*Is it possible that “All Particles Naturally Oscillate”, and that their “Frequency of Oscillation” determines how much mass they have and how fast they can move?…*

The central problem of modern physics is that while its two great pillars General Relativity (a theory of the very fast and the very large) and Quantum Mechanics (a theory of the very small) are independently incredibly successful, they are however, at their core, fundamentally incompatible. One theory says that empty space should be empty, the other says that at the microscopic level empty space is a cauldron of *“quantum fluctuations”*.

Here we will consider the possibility that, while the linear mathematics of quantum mechanics produces spectacularly accurate predictions it might not be because of quantum fluctuations. In this paper we will consider the possibility that maybe we live in a *“discrete space-time”* universe and that

*Quantum Mechanics and General Relativity are geometric manifestations of the mathematical beauty found within Euler’s Formula.**..*

**Part 1 – Discretization of Euler’s Formula**

__Euler’s Formula__

Euler’s Formula ( *e ^{ix} = cos(x) + i sin(x) *

_{where x is an angle measured in radians}) is a mathematical tool used to describe how a

*“vector rotates”*through an angle in the complex plane, but it can also be used to describe a

*“periodic oscillation”.*

The equation for an oscillation can be written as *Ae ^{(i}*

^{ω}*where*

^{t)}*(A)*is the amplitude of oscillation, and

*(e*

^{(i}

^{ω}

^{t)}*)*is the

*changing phase of oscillation over time*. In this phase term, the angle

*(x)*in Euler’s Formula has been replaced with

*angular-frequency (*

*ω*

*)*multiplied by

*time*

*(*

*t*

*)*in order to allow for the position of the oscillating body to change over time.

Angular frequency is usually expressed in radians per second and is normally assumed to be *“continuous change over continuous time”.* But what if, in reality, *“Time”* is NOT *“continuous”*, but actually occurs in tiny *“discrete incremental steps”*; what if there is actually some *“smallest fundamental unit of time”?*

———-

__Fundamental Units__

Let’s take a guess that the fundamental unit of time might have the same magnitude as the value of Planck’s Constant.

(1 h-second = 6.626 × 10^{-34} seconds)

Furthermore let’s define a *“unit of angular displacement”* using Euler’s Formula. Let’s define this unit to have the same magnitude as the value of Planck’s Reduced Constant.

(1 h-cycle = 6.626 ^{x} 10^{-34} cycles)

(1 (^{h}/_{2}_{π})-cycle = 1 ħ-cycle = 1.054571726 ^{x} 10^{-34} cycles)

(1 ħ-radian = 1.054571726 ^{x} 10^{-34} radians)

e^{i}^{x} = cos(x) + i sin(x)

e^{i} = cos(1) + i sin(1)

e^{i}^{ħ} = cos(ħ) + i sin(ħ)

e^{i}^{ħ} = (0.999999 + iħ)

Given this *“defined”* *“**ħ**-unit of angular displacement”, *we can now think of every other size of angular displacement as *an integer number of units of (1 +i**ħ**);*

** **(e^{i}^{ħ})^{n} = (e^{i})^{ħ}^{n} = e^{in}^{ħ} = (1 + iħ)^{n} = n ħ-radians _{(where n = 1,2,3….)}

And when n = Ħ = 9.48252238653324 ^{x} 10^{+33} (i.e. capital-h-bar : the inverse of ħ) we get an angle of

^{ } (e^{i}^{ħ})^{Ħ} = (e^{i})^{ħ}^{Ħ} = e^{i} = (1 + iħ)^{Ħ} = Ħ ħ-radians = 1 radian

———-

__Particle-Oscillation__

Let us now look at a possible physical realization of this discrete mathematics. Let’s look at an up and down oscillation (on the y axis) that occurs in discrete incremental steps of size (n ħ-radians).

Initial Displacement: y = A sin(nħ) (equation 1)

In order to evolve this initial displacement over *incremental time*, we will replace (nħ) with (ωt) — where (ω = ^{n}^{ħ}/_{t} ) and where time (t) is measured in h-seconds, and can only have integer values (i.e. t = 1,2,3… h-seconds.)

Displacement over Time: y = A sin(ωt) (equation 2)

Associated with this oscillation is an *incremental velocity*

v = Aω cos(ωt) (equation 3)

The momentum associated with this velocity is given by

p = mAω cos(ωt) (equation 4)

If we multiply this momentum by velocity we get

pv = mA^{2}ω^{2} cos^{2}(ωt) (equation 5)

We can rewrite equation (5) in angular form replacing (mA^{2}) with the moment of inertia (I)

pv = Iω^{2 }cos^{2}(ωt) (equation 6)

And since (Iω) represents angular momentum, we could guess that maybe it actually represents a *single discrete ħ-unit of angular momentum* which is represented by (ħ)*.*

pv = ħω cos^{2}(ωt) (equation 7)

