# PAN-Theory

A unified theory of fundamental interactions represents today one of the most advanced research sectors. To it are linked the expectations about new worldviews and new implications on the technological and industrial processes for the evolution of the human species. At this unification process, it is assigned a general understanding of phenomena. In this article, we present an alternative theory we call PAN-Theory. The PAN-Theory provides a simple mechanical-algebraic model that proves to be exactly the Coulomb’s interaction; therefore we demonstrate the elementary charge *e* and Planck’s constant *h* to be expression of other constants with negligible errors. By the definition of a mass–time relation the presented Theory obtains and interprets in a new way the fine-structure constant *α* and estimate the proton radius in an agreed extent with the most complex and current models. By the use of PAN-Theory a relativistic mass-energy relation is demonstrate and the wave-particle duality is presented in a new aesthetic. The final product of the presented theory is the expression of Newtonian gravity model as isomorphism of Coulomb model with negligible errors. In appendix a short demonstration for Doppler effect and for quantum entanglement is gave.

## Carmine D’Errico – *free researcher*

## 22.11.2016

# 1 The PAN-Theory axioms

- The generic subatomic particle, named “
”, is a cluster (conglomerate) of two types of matter-bricks called*primary matter**PAN*^{ + }and*PAN*^{ − }with the same mass*m*_{0}and the same radius. - Exists a minimal cluster for each type of PAN (both + and -) that interacts with matters. We’ll call this minimal cluster
he’s formed by a mean number*PANONE:**ᾰ*of PANs and it has radius is (*λ*_{0})/(2). - Every type of cluster has no fixed geometry but it is built up maximizing order and minimizing vacuum within it.
- We assume the mass
*m*_{e}of the primary matter considered here to be much larger than the mass*ᾰ**m*_{0}of the single PANONE, i.e.,*m*_{e}≫*ᾰ**m*_{0}. - The primary matter is intrinsically unstable and emits PANONEs in space at the rate (emission frequency)
*ν*_{0}= (1)/(*t*_{0}). The velocity of each emitted PANONE is*v*_{0}, see eqs. (3↓ 4↓ 5↓). - The total amount of emitted mass in each emission is
*n*≥ 1 ∈ {1, 2, 3, …} ⊆ N^{ + }except the emissions in condition as in axiom 9. - For each emission the conservation of motion quantity is respected.
where is the null vector and

*r*is the versor of_{i}̂*i*–th emitted PANONE. - We further assume that no external force drives the emission mechanisms among PAN and other matter. This amounts to consider momentum conservation as the basic mechanism ruling the interaction among bodies mediated by PANs.
- Each emitted PANONE from a primary matter object may impinge on the surface of a second object of primary matter. Each collision is assumed to be:
**perfectly inelastic**in case of colliding matter of the same type giving rise to the mass*m*_{e}of the primary matter to increase by the total mass of the PANONEs impinging on the surface.**elastic**in case of collision interesting two different types of matter, in this case surface part of*m*_{e}interested from the collision with a different type of PAN became inactive and it can be reactivate only by the collision with a further PANONE of the same type of primary matter.- For the sake of simplicity, in the derivation of the most important consequences of the existence of PANs we consider two spherical bodies of primary matter with equal radius
*R*and mass*m*_{e}, at a distance*d*≫*R*

# 2 Definition of a clock

*λ*

_{0}be the smallest length that we are able to measure, i.e. the inferior limit in any length measurement process. If we further assume the PANONE to move in an isotropic multidimensional space, let

*λ*

_{0}represent the diameter of the smallest sphere S limiting the measurable space occupied by PANONE. If the motion of PANONE is restricted to S then it will consist, in absence of external forces, in a back-and-forth displacement that we can describe by a full oscillation of length 2

*λ*

_{0}and period

*t*

_{0}. We take

*t*

_{0}as the shortest time we are able to measure in the sense that any time measurement must give integer multiples of

*t*

_{0}. We call this system the

**. The speed of PANONE is then given by :**

*elemental clock for a natural phenomenon*and represents the upper limit speed of any similar clock since, according to the definition of the minimal clock and 3↑ we have for a generic speed *v*_{k}:

where *n*_{k}, *m*_{k}, are positive integers restricted by *m*_{k} ≤ *n*_{k}. The equality sign just describes a slower minimal clock with the same velocity in eq. 3↑ for the natural phenomenon stating that the velocity of the natural clock is the same for all clocks and is an even multiple of the ratio between the two fundamental quantities *λ*_{0} and *t*_{0} characteristic of the phenomenon. We assume the limit speed for phenomena measured by the elemental clock to be the light velocity

according to the 2014 CODATA value[11].

