corrado malanga - luciano pederzoli sstcorrado malanga - luciano pederzoli sst superspin theory part...

SST- SuperSpin Theory – Part One

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Corrado Malanga - Luciano Pederzoli

SST SuperSpin Theory

PART ONE

DIMENSIONAL RELATIONS AND UNCERTAINTY

Rev. 1.0 - november 27, 2003 Rev. 1.0.1 - march 10, 2004

Original work registered on: December 1st, 2003

ALL RIGHTS CONCERNING THIS WORK ARE RESERVED. Copy, transmission or memorization of this work are subjected to the following conditions: • This work is free to use for non profit purposes and respecting the condition that

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PRESENTATION OF THE SST - PART ONE

When, as for us, it is not possible to access sources of rich funds for own researches, the first question to ask is: How is it possible, without having suitable funds, to perform the long series of expensive experiments that could bring to new discoveries? The answer, as we will see, is this: • You must use the experiments already performed so many times to give absolutely

certain results, • looking for new interpretations of them and • blocking the validity of all that, until now, has officially been deducted from such

experiments • without forgetting that it is very probable (practically sure) that reality extends itself

beyond the fields explored till now. The fundamental experiments are all well reported in the literature and from them have been obtained the few fundamental quantities which allow measuring, and therefore using, all that we know. The synthesis is represented by the so-called Systems of Measures, among which one (the International System) has been considered the world standard for over forty years. Nevertheless, it isn’t stated anywhere that we need to use the fundamental unities of that System of Measure, on the contrary a useful and economic (in monetary, not temporal, terms) experiment really consists in replacing such unities and drawing a new description of the reality known to us. It is true that the new description cannot indeed contain anything new, compared to that from which it has been drawn, but it is also true that it describes the reality from another point of view, therefore it can suggest new interpretations or can make spin innovative ideas. The SST (SuperSpin Theory) - Part One shows the results of one of such experiments and the innovative ideas that have sprung of it. They give origin to a wider description of the currently approved reality. In comparison to this view the actual description represents only a particular case, even though unequivocally correct.


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α) UNEXPECTED DIMENSIONAL RELATIONSHIPS [in the text this character is reserved to the dimensional analysis and to the

concerning comments] The dimensional equations establish the existing relationships among the quantities appearing in a physical formula, apart from possible adimensional constants; it is well known that the respect of the dimensional equations is the first rule to observe when physical laws are to be applied. The Systems of Measure, in turn, represent what is most consolidated and unanimously approved in the technical-scientific field. Comparing the measuring systems before and after the 60’s, and particularly the actual International System (IS), of general use, with its main predecessor for more than eighty years (the CGS Electrostatic System), we can see that the fundamental difference and the fullest of consequences too, between these two Systems, is the different definition of electric charge. For the old CGS Electrostatic System the electric charge is static and shows the following dimensions:

α-01) [l 3 m t -2] 1/2

NOTE α-α In fact, the Coulomb’s law says that:

F = cq*(Q1*Q2)/r2 Where Q1 and Q2 are punctiform electric charges, cq is a constant, assumed equal to 1 in the CGS System, r is the distance dividing the charges and F the strength with which they are attracted or rejected, according to their signs. Assuming as equal the two charges, it becomes:

F = Q2/r2 from which we have that:

Q = (F*r2)1/2 or, being F = m*a , also:

Q = (m*a*r2)1/2 , whose dimensions exactly are:

[l 3 m t -2] 1/2

For the International System, the charge is instead in movement and has dimensions:

α-02) [t i] Making equal the two charges with appropriate calculations and, accordingly, the relative dimensional expressions, we have:

α-03) i = [l 3/2 m 1/2 t -2] = [l 3 m t -4] 1/2


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Substituting this dimensional expression in place of the electrical current of the International System and bringing only the quantities of greatest interest to this work (the dimensions brought in substitution are cyclamen-coloured) we obtain the following Table α-a:

TABLE α-a

MODIFIED INTERNATIONAL SYSTEM

Physical quantity

Dimensions

l = length [ l ] t = time [ t ] m = mass [ m ] f = frequency [ t -1] v = speed [ l t -1] a = acceleration [ l t -2] F = force = m*a [ l m t -2] U = energy [ l 2 m t –2] P = power [ l 2 m t -3] i = electric current (IS)

i = electr. curr. (from CGS)[ i ]

[l 3 m t -4] 1/2 ε0 = dielectric constant [ l -3 m -1 t 4 i 2]

