𝑥 /ℏ∞

−∞𝑑𝑥

The wave function in position space is obtained via the inverse Fourier transform of the wave function in

momentum space

2) Ψ(𝑥) =1

2𝜋ℏ Φ(𝑝)𝑒𝑖𝑝𝑥 /ℏ∞

−∞𝑑𝑝

Where, Φ 𝑝 is the amplitude of the momentum space wave, Ψ(𝑥) is the amplitude of the position space

wave, 𝑝 is the angular momentum of the wave and ℏ is the reduced Planck constant 𝑕/2𝜋.

Planck’s constant

Planck’s constant —a quantity called action, with units of 𝐸𝑛𝑒𝑟𝑔𝑦 ∙ 𝑇𝑖𝑚𝑒— is introduced in the above

transform equations, to neutralize the units of 𝑃𝑜𝑠𝑖𝑡𝑖𝑜𝑛 ∙ 𝑀𝑜𝑚𝑒𝑛𝑡𝑢𝑚 —with units of 𝑀𝑎𝑠𝑠 ∙ 𝑉𝑒𝑙𝑜𝑐𝑖𝑡𝑦2 ∙

𝑇𝑖𝑚𝑒—, because by definition, the Fourier transform requires unitless exponents.

Switching to velocity space from momentum-space, I propose

3) Φ 𝑣 =1

2𝜋ℏ𝑣 Ψ(𝑥)𝑒−𝑖𝑣𝑥/ℏ𝑣∞

−∞𝑑𝑥

And by the inverse transform, we get the wave function in position space

4) Ψ(𝑥) =1

2𝜋ℏ𝑣 Φ(𝑣)𝑒𝑖𝑣𝑥/ℏ𝑣∞

−∞𝑑𝑣

Where ℏ𝑣 = 𝑕𝑣/2𝜋 is the unit neutralizing factor for the product 𝑣 ∙ 𝑥, which has units of 𝑉 ∙ 𝐿 = 𝐿2 ∙ 𝑇−1,

the velocity-space equivalent to Planck’s constant, ℏ𝑣 = ℏ/𝑚, where 𝑚 is the mass property. As can be seen

𝑕𝑣 is not a universal constant as Planck’s constant is claimed to be and is more of a parameterized factor for a

given wavicle. In the rest of this monograph I will refer to 𝑕𝑣 as the scale factor and will symbolize it by 𝒮. As

will be shown below, 𝒮 is a variable factor that is dependent on wave-properties only.

Particle wave-properties We have, in the conjugal position/velocity domains, eliminated the concept of mass so that we can proceed to

define all the wave properties of objects in terms of position and velocity only. The point of getting rid of the

concept of mass is not just to simplify the physics but to eliminate the paradoxical statements and confusing

questions that result from our attachment to materialism, besides, an immaterial model, such as Infrarealism,

by definition needs to be massless.

At first the concept of a massless world seems impossible, but we should realize that mass is nothing but the

measurement of an abstract concept devised to satisfy physical equations of motion. The concept of mass is

not intuitive; we confuse mass with weight, the latter being a force and a more natural observable due to

gravitational acceleration. By getting rid of the mass property of objects, we are left with waves only, as we

enter the realm of Infrarealism. We will refer to them as spacetime waves or st-waves because they

propagate trough space.

Before we go on, let’s define the two observable types of objects, particle-waves or wavicles, or simply lv-

waves (lower-velocity waves), and electromagnetic waves, which I will refer to as nv-waves or Nyquist velocity

waves or photons, which are defined as follows:

Lv-waves (wavicles) are spherical standing spatial wave packets with group velocity 𝑣, where 𝑣 has the

range 0 ≤ 𝑣 < 𝑐. Although the standing wave packet has group velocity 𝑣, its phase velocity is the speed

of light 𝑐, so the frequency to wavelength relationship remains as 𝜆𝑓 = 𝑐, where 𝜆 is the wavelength in 𝐿

units and 𝑓 is the temporal frequency in 𝑇𝑖𝑚𝑒−1 units, leaving 𝑐 with units of 𝐿 ∙ 𝑇𝑖𝑚𝑒−1, as expected.

Nv-waves are transverse spatial waves with group and phase velocity equal to 𝑐.

St-waves, spacetime waves or wavicles, are the generic terms when we refer to both.