# 3 First consequence of the PAN-axioms: the number ℵ of emitted PANONEs

*R*≫

*λ*

_{0}. In the simplest picture of isotropic emission mechanism, we consider

*λ*

_{0}just equal to the thickness of the outermost shell containing ℵ PANONEs to be emitted and (

*π*)/(6)

*λ*

^{3}

_{0}as the loss of volume for each emitted PANONE.

*n*= (ℵ)/(4

*π*

*d*

^{2})

The body will, therefore, undergo a volume loss *V*_{d}, due to the “peel off” of the ℵ emitted PANONE, amounting to:

Considering that *R*≫*λ*_{0}, *V*_{d} can be approximated by:

The number ℵ can be thus easily obtained dividing *V*_{d} by the volume occupied by one PAN and considering *R* = *n*_{R}*λ*_{0}, this simply yields:

The last equation automatically fully respects the axiom (6) in his equation 1↑. Then the eq. 9↑ is a consequence of axiom (6) and vice-versa.

# 4 PAN-Theory: secondary consequence

*m*

_{e1}and

*m*

_{e2}which interact each others by exchanging PANONEs as defined in the hypothesis.

*m*

_{e1}impact on mass

*m*

_{e2}The mass

*m*

_{e2}in turn emits PANONEs that are intercepted by the mass

*m*

_{e1}. It shows that for the conservation of momentum and in a totally intuitive we can affirm that the bodies tend to approach (move away, respectively) along the straight line joining the two centers of gravity if the two masses of primary field are different (equal, respectively) . The momentum per unit of surface at any distance d emitted from the generic particle is expressible as follows:

\strikeout off\uuline off\uwave offwhere *m*_{0} is the mass of a single PAN, and where *v*_{0} is the speed of the PANONEs.\uuline default\uwave default The change in the amount of motion because of collisions with PANONEs intercepted by *m*_{e2} will then:

If we multiply both quantity for the emission frequency *ν*_{0} and we take count of the double effect (*m*_{e1} on *m*_{e2} and vice-versa) we get the total variation of amount of motion is:

The equation 12↑ is the dimension of a force. It also notes that the quantity *ᾰ*ℵ*m*_{0} is the amount of total mass emitted into free space from the body *m*_{e1}.

# 5 A new model for the interpretation of the Coulomb forces and the Sommerfeld’s constant

*h*is a short form for PANONE’s mass, number of emitted PANONEs, primary matter minimal surface and the emission frequency:

We must here emphasize that the axioms (6) and (7) imply the Planck’s constant and vice versa. By considering the Sommerfeld’s expression of fine–structure constant[9] we can write:

where *α* is the *fine structure constant*, *k* is Coulomb’s constant and *e* is the elementary charge. By dividing both quantities for a distance *d*^{2} and moving *c* to the left side of equation 14↑:

what we can observe on the left side of eq. 16↑ is exactly the equation 12↑ of PAN-Theory model. By placing:

*v*_{0} = *c*

we demonstrate that the PAN-interaction between two masses of primary matter is exactly the Colombian interaction between two elementary charges:

Now we can give a microscopic interpretation of the Sommerfeld’s constant*: α^{ − 1} is the average number of PANs in an emitted PANONE. And the speed of each PANONE is c.*

*γ*

_{c}as following:

We can write an *identity of fine structure*:

# 6 The Mass–Time relation and a redefinition of the Planck’s constant

*h*= ℵ(

*m*

_{0})/(

*α*)Σ

_{e}

*ν*

_{0}

where we used the relation Σ_{e} = *π**R*^{2} = *π*(*n*_{R}*λ*_{0})^{2}. Multiplying both members by 8*ν*_{0}:

*n*

^{2}

_{R}ℵ

*m*

_{0}= 16(

*ke*

^{2})/(

*c*

^{3})

*ν*

_{0}

*n*

^{2}

_{R}ℵ

*m*

_{0}we come to a constituent relation for mass–time

**:**

The eq. 22↑ expresses a mass–time relation, its value is so close to the fine–structure constant, so that we could redefine the fine–structure as following:

Where χ = 0, 999736133⋅10^{ − 54}[*kg*⋅*s*]≅10^{ − 54}[*kg*⋅*s*]. This additional definition of *α* is fully consistent with the meaning expressed in eq. 17↑. By comparing the equations 20↑and 23↑:

(1)/(*α*) = 16(*ke*^{2})/(*c*^{3}χ)

by considering *k*^{2} = (*c*^{4}*μ*^{2}_{0})/(16*π*^{2}) we can express Planck’s constant in a new form:

*h*_{b} = (*π**c*^{2})/(8*α*^{2})χ

the mean value for *h* in eq. 25↑ give us an error *ε*_{h} on the CODATA value of Planck’s constant[11] as following:

# 7 A new definition of the elementary charge and the mass–energy equivalence

the precision of the obtained value for the elementary charge to the CODATA values[11] is:

we observe also that the positive and negative charges is only a consequence of PAN-Theory axioms, then the equation 27↑ shows that PAN-Theory is a more general isomorphism who could replace the classical electromagnetic theory. By the use of simple substitutions it is also observed that:

it points out that:

The pseudo–mass (*π*)/(2*α*)♏ expressed in eq. 30↑ represents exactly the total mass released in free space at speed *c* from a body in form of PANONEs multiplied by the impinging surface and normalized by *λ*^{2}_{0}.

*λ*

_{0}as a wavelength of a wave who propagates with speed

*c*, then

*λ*

_{0}= (

*c*)/(

*ν*), it points out:

_{0}̌Where *ν _{0}̌* is the frequency associated to the wave. Then for a general amount of mass

*m*we can write more generally:

Last equation could be considered the final equation of wave-particle duality, it expresses at the same time the mass-energy principia and the fundamental behavior of matter in PAN-Theory. The deeper meaning of the equation 30↑ is a synthetic form that encloses the model assumed: the matter (in bulk form) is dispersed disintegrating itself into PANONEs. In eq. 31↑ the member to the left side is the maximum energy that can have a classic photon when considered to be a wave. The member to the right represents the* equivalent mass of the photon* when it is regarded as a particle. So in general by the eq. 32↑ we can give mass to any photon with general frequency *ν* as a “** mass equivalent particle**” equal to:

By placing *n*_{R} = 1 we can calculate the minimum photon–frequency associated to a PANONE:

**, photoelectric effect and photon-gravity interaction[5]), the same relation reveals that it’s possible the transformation of matter into light and vice versa. For electron-positron pair generation by gamma ray interaction, it can be observed that this theory is fully compatible with actual model. Then we expect now that the gravity model in far field conditions is a further expression of PAN-Theory model. We’ll demonstrate it.**

*[10]*# 8 By Coulomb model to gravity model

multiplying and dividing by *n*_{1}*n*_{2}(*α*^{2})/(*π*^{2})*λ*_{0}♏ with *n*_{1} and *n*_{2} two positive integer numbers:

where *n*_{1}*n*_{2} = *n*_{12}. Aesthetically and dimensionally there seems to be a certain equivalence between the Coulomb and the gravitational equations. We can write following:

where the second factor can be wrote as:

so we can express the gravitational constant as:

We call *Ǧ* “** Electrogravitational Constant**“: the constant who ties Coulomb model and Gravitational model. The constant 39↑ is an equation with two unknowns that has an infinite set of solutions. Of all the infinite possible solutions we choose the one obtained through the pseudo-inverse. So you get as a result:

*Ǧ* = 1([*m*^{4}])/([*kg*∙*s*^{3}])

As defined *n*_{12} must be the product of two nonzero integers, we can approximate (accepting an error of *ε*_{G}≅419⋅10^{ − 6}) *n*_{12} = 25. Consistent with all the assumptions that we have so far conducted, the searched *n*_{12} number is a perfect square. We can definitely rewrite the 36↑ as:

The equations seem to point out that the gravitational forces are a representation in far field of PANONEs who spreads out from primary matter bodies and impacts on others. The natural attraction forces between bodies in gravitational fields indicates that the electrons are responsible of the big part of PANONEs emission but the big role in the PANONEs interception is played from protons.

# 9 PAN-Theory model solutions

(♏)/(*ν*_{0}) = 16(*α*ℏ)/(*c*^{2})

*α**c*ℏ = *G*⎛⎝(5*π*♏)/(2*α*)⎞⎠^{2}

The solutions are:

*ν*_{0} = (1)/(40*π*)√((*α**c*^{5})/(ℏ*G*)) = 1, 260929⋅10^{40}[*s*^{ − 1}]

♏ = (2*α*)/(5*π*)√((*α**c*ℏ)/(*G*)) = 1, 727471⋅10^{ − 12}[*kg*]

the expression of the solutions is exceptionally similar to the expression of “** Planck’s units**“[7].