1 (CGS typ. value) µ0 = absolute permeability

µ0 = 1/v2 [ l m t -2 i -2]

[ l t -1] –2 G = gravitation constant [ l 3 m -1 t -2] h = Planck constant

H = Q 2 / v = Φ 2 * v [ l 2 m t -1]

K = electric field strength [ l m t -3 i -1]

[ l -1 m t -2] 1/2 H = magnetic field strength [ l -1 i ]

[ l m t -4] 1/2 Q = electric flux (electric charge)

Q2 = Energy * Length

[ t i ] [ l 3 m t -2] 1/2

[ l 3 m t -2] Φ = magnetic flux

Φ = Q/v

Φ2 = Space * Mass

[ l 2 m t -2 i -1] [ l m ] 1/2

[ l m] The substitution already allows to glimpse at relationships among electricity, magnetism, space, time, mass and energy, but let’s try to see what happens if it is adopted, as fundamental physical quantities, the energy instead of the mass.


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We get the following TABLE α-b (we will call the new System of Measure "S-T-U", from S = Space, T = Time and U = Energy): TABLE α-b

S-T-U SYSTEM

Physical quantity Notes and processings Dimensions

l = length (monodimensional Space)

[l]

t = Time (1/f = T = period) [t] U = Energy [u]

m = mass From U = m*v2/2 comes the m = 2U/v2 [l -2 t 2 u]

f = frequency 1/T = f = frequency [t -1]

V = volume [ l 3] v = speed [l t -1] a = acceleration [l t -2] F = force = m*a [l m t -2] = [l l -2 t 2 u t -2] [l -1 u] P = power [l 2 m t -3] = [l 2 l -2 t 2 u t -3] [t -1 u]

h = Planck’s constant [l 2 m t -1] = [l 2 l -2 t 2 u t -1] [t u]

µ0 = absolute permeability [l –2 t 2] ε0 = dielectric constant 1 G = gravitation constant [l 3 m -1 t -2] = [l 3 l 2 t -2 u -1 t -2] [l 5 t -4 u -1]

i = electric current [l 3 m t -4] 1/2= [l 3 l -2 t 2 u t -4] 1/2 [l t -2 u] 1/2

Q = electric charge [l 3 m t -2] 1/2= [l 3 l -2 t 2 u t -2] 1/2 [l u] 1/2 K = electrical field strength [l -1/2 m 1/2 t -1] = [l -1/2(l -2 t 2u) 1/2 t -1] [l -3 u] 1/2 Φ = magnetic flux [l m] 1/2 = [l l -2 t 2 u] 1/2 [l -1 t 2 u] 1/2 H = magnet. field strength [l m t -4] 1/2= [l l -2 t 2 u t -4] 1/2 [l -1 t -2 u] 1/2 The dimensional expressions listed in the right column don't contain references to the mass anymore, but only to length, time and energy. All the relationships contain small values of raising to power of the aforesaid three quantities, aside from the gravitational constant. From the TABLE α-b it is possible to derive the TABLE α-c, which, starting from the expressions of Q, K, Φ and H drawn just now and examining all their products and ratios, as well as, later, some other combinations, shows us unexpected relations among electric charge, electric field strength, magnetic flux, magnetic field strength, time, energy, force, power, length, volume and mass.


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TABLE α-c

Physical quantities

Processings

Dimensions Q Length * Force1/2 [l] [l -1 u] 1/2 [l u] 1/2 K Length-1 * Force1/2 [l] -1[l -1 u] 1/2 [l -3 u] 1/2 Φ Time * Force1/2 [l u] 1/2 [l t -1] -1 [l -1 t 2 u] 1/2 H Time-1 * Force1/2 [l -1 t 2 u] 1/2 [t] -2 [l -1 t -2 u] 1/2

Q2 = Q*Q Energy * Length [l] [u] [l u] K2 = K*K Force / Length2 [u] [l] -3 [l -3 u] Φ2 = Φ*Φ Force * Time2 [l u] [l t -1] -2 [l -1 t 2 u] H2 = H*H Force / Time2 [l -1 u] [t] -2 [l -1 t -2 u]