Lv-wave properties

Let’s look at the de Broglie relativistic wavicle relationship for temporal frequency 𝑓

𝑓 =𝛾𝑚0𝑐

2

𝑕= 𝛾𝑓0 =

𝑓0

1 −𝑣2

𝑐2

This is the relativistic equation for the temporal frequency of a de Broglie wavicle as its group velocity 𝑣

approaches the velocity of light 𝑐 as a function of its stationary temporal frequency 𝑓0. From a rudimentary

examination of the above equation we can elaborate the following statements about a wavicle:

Its frequency approaches infinity as its velocity approaches the speed of light.

Its wavelength contracts with increasing velocity, which can be seen as the direct result of relativistic

length contraction. But how does a vacuum contract? The answer is, it doesn’t. Space does not contract.

Space by itself is not observable, what we observe are its perturbations. What contracts is the wavelength

of the wavicle with increasing velocity. In other words, length contracts, which is observed as a

contraction of spacetime. Let’s say it in a different way; spacetime is manifested as a perturbation of

space because spacetime is the synthesis of space and motion. What we observe is a contracted

manifestation of spacetime, which is equivalent to the contraction of the wavicle’s wavelength as its

velocity is increased. Space plus a wavicle is the manifestation of spacetime.

Here is some indication that wavicles are longitudinal compression waves, very much like sound waves.

Spatial density increases with increasing linear velocity in a relativistic fashion, thus increasing the total

density-energy of the wavicle.

The concept of velocity-inertia is clearly conceived as the resistance manifested by wavicles to accelerate

their oscillatory frequency when their linear (group) velocity is increased. Imagine trying to push a moving

automobile while in first gear, increasing its velocity makes the motor turn faster, resisting the attempt to

increase its velocity.

Density-inertia or inertial reaction density-force is the manifestation of a negative force proportional to an

induced acceleration on a wavicle –𝐹𝑑 =∝ 𝑎, where ∝ is a proportionality factor.

The force required to accelerate a wavicle is 𝐹𝑑 =∝ 𝑎. This is Newton’s second law in infophysical terms.

The wavicle’s density-energy 𝐸𝑑 and its inertial reaction wave-force – 𝐹𝑑 increase according to 𝛾, the

Lorentz factor.

Mass is out of the equation and so is the “universal” constant 𝑕.

The scale factor

Using de Broglie’s and Einstein’s relativistic mass-energy equations, we find that

5) 𝑕𝑓 = 𝑚𝑐2

This equation represents the stationary energy of a wavicle set equal to the kinetic energy of an equivalent

photon. The equation holds true because the photon’s kinetic energy is equal to its total energy

Dividing by 𝑚

6) 𝒮 =𝑕

𝑚=

𝑐2

𝑓= 𝜆2𝑓 = 𝜆𝑐 =

𝑐

𝜎, is the scale factor.

7) 𝑕 =𝑚𝑐2

𝑓=

𝐸

𝑓, is the conversion factor relating energy and temporal frequency.

8) 𝜎 = 𝑐

𝑕 𝑚, is the conversion factor relating mass and spatial density.

9) 𝑚 = 𝜎 𝑕

𝑐 , is the conversion factor relating spatial density and mass.

10) 𝑝 = 𝑚𝑣 = 𝑕

𝑐 𝜎𝑣 =

𝑕

𝑐2 𝑓𝑣, is momentum in terms of spatial density and group velocity.

11) 𝐸 = 𝑕

𝑐 𝜎𝑐2 = 𝑕𝑐𝜎, 𝑕𝑐 is the conversion factor relating energy and spatial density.

From the above equations we can summarize that:

From Eq. (6), the scale factor takes the value𝑠 𝒮 =𝑐2

𝑓= 𝜆2𝑓 = 𝜆𝑐 =

𝑕

𝑚=

𝑐

𝜎 for all wavicles, including

photons. Notice that the scale factor has a constant value for wavicles and that it is dependent only on

frequency and the speed of light 𝑐, the Nyquist5 velocity of our reality. For a wavicle the scale factor

𝒮 =𝑕

𝑚= 𝜆𝐶𝑐 =

𝑐

𝜎𝐶 is the Nyquist velocity 𝑐 divided by its Compton spatial frequency. As expected, the

scale factor has units of 𝑉2 ∙ 𝑇 = 𝐿 ∙ 𝑉 = 𝐿2 ∙ 𝑇−1, spatial area per unit time, spacetime units only. I show

in Spacetime Unveiled (2) how the scale factor 𝓢 =𝒉

𝒎=

𝒄

𝝈𝑪= 𝝀𝑪𝒄 establishes the relationship between

space and time for a given dimensional scope. The ratio 𝑕

𝑚 is equivalent to the ratio

𝑐

𝜎𝐶 showing that 𝑕 is to

𝑐 as 𝑚 is to 𝜎𝐶 , which in turn shows that 𝒉 is not a fundamental constant, but just a different way of

referring to the speed of light 𝒄, depending on the units chosen for 𝒎. The units of 𝑕

𝑚 are

𝐸∙𝑇

𝑀=

𝑀∙𝐿2 ∙𝑇

𝑇2 ∙𝑀=

𝐿2 ∙ 𝑇−1, as expected, spatial area per unit time.