# 10 Determination of the PAN mass *m*_{0} and of the proton radius *R*_{p}

by considering the PANONE’s natural mass density *ρ*_{0} and natural diameter*λ*_{0} we can write:

for the general primary matter mass we can write:

using the proton mass as *m*_{e} in the eq. 46↑, we can define the mass of the PAN *m*_{0}, the number *n*_{R}, the proton (and electron) radius. Substituting the 44↑ in the 46↑ we get:

then:

*m*_{0} = (♏)/(48*n*^{4}_{R})

*R*_{e} = *n*_{R}*λ*_{0}

*m*_{e} = *m*_{proton}

It points out:

By this relation we can express the energy of a photon in a particular form:

Eq. 51↑ is very similar to the Boltzmann’s energy relation for a perfect mono-atomic gas. Could be the Temperature only another way to represent photons who take apart of atomic equilibrium? The numeric results obtained by solving the system 49↑ are the following:

*ρ*_{0} = 1, 811007⋅10^{19}[*kg*⋅*m*^{ − 3}]

*n*_{R} = 2, 358829⋅10^{16}

*R*_{e} = 2, 804119⋅10^{ − 16}[*m*]

where:

*m*_{0}is the mass of a PAN;*ρ*_{0}is the natural mass density;*n*_{R}is the number of PANONEs who form a proton radius;*R*_{e}is the proton’s radius. This value for proton radius accords to the accepted value for proton radius. Recent experiments carried out for the determination of the radius of the proton agree with the CODATA value[11] which establish that radius has to be*R*^{CODATA}_{p}= 0, 8751⋅10^{ − 15}[*m*].

and then we can express with negligible error the CODATA proton radius calculated with actual theories as following:

This value is so close to last esteem about proton magnetic radius[2]. If the actual CODATA value for proton radius is the correct one and we made a journey back replacing in eq. 54↑ the definitions eq. 13↑, 46↑ it points a paradox:

Could this paradox hide a geometry problem for near field interactions used by actual measurement theory? Could the existence of a an indivisible universal brick break the euclidean space fundamentals in small scale / near field? Or is the actual CODATA proton Radius value wrong?

# 11 Appendix: Doppler Effect and Quantum Entanglement

## 11.1 Doppler effect due to the observer motion

*v*

_{m}= 0 . The PANONEs emission fronts are distant each other of exactly

*d*= 2

*λ*

_{0}. An observer who approach the (resp. moves away from) body with speed

*v*

_{o}will meet the fronts in a reduced (resp. expanded) distance

*d*’ =

*d*±(

*v*)/(

*ν*

_{0}) , for this motivation the frequency measured by the observer will be:

because of 3↑ :

where we adopted *f*≔*ν* in order to improve readability.

## 11.2 Doppler effect due to the body motion

*v*

_{m}≠ 0 and he must respect the emission’s lows 1↑ and 2↑, then the PANONEs have to move with speed

*c*respect to the body’s external surface. For the sake of simplicity we’ll consider the variation of emission phenomena on the elementary surfaces Σ

_{e}who locate itself along the direction of motion. For a stationary body the equilibrium condition is:

*c*but this behavior is not allowed. A change of emitted PANONEs mass is also not allowed (violation of

*α*

^{ − 1}) because of 1↑. The only parameter who can changes is the emission frequency:

and we obtain exactly the equation57↑.

## 11.3 Photonic slime and quantum entanglement

*ν*

_{0}, each emitted PANONE has to move with speed

*c*respect to the photon: So these two emitted PANONEs are issued with a null speed compared to a stationary outside observer to the system, then they are therefore deposited in the photon’s rear relative to the direction of photon’s motion.

**.**

*photonic slime**c*leaves behind a trace of its path. So if two photons at two different times moves along the same trajectory up to a division point in which they separate, the path that connects their topological position in the future is connected into to division point: The photons are physically connected through this drivel as inextensible rod with negligible mass. The PANONEs rotate with respect to each other as in a huge row of toothed wheels and in harmony with both photons. Since the separating point is common to the two photons they may have typically opposite spin.

# 12 Conclusions

# References

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