Equation (7) is effectively an equation for the kinetic energy of an oscillating particle, which means there must be a potential energy associated with it, and consequently the total energy of the oscillation (at any given moment in time) is distributed between kinetic and potential energy and expressed as a function of time

E(t) = ħω(cos^{2}(ωt) + sin^{2}(ωt)) (equation 8)

Lastly to obtain a travelling, or moving, version of this oscillation equation we use the relationship between angular frequency and the linear velocity of the particle (i.e. ω=kv _{w}_{here k is the so-called wave number}_{ and v is the linear velocity of the particle}) and so equation (8) can now be expressed as a function of both time and distance

**E(x,t) = ****ħ****ω(****cos ^{2}(kv**

**t**

**) + sin**

^{2}(kv**t**

**))**(equation 9)

__Planck-Einstein Relation__

The Planck-Einstein Relation (E=hf) tells us that energy is proportional to frequency. The discretization of Euler’s Formula suggest that different frequencies represent, different amounts of *“oscillation units”* in a particle. It suggests that (h) is a unit of angular momentum, that (f) is the number of these units in a single h-second, which makes (E) equal to an amount of angular momentum per unit time.

Equation (9) is an extended version of the Planck-Einstein Relation that incorporates *an oscillation *within a moving particle. This equation bears some resemblance to the Schrodinger Wave Equation which has a *wave function* of the form Ae^{i(kx+}^{ω}^{t)} incorporated into a conservation of energy equation. Equation (9) however is both an *oscillation function* and a conservation of energy equation, where phase-over-distance and phase-over-time are * combined* in a single term Ae

^{i(kvt)}. This is an equation for a travelling

*“oscillating particle”*

__not__a*“probability wave”…*

**Part 2 – Oscillating Particle**

__No Medium Required__

In the classical world *“waves”* travel through a medium, because classical waves involve the transport of energy without the transport of matter. In a water wave for instance, particles oscillate in situ while the wave moves from one particle to the next. In the quantum world however maybe particles oscillate as they move –- if so, no medium is required.

__Phase ____Information__

An oscillating particle would carry with it its own “*phase-information”. *This means that a particle always knows where it is in its own cycle, and therefore how much of its total energy is being expressed as kinetic energy at any moment in time. Interaction with multiple particles at different points in their individual cycles would consequently yield wave-interference-like results because the particles would, on interaction, either be absorbed or not absorbed depending on where they are in their own individual cycle.

__Phase-Size + Number-of-Phases__

In a discrete oscillation angular frequency effectively describes the size of a *“discrete phase-step”* over time. We know that the size of the phase-step (nħ) is effectively the step-size of oscillation. We also know that the step-size times the number-of-steps in a cycle is always a constant 2π radians. This means that, in a discrete oscillation, as the amount of change that occurs in one unit of time goes up, the number-of-steps/number-of-phases must comes down.

__Wave-like vs. Particle-like__

The number-of-phases determines the degree of wave-like behaviour. Smaller phase-size means more phases per cycle; and consequently more wave-like the behaviour. But as the frequency/phase-size goes up the number-of-phases comes down, and so the behaviour would become less wave-like, more particle-like…

__Euler’s Identity__

The ultimate instance of this occurs when the phase-size reaches πĦ ħ-radians or 180 degrees per h second. 180 degrees is the maximum phase-size, because any excess of 180 degrees in one direction is the same as a deficit of 180 degrees in the opposite direction (i.e. 181 degrees anti-clockwise is exactly the same as 179 degrees clockwise). This maximum phase-size corresponds to a pure *binary oscillation —* an oscillation in 2 discrete steps, each step of which is mathematically described by Euler’s Identity *(**e ^{iπ} = -1)*.

__Max Speed/Frequency of Oscillation__

Angular frequency and angular speed are one and the same thing. Euler’s Identity suggests a *quantum limit to the angular speed of oscillation*.

We can calculate this possible *quantum limit of angular speed (**Ω)* by simple scaling the quantum-angular-frequency back up to our everyday scale measured in Hz (i.e. cycles per second)

**f**** = ½ cycle per h-second = (½H) cycles per second = 7.5 ×**** 10 ^{32} Hz**

^{[ where H} ^{= 1.5 × 10^33}^{ (i.e. H = }^{the inverse of h}^{) ]}

**Ω ****= 2****πf**** = (****π****Ħ****) radians per second = 4.7 ×**** 10 ^{33} radians per second**

**Part 3 – Energy Mass Space and Time**

__The Speed of Light__

So if the universe does indeed have a limit to angular speed, it seems logical to assume that this speed limit might have some relationship to the limit of linear speed (i.e. to the speed of light).

We know that for the speed of light, the frequency times the wavelength is a constant. Equally we also know that, for an oscillation, the phase-size times, the number of phases in a cycle, is also a constant. This would seem to imply that just as frequency relates to phase-size, so wavelength might relate to number-of-phases.