The scale factor, the quantity 𝑕/𝑚 , is the layer we need to peel away to reveal the immaterial reality we

live in, leaving as frequency and velocity the only action properties of our observables, both of which are

functions only of time and space, respectively, leaving spacetime as our only reality, nothing else.

From Eq. (7), Planck’s constant reveals itself as a unit conversion factor from temporal frequency to energy

for all wavicles.

Eq. (8) and Eq. (9), reveal the relation between, what we observe as the mass property of wavicles, to their

density property, to be the constant 𝑕

𝑐= 1.9864456833x10−25 𝐽𝑜𝑢𝑙𝑒𝑠 − 𝑠𝑒𝑐𝑜𝑛𝑑2/𝑚𝑒𝑡𝑒𝑟, this means

that the mass property is directly proportional to the spatial density property and that we can use 𝜎 in all

physical relationships to replace 𝑚 and still retain their usefulness.

Eq. (10) shows the relationships between momentum to spatial density and to temporal frequency.

Eq. (11) reveals the relation between, what we observe as the energy of wavicles, to their spatial density

property, to be the constant 𝑕𝑐 = 1.9864456833𝑥10−25𝐽𝑜𝑢𝑙𝑒𝑠 −𝑚𝑒𝑡𝑒𝑟𝑠, this means that the energy

property is also directly proportional to the spatial density property.

Planck’s constant 𝑕 = 𝑚0𝑐2/𝑓0 remains as a conversion factor back to SI units.

A simpler conversion factor back to SI units is the stationary mass to stationary frequency ratio 𝑚0/𝑓0 =

𝑕/𝑐2, that is, which has the value 7.372497 ∙ 10−51 𝐾𝑔 ∙ 𝑠. As you can see it is a very small number

relating our observables mass and time.

The relationship between wavelength and temporal frequency for photons and wavicles is 𝑐 = 𝜆𝑓.

5 Because of our bandlimited Reality, the Nyquist construction velocity 𝑐 has been shown by the author, in his essay on Infrarealism,

to be the maximum expressible velocity of a Real object.

The relationship between wavelength and density 𝜎 (Greek letter Sigma) for photons and wavicles

is 𝜎 = 1 𝜆 .

The density and position domain conjugate pair

Reintroducing the scale factor 𝜆𝑐 back into Eq. (3), we get the wave function in the density domain

12) 𝛷 𝜎 = 𝜎𝑐 𝑐 𝛹(𝑥)𝑒−𝑖2𝜋 𝑣

𝑐 𝜎𝑐𝑥∞

−∞𝑑𝑥

, where, the relativistic ratio 𝛽 = 𝑣

𝑐 shows up as a result of the circular and orthogonal nature of the Fourier

transform. It must be pointed out, that the property 𝑣 shown above is the tangential velocity of the wavicle.

By the inverse transform, we get the wave function in the position domain

13) Ψ(𝑥) = 𝜎𝑐 𝑐 Φ(𝜎)𝑒𝑖2𝜋

𝑣

𝑐 𝜎𝑐𝑥∞

−∞𝑑𝜎

Eq. (12) and (13) are the exact equivalents of Eq. (1) and (2), where the momentum property is replaced by the

density property of the wavicle, leaving only spacetime wavefunctions.

Density-momentum

The definition of relativistic density-momentum for wavicles becomes

14) 𝑝𝑑 = 𝜎𝑣 = 𝑣

𝜆= 𝑓

𝑣

𝑐

15) lim𝑣→𝑐 𝑝𝑑 = 𝑓

The discontinuity as 𝑣 approaches 𝑐 is avoided in Eq. (15) because 𝑣 has integer values (2) and the kernel of

the inverse Fourier transform in Eq. (13) is defined for 𝑣/𝑐. This can be interpreted as a wavicle transitioning

into a photon by gaining a finite energy.

For photons

16) 𝑝𝑑 = 𝜎𝑐 =𝑐

𝜆= 𝑓

Notice that:

𝜎, the density, replaces the mass property. The behavior of 𝑝𝑑 as a function of 𝑣 is identical to 𝑝 the mass-

momentum, which amounts to substituting the value of the mass property with the density property. The

density-momentum 𝑝𝑑 = 𝜎𝑣, has units of 𝑇−1, the units of temporal frequency.

For wavicles the density-momentum is 𝑝𝑑 = 𝑓 𝑣 𝑐 = 𝛽𝑓. The wavicle’s inward motion reduced by 𝛽 the

ratio of its displacement velocity to the speed of light.

For photons the density-momentum is 𝑝𝑑 = 𝑓. This means the density-momentum of a photon reduces

to the same property as its temporal frequency.

The density-momentum 𝑝𝑑 can be thought of as inward motion.

The concept of density-momentum applies naturally to photons, in contrast to the concept of mass-

momentum for light, because there is no conflict with the concept of massless photons.

By removing the mass property, the relativistic wave-momentum equation for wavicles, Eq. (14), reveals a new

term 𝛽 in the numerator showing the convergence 𝑝𝑑 = 𝑓 as 𝑣 → 𝑐, losing the discontinuity as 𝑣 approaches

the speed of light for the reason state above.

Inertial reaction force

As presented above, inertial reaction force is the manifestation of a negative force proportional to an induced

acceleration on a wavicle – 𝐹𝑑 =∝ 𝑎. The force required to accelerate a wavicle is 𝐹𝑑 =∝ 𝑎. This is Newton’s

second law in infophysical terms. If follows, that if an inertial force is caused by the increase in density of a

wavicle, then that force must be dependent only on the increase of its density and its velocity, which means

that the force

𝐹𝑑 =𝑑(𝜎𝑣)

𝑑𝑡=

𝑑

𝑑𝑡 𝜎0𝛾𝑣 = 𝜎0

𝑑

𝑑𝑡 𝛾𝑣 = 𝛾3𝜎0𝑎 = 𝛾2𝜎𝑎

, where 𝜎 ∙ 𝑣 is the density-momentum 𝑝𝑑 . The proportionality factor for inertial force turns out to be 𝜎𝛾2,

which has units of 𝐿−1. The density-force 𝐹𝑑 has units of 𝑇−2, (in SI units, cycles/sec2) the units of temporal

frequency acceleration.

For non-relativistic objects, the density-force 𝐹𝑑 reduces to 𝐹𝑑 = 𝜎0𝑎, where 𝜎0 is the stationary density of a

wavicle and 𝑎 is its acceleration, which is Newton’s second law of motion in infophysical terms.

Density-energy

Paralleling Einstein’s mass-energy to momentum relationship for photons 𝐸 = 𝑐𝑝

17) 𝐸𝑑 = 𝑐𝑝𝑑 = 𝑐𝑓 =𝑐𝑐

𝜆= 𝜎𝑐2

, where 𝐸𝑑 is taken as the total density-energy because photons do not have stationary energy. This result

shows very clearly how superfluous the concept of mass can be. Density-energy 𝐸𝑑 , has units of 𝑉2 ∙ 𝐿−1 = 𝐿 ∙

𝑇−2, meter-cycles/sec2 the units of linear velocity times temporal frequency, also the units of acceleration.

In other words, density-energy is the measure of density-momentum multiplied by linear velocity for all

wavicles, including photons.

Continuing with energy, using Einstein’s equation for energy 𝐸 = 𝑚𝑐2and substituting mass by density

𝐸𝑑 = 𝜎𝑐2 = 𝛾𝜎0𝑐2

For wavicles, 𝐸𝑑 = 𝜎𝑐2 is the total density-energy relationship, including 𝐸𝑑 = 𝑐𝑝𝑑 = 𝜎𝑐2, for photons. The

concept of density-energy is the same for all wavicles. By substituting the spatial density property for the

mass property in the energy equations, the concept of density-energy becomes identical for wavicles and

photons.

Continuing with energy, kinetic mass-energy 𝐸𝑘 = 𝐸𝑡 − 𝐸0 where 𝐸𝑘 is the kinetic energy and 𝐸0 is the

stationary energy, for wavicles

𝐸𝑑𝑘 = 𝐸𝑑𝑡 − 𝐸𝑑𝑜

𝐸𝑑𝑘 = 𝛾𝜎0𝑐2 − 𝜎0𝑐

2

𝐸𝑑𝑘 = 𝜎0𝑐2(𝛾 − 1)

From the above density-energy equations we can conclude that:

The concept of kinetic density-energy 𝐸𝑑𝑘 remains of some practical use for wavicles in general since the

total density-energy for photons is kinetic. Photons are never stationary

For classical wavicles 𝐸𝑑 ≅1

2𝜎0𝑣

2.

The density-energy equations for wavicles are identical to the mass-energy equations, except for

the substitution of mass by density.