Furthermore, since the linear speed of light (v= fλ) is the same for all frequencies, so maybe the maximum linear speed of oscillation (v=ωA) is the same for all angular frequencies. This would then mean that the amplitude of oscillation (A) would be equivalent to the number-of-phases in a cycle, and is the angular equivalent to ^{λ}/_{2}_{π }. This idea suggests that

**at higher frequencies what contracts**

**is not just the wavelength**

**but space-time itself…**

It suggests that in a quantum oscillation amplitude (A) represents *spatial distance in space-time _{(i.e. as the amount of change in one unit of time goes up, the distance in space-time comes down)}*; and moreover the fact that maximum linear speed of oscillation (v=ωA) is a constant further suggests that it is the angular equivalent to the linear speed of light; in other words that they are one and the same linear speed (i.e. ωA = c).

__Does the Inertia of a body depend on its Energy Content?__

If (ωA) is both a constant and a constituent of our unit of angular momentum (ħ = mωAA) this would mean that (mA) must also be a constant and so when the energy (ħω) of an oscillating particle goes up, space-time amplitude (A) comes down which requires that mass (m) must go up.

__Conservation of Mass-Energy and Space-Time__

Photons are traditionally believed to be massless, but there has never been any experimental proof of this belief. So maybe low energy photons simply have imperceptible mass, too small to be measurable, but maybe at higher energies/frequencies measurable mass might actually emerge.

If so, then by the conservation of momentum, this would mean that the increase in mass would be accompanied by a decrease in linear velocity. And this idea, when extended from momentum to energy, would mean that total energy in a photon is in fact the sum of Kinetic Energy * and* a form of

*“Kinetic Mass”*(i.e. Mass Energy); and by adapting the Lorentz transformation (from linear to angular) the breakdown between the two would be given by

**E = ****ħ****ω (( ^{ω}/**

_{Ω}**)**

^{2}**+**

**(1 – (**

^{ω}/

_{Ω}**)**

^{2}**))**(equation 10)

Where (ħω(^{ω}/_{Ω})^{2}) represents mass-energy and (ħω(1 – (^{ω}/_{Ω})^{2})) represents kinetic-energy. Equation (10) is a *“falsifiable statement”;* it suggests that extremely high frequency photons should gain *“measurable”* mass and therefore travel *“measurably”* slower than the speed of light. Admittedly this equation would require very high energies to test, but nevertheless it is potentially testable, and therefore falsifiable.

Furthermore, if (ωA = c), then

**EA = ****ħc**** (( ^{ω}/**

_{Ω}**)**

^{2}**+**

**(1 – (**

^{ω}/

_{Ω}**)**

^{2}**))**(equation 11)

Equation (11) suggests some form of conservation of Mass-Energy and Space-Time; where the first term (ħc(^{ω}/_{Ω})^{2}) might represent the Mass-Energy and the second term (ħc(1 – (^{ω}/_{Ω})^{2})) represents Space-Time. This suggests that the more massive a particle becomes, the more space-time contracts; and it is the frequency of oscillation _{(i.e. the quantum amount of angular momentum per unit time)} that determines the balance between the two.

__Singularity__

Euler’s Identity occurs when (ω=Ω).** **According to equation (11) Euler’s Identity effectively describes an energy black-hole; a quantum phase-size maximum where effectively all energy is kinetic-mass, and each unit of time simply executes one of the two legs of a discrete binary oscillation._{. }

Equations (10) say that any further increase in energy will not increase the frequency of oscillation, it will simply increase the mass of this energy black-hole. Equations (11) suggests that Euler’s Identity is representative of an oscillation where all space and time get condensed into mass in a *“Singularity”*.

Furthermore it suggest that this singularity is not an infinitesimally small point but instead a smallest unit of space. We calculate the size of this unit as

Amplitude of Singularity = c/Ω = (3 × 10^{8}) / (4.71 × 10^{33})

**Amplitude of Singularity = 0.637 x 10 ^{-25} meters**

———-

__Quantum Relativity__

In the latter part of the 19^{th} century, classical physics used the concept of the *“equipartition of energy”* to conclude that all possible wavelengths and frequencies were equally likely. Consequently, when applied to the problem of *“blackbody radiation”* it was assumed that all possible standing waves inside the cavity of a *“black body”* were merely *“degrees of freedom”.* This line of thinking ultimately lead to the so-called *“ultraviolet catastrophe”*. Planck’s quantization of energy appeared to solve the problem, *but the original idea that wavelengths can be infinitely small, and frequencies infinitely large, nevertheless managed to live on.*

The argument for the existence of a possible maximum angular frequency suggests we *abandon the idea of infinitely small wavelengths*. The discretization of Euler’s Formula suggests both a smallest frequency __and__ a smallest wavelength.

It allows us to think of frequency as *“phase-size”* and wavelength as the *“numbers-of-phases”*, and in doing so suggests a possible basis for the unification of Quantum Mechanics and General Relativity; it suggests 3 fundamental principles of *“Quantum Relativity”…*