𝛾 Acts as a “compression” resistance factor. The resistance to linear acceleration and the

accumulation of density-energy are a direct result of its non-linear behavior 1/ (1 − 𝑣2/𝑐2 ) as a

function of velocity.

Density-energy has units of 𝐿 ∙ 𝑇−2, the units of linear acceleration.

Under the veil

From Einstein’s relationship between mass-energy and photon-energy 𝑚𝑐2 = 𝑕𝑓, dividing both sides by 𝑚 we

get:

𝑐2 =𝑕𝑓

𝑚= 𝒮𝑓 = 𝜆𝑐𝑓

𝑐 = 𝜆𝑓

, which is the frequency-wavelength to phase velocity relationship for photons. This may come as a surprise,

once the materialistic veil (𝑕/𝑚) has been removed, but it makes sense because infophysical energy 𝐸𝑑 = 𝜎𝑐2

for all wavicles is only a function of density and the velocity of light, making Einstein’s mass-energy

relationship a restatement of the frequency-wavelength relationship.

Let’s probe a little more below the veil. Setting infophysical mass equivalent to spatial density,

18) 𝑚 → 𝜎, amounts to setting,

19) 𝑕

𝑐→ 1 in Eq. (9), leading to,

20) 𝑕 = 𝑐, in infophysical spacetime only units.

Going back to Equations (9) trough (11) and substituting for 𝑕, we get,

21) 𝑚 = 𝜎, mass in units of spatial density,

22) 𝑝 = 𝜎𝑣, momentum in units of spatial density.

23) 𝐸 = 𝜎𝑐2 = 𝑐𝑓, energy in units of temporal density and

As you can see, these are identical to the infophysical definitions of mass, momentum and energy respectively.

Again, going full circle,

24) 𝐸 = 𝑚𝑐2 = 𝜎𝑐2 = 𝑐𝑓, results in

25) 𝑓

𝜎= 𝑐, or 𝑐 = 𝜆𝑓, the spatial density to temporal density relationship of light, which is what we expected,

because:

Energy is our sensory perception of the temporal density of wavicles.

Mass is our sensory perception of the spatial density of wavicles.

Einstein’s 𝑬 = 𝒎𝒄𝟐 relationship is a re-statement of the spatial density to temporal density relationship

in SI units.

Planck’s constant 𝒉 is the unit conversion factor from infophysical spacetime units back to SI units, and

is equivalent to the speed of light in a vacuum 𝒄 .

Planck’s constant 𝒉 is not a fundamental constant.

Energy-momentum relationship

From the relation between mass-energy and mass-momentum 𝐸2 = (𝑝𝑐)2 + (𝑚0𝑐2)2, we get

𝐸𝑑2 = (𝑝𝑑𝑐)2 + (𝜎0𝑐

2)2

Substituting for 𝐸𝑑 and 𝑝𝑑

(𝜎𝑐2)2 = (𝜎𝑣𝑐)2 + (𝜎0𝑐2)2

Using the relativistic form for density

(𝛾𝜎0𝑐2)2 = (𝛾𝜎0𝑣𝑐)2 + (𝜎0𝑐

2)2

, and dividing by 𝜎02 and by 𝑐4

𝛾2 =𝛾2𝑣2

𝑐2+ 1

Substituting for 𝛽2 = 𝑣2/𝑐2

𝛾2 = 𝛾2 𝛽2 + 1

Factoring out 𝛾, taking the square root of both sides and solving for 𝛾

𝛾 =1

1 − 𝛽2

, which is the Lorentz factor.

But we already knew this; we can clearly see from the above that the Pythagorean relationship between

energy and momentum is a direct result of Lorentz time dilation, which in turn, as shown by infophysics, is a

direct result of the expression process and its Nyquist velocity 𝑐, which is the limiting velocity of our

bandlimited reality, the speed of light. The energy to momentum relationship is equivalent, as derived in

Infrarealism (1) by the Pythagorean Theorem, to

𝑡′2 = (𝛽𝑡′)2 + 𝑡2

Where 𝑡’ is time in the moving frame, 𝑡 is time in the stationary frame and 𝛽 = 𝑣/𝑐.

In summary we conclude that for wavicles:

The square of the total density-energy 𝐸𝑑2 is related to the square of time 𝑡′2 in the moving frame.

The wave-momentum times the velocity of light squared (𝑝𝑑𝑐)2is related to (𝛽𝑡′)2 in the moving frame.

The square of the stationary density-energy 𝐸𝑑02 is related to the square of time 𝑡2 in the stationary

frame.

All of course, a direct result of Lorentz time dilation. There is more time for cycles, so the wave frequency

increases and so does the density-energy.

Figures 1 and 2 clearly show how the Pythagorean density-energy/momentum relation

𝐸𝑑2 = (𝑝𝑑𝑐)2 + (𝜎0𝑐

2)2, is related to the velocity of a moving wavicles as follows:

Kinetic density-energy 𝐸𝑑𝑘 is proportional to [due to] the increase in time Δ𝑡 of the moving frame

Density-momentum is proportional to 𝛽𝑡′, where 𝛽 = 𝑣/𝑐

Stationary density-energy is proportional to time in the stationary frame.

Conclusions In order to conform to Infrarealism, which is a form of immaterialism, the need to eliminate the concept of

mass, becomes a logical consequence.

Figure 1. Energy Momentum Relationship

Figure 2. Time Dilation/Velocity Relation

The massless Gaussian wavefunction

This monograph removes the mass property from the energy equations thus removing the concept of mass-

energy, substituting it with the density property of objects and thus defining the concept of density-energy 𝐸𝑑 .

In the process of removing the mass property, we needed to remove the concept of solid matter and of

particle objects. We have also removed Planck’s constant, which turns out not to be a universal constant and

replaced it with the scale factor 𝒮, which relates only two infophysical wave properties of objects, their

Compton wavelength property 𝜆𝐶 , and the Nyquist velocity 𝑐. The one-dimensional quantum mechanical

wavefunction in velocity space at particular time point 𝑡 reduces to:

Φ 𝑣 =1

𝜆𝑐 Ψ(𝑥)𝑒−𝑖2𝜋𝑣𝑥/𝜆𝑐

∞

−∞

𝑑𝑥

And by the inverse transform, we get the wave function in position space

Ψ(𝑥) =1

𝜆𝑐 Φ(𝑣)𝑒𝑖2𝜋𝑣𝑥 /𝜆𝑐

∞

−∞

𝑑𝑣

Where, 𝜆𝑐 = 𝒮 is the scale factor for a wavicle, 𝜆 is its wavelength and 𝑐 is the Nyquist velocity of reality. As

you can see the kernel of the Fourier transform has units of spatial density and length which means that the

transforms are conjugates in the density/spatial (𝜎 − 𝑥) domains and can be expressed as follow:

The wave function in density space

Φ 𝜎 = 𝜎𝑐 𝑐 Ψ(𝑥)𝑒−𝑖2𝜋(𝑣𝑐

)𝜎𝑐𝑥

∞

−∞

𝑑𝑥

And by the inverse transform, we get the wave function in position space

Ψ(𝑥) = 𝜎𝑐 𝑐 Φ(𝜎)𝑒𝑖2𝜋(𝑣𝑐

)𝜎𝑥

∞

−∞

𝑑𝜎

There is no change to the wave function equations except for the introduction of the density variable 𝜎 = 1 𝜆

and the rearrangement of the constants leaving the conjugate transforms in the spatial-density/position

domains.

Additional conclusions

Removing the mass property also facilitates the concept of velocity-inertia, which is conceived as the

resistance manifested by wavicles to accelerate their oscillatory frequency when their linear velocity is

increased, due to relativistic effects. This concept alone is a very large step towards the unification of Special

Relativity and Quantum Theory.

The discourse of this monograph leads to the following additional conclusions:

There is no such thing as mass-energy conversion because Real objects are waves only, matter does not

exist, there is no mass-energy and we are left with density-energy only. In other words, everything is

density-energy leaving nothing to convert to.

All real objects are expressed by us as spacetime wave packets (st-waves) which in turn are observed as

what we call matter or particles. Under this model, an electron, for example, has no mass and it’s just a

spacetime wave structure observed by us as a particle.

Neither the expressed spacetime wave packets nor what we call matter have permanent existence.

Physics becomes Infophysics. All concepts of the physical sciences need to be revised in terms of wave

mechanics and information theory.

Nuclear reactions are wavicle transitions.

The Standard Model of particles also needs to be revised to exclude particles and to consider only

Infophysical wave structures.

Photons are massless but so are all the objects we call particles, such as electrons, protons, neutrons etc.

For example, there is no fundamental difference between an electron and a photon, except for their

density-energy and their group velocity. An electron’s group velocity is always less than the Nyquist

velocity 𝑐, a photon’s group velocity is always equal to 𝑐. Photons are Nyquist velocity wavicles, electrons

are wavicles. Yet they are both spacetime wave packets that differ only in their density-energy and group

velocity properties.

The density-momentum 𝑝𝑑 for a wavicle at rest can be interpreted as the combination of two types of

motion —as described by Bohm6—, inward and outward motion, which relate to inward (stationary)

energy and displacement (velocity) energy, respectively. But in infophysical terms we look at the two

types of motion as oscillatory motion and linear motion, which correspond to inward and outward motion

respectively, independently of whether the wavicle is at rest or not. The two types of motion are

considered but there is only one type of energy, density-energy, which is attributed to the oscillatory

motion. The displacement motion contributes to the density-energy because of its relativistic effects on

the oscillatory motion. Displacement energy is done away with.

Based on the previous reasoning, mass-energy conversion —for the rest mass, as devised by Einstein—

amounts to the transition from wavicle to photon. Notice that there is no energy conversion because

there is only one type of energy.

Einstein’s mass-energy relationship 𝐸 = 𝑚𝑐2 = 𝑕𝑓 is shown to be a restatement of the wavelength-

frequency relationship 𝑐 = 𝜆𝑓 for wavicles.

Planck’s constant 𝑕 is a unit conversion factor, not a fundamental constant.

The concept of kinetic density-energy moves awkwardly into infophysics because of its material classical

origin. I suspect we will eventually abandon the concept of kinetic energy completely so as to avoid

confusion.

One way wavicles can transform into photons is by what we presently call matter/anti-matter annihilation,

which implies the concept of anti-wavicles. But anti-wavicles are probably the same as wavicles with some

differentiating fundamental property —which could be some form of oscillatory property— that must be

responsible for what we observe as charge.

One important conclusion that follows from the previous reasoning is that, if wavicles and anti-wavicles

are fundamentally the same, then so is the case for matter and anti-matter, which in turn implies that

6 Pgs 142-143 in Bohm, Special Relativity

matter anti-matter annihilation is just a fundamental property interaction leading to a wavicle

transformation. Answering the question of how a wavicle transforms into a photon is beyond the scope of

this monograph.

I believe the most important result of this monograph is that due to the immaterial nature of our reality

there is only one type of energy thus making mass-to-energy conversion unnecessary. Energy generation

then becomes a process of transformation from:

1. Wavicles into photons for large amounts of heat energy generation. High energy nuclear reaction

(HENR).

2. Large composite wavicles into smaller wavicles. Low energy nuclear reaction (LENR).

3. Larger composite wavicles into smaller wavicles and free electron wavicles. Electron emission (EE).

The following table summarizes the infophysical relations derived in this monograph:

Particles lv-waves (wavicles) nv-waves (photons) Wave relationships:

𝑣 = 𝑣𝑐𝑙𝑎𝑠𝑠𝑖𝑐𝑎𝑙 . Their group velocity is equal to their classical velocity

𝑣 = 𝑣𝑐𝑙𝑎𝑠𝑠𝑖𝑐𝑎𝑙 . Their group velocity is equal to their classical velocity

Velocity of light is the constant 𝑐

𝑣 ≠ 𝑣𝑝𝑕𝑎𝑠𝑒 . Their group velocity is not

equal to their phase velocity

𝑣 ≠ 𝑣𝑝𝑕𝑎𝑠𝑒 . Their group velocity is not

equal to their phase velocity.

𝑣 = 𝑣𝑝𝑕𝑎𝑠𝑒 = 𝑐. Their group velocity is

equal to their phase velocity and equal to the velocity of light

𝑓 = 𝛾𝑓0. Wave temporal frequency 𝑓. Wave temporal frequency

𝜆𝑓 = 𝑐. Frequency to wavelength relation

𝜆𝑓 = 𝑐. Frequency to wavelength relation

Mass:

Mass property 𝑚 𝑚 ↔ 𝜎(𝑕 𝑐) = 𝜎𝑕𝑐 . Mass ↔ to density, where 𝑕𝑐 = 𝑕 𝑐 = 2.210218901x10

-35 ergs/meter, the

equivalence factor

𝑚 ↔ 𝜎𝑕𝑐 . Mass ↔ to density 𝑕𝑐 = 𝑕 𝑐 = 2.210218901x10

-35 ergs/m

Units are 𝑀, in Kg 𝐸 ∙ 𝐿−2, ergs/m2, energy density 𝐸 ∙ 𝐿−2, ergs/m

2, energy density

Momentum:

Planck’s constant 𝑕. Scale factor 𝒮 = 𝑐2/𝑓0 = 𝜆0𝑐 Scale factor 𝒮 =𝑐2

𝑓= 𝜆𝑐.

Mass-momentum 𝑝 = 𝑚𝑣 = 𝛾𝑚0𝑣. Density-momentum 𝑝𝑑 = 𝜎𝑣 = 𝛾𝜎0𝑣. Density-momentum 𝑝𝑑 = 𝑓.

De Broglie’s mass-momentum 𝑝 = 𝑕/𝜆.

Density-momentum 𝑝𝑑 = 𝛾𝑣/𝜆0

Density-momentum 𝑝𝑑 = 𝜎𝑐 =𝑐

𝜆= 𝑓

Units are 𝑀 ∙ 𝐿 ∙ 𝑇−1, Kg-m/sec 𝑇−1, temporal frequency 𝑇−1, temporal frequency

Mass-momentum ↔ density-momentum

𝑝 ↔ 𝑕𝑐𝑝𝑑 = 𝑕𝑐𝜎𝑣, in terms of density 𝑝 ↔ 𝑕𝑐𝑝𝑑 = 𝑕𝑐𝑓, in terms of temporal frequency

Force:

Newton’s second law 𝐹 = 𝛾2𝑚𝑎. Density-force 𝐹𝑑 = 𝛾2𝜎𝑎. Photons cannot be accelerated.

Units are 𝑀 ∙ 𝐿 ∙ 𝑇−2, Kg-m/sec2

𝑇−2, cycles/sec2, temporal frequency

acceleration

Energy:

Total mass-energy 𝐸 = 𝑚𝑐2. Total density-energy:

𝐸𝑑 = 𝜎𝑐2 = 𝛾𝜎0𝑐2 = 𝛾𝑐2/𝜆0

Total density-energy:

𝐸𝑑 = 𝜎𝑐2 = 𝑐2 𝜆 = 𝑓𝑐 Units are 𝑀 ∙ 𝐿2 ∙ 𝑇−2, joules 𝐿 ∙ 𝑇−2, meter-cycles/sec

2,

displacement times frequency acceleration

𝐿 ∙ 𝑇−2, meter-cycles/sec2,

displacement times frequency acceleration

𝐸0 = 𝑚0𝑐2. Stationary mass-energy 𝐸𝑑0 = 𝜎0𝑐

2. Stationary density-energy photons do not possess stationary energy

Kinetic mass-energy 𝐸 = 𝑚0𝑐

2(𝛾 − 1). Kinetic density-energy

𝐸𝑑 = 𝜎0𝑐2(𝛾 − 1).

Kinetic density-energy is equal to total

energy 𝐸𝑑𝑘 = 𝐸𝑑 .

Relation of mass-energy to momentum 𝐸2 = (𝑝𝑐)2 + (𝑚0𝑐

2)2 Relation of density-energy to

momentum 𝐸𝑑2

= (𝑝𝑑𝑐)2 + (𝜎0𝑐2)2.

Einstein’s energy to momentum relation 𝐸𝑑𝑘 = 𝑝𝑑𝑘 𝑐.

Relation of wavelength to mass-momentum 𝑝 = 𝑕/𝜆.

Relation of wavelength to density wave-momentum 𝑝𝑑 = 𝑐/𝜆

Relation of wavelength to wave-momentum 𝑝 = 𝑐/𝜆

References:

1. Sotomayor V., Bernardo. Reality Unveiled - A Collection of Monographs on the Infrastructure of

Reality. The Consensual Beliefs Foundation. Jinotepe : s.n., 2011. Essay. RPI No:000244.

2. —. Spacetime Unveiled. The Consensual Beliefs Foundation. Jinotepe : s.n., 2014. Infophysics

Monograph. RPI No:000749.

Other monographs written by the author

If you enjoyed this monograph, you are welcome to download the following monographs belonging to

the Reality Unveiled Collection:

Reality Unveiled. The defining monograph for the Reality Unveiled Collection.

Spacetime Unveiled. The defining monograph for the Infophysical Spacetime Model (ISM).

Special Relativity Unveiled. A new derivation of Einstein’s Special Relativity theory using the ISM.

About the Author

Bernardo Sotomayor Valdivia is an independent scientific researcher born in León, Nicaragua. He has

degrees in Physics and Systems Engineering, as well as advanced studies in Information Systems. He

participated in the US space program, including the Viking program at Jet Propulsion Laboratory, NASA,

in Pasadena CA and was for many years Chief Technology Officer for various start-ups in e-commerce

within the US. He now writes on Infrarealism and Infophysics.

Feedback

The author is always grateful for constructive feedback through the network where you obtained this

document. If you detected any errors during your reading, before you send your feedback, please

make sure you have downloaded the latest version to make sure they have not been corrected already.

Thanks for your support,

Bernardo Sotomayor Valdivia.

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