Q * K Energy / Length = F [l u] 1/2 [l -3 u] 1/2 [l -1 u] Q * Φ Time * Energy = h [l u] 1/2 [l -1 t 2 u] 1/2 [t u] Q * H Energy / Time = P [l u] 1/2 [l -1 t -2 u] 1/2 [t -1 u] K * Φ Power / Speed2 [l -3 u] 1/2 [l -1 t 2 u] 1/2 [l -2 t u] K * H Power / Length2 [l -3 u] 1/2 [l -1 t -2 u] 1/2 [l -2 t -1 u] Φ * H Energy / Length = F [l -1 t 2 u] 1/2 [l -1 t -2 u] 1/2 [l -1 u] Q / K Length2 [l u] 1/2 [l -3 u] -1/2 [ l ] 2 Q / Φ Length / Time = v [l u] 1/2 [l -1 t 2 u] -1/2 [l t -1] Q / H Length * Time [l u] 1/2 [l -1 t -2 u] -1/2 [l t] Φ / K Length * Time [l -1 t 2 u] 1/2 [l -3 u] -1/2 [l t] K / H Time / Length = 1/v [l -3 u] 1/2 [l -1 t -2 u] -1/2 [l -1 t] Φ / H Time2 [l -1 t 2 u] 1/2 [l -1 t -2 u] -1/2 [ t ] 2

K * Q3 Energy2 [l -3 u] 1/2 [l u] 3/2 [ u ] 2

Q / T i = electric current [l u] 1/2 [t] -1 [l t -2 u] 1/2

m * a Energy / Length = F [u] [l] -1 [l -1 u] µ0 Time / Length = 1/v2 [l -2 t 2 u] [u] -1 [l –1t] 2 ε0 Pure number 1 1

You can see that some expressions are equivalent. For instance:

α-04) Φ*H = Q*K = m*a = F

or also:

α-05) Q/Φ = H/K= v

There is no need to underline the existence of meaningful relationships among length (space), time, energy, electric position, strength of the electric field, magnetic flow and strength of the magnetic field. It is evident the utility to deepen such relationships, both from the theoretical and the experimental points of view. There are, for instance, three expressions of the strength, respectively functions of the electric charge Q, of the magnetic flux Φ and of the mass m:


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α-06) F = Q*K α-07) F = Φ*H α-08) F = m*a

From them it is inferred that electric charge Q, magnetic flux Φ and mass m are equivalent to each other, representing the sources of the respective fields; in fact, as it has been already said, we define “K” the strength of the Electric Field, “H” the strength of the Magnetic field and “a” the strength of the Gravitational Field (in fact it is well known to everybody that the acceleration of gravity, on the terrestrial surface, is equal to about 9.81 m/s2). Dimensionally “a” is:

[ l t -2] But, being:

Φ/H [ t ] 2 and:

Q/K [ l ] 2 it follows that, always in dimensional terms, it is true:

α-09) a = strength of the Gravitational field = [ l t -2] = [ i 2 u -1] = (Q/K)1/2 * (H/Φ) As it is seen, from the U = ½ m*v2 (or from U = m*c2) it is drawn that, in dimensional terms, m = U/v2. It is:

[ l -2 t 2 u ] But, since:

[ l ] 2 Q / K [ t ] 2 Φ / H [ u ] 2 K * Q3

it follows that:

α-10) m = [ l -2 t 2 u ] = K/Q * Φ/H * (K*Q3)1/2 = (Φ/H) * (K3*Q)1/2 Nevertheless they are also worth:

from which derives the more manageable:

α-11) m = mass = [ l -2 t 2 u ] = Φ2* (K/Q)1/2 From α-05) it is had, then, Q/Φ = H/K= v , therefore also a new expression of the mass:

α-11) m = mass = [ l -2 t 2 u ] = (K/H)2* Q * (K*Q)1/2 The verification effected on the product m * a brings, in both cases, to the same result (F):

α-12) m * a = Φ2* (K/Q)1/2 * (Q/K)1/2 * (H/Φ) = Φ*H = F α-13) m * a = (K/H)2 * Q * (K*Q)1/2* (Q/K)1/2 * (H/Φ) = (K2*Q2) / (H*Φ) = F2/F = F

[ l ] -1 (K / Q)1/2 [l -1 t 2 u] Φ2


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The relationships brought in the following TABLE α-d are particularly interesting: TABLE α-d

(Q / K)1/2 [ l ] [l 1/2 u 1/2] 1/2 [l 3/2 u -1/2] 1/2 Length (Φ / H)1/2 [ t ] [l -1/2t u1/2] 1/2[l 1/2t u-1/2] 1/2 Time

Q * (K * Q)1/2 [ u ] [l -3/2 u 1/2] 1/2 [l 3/2 u3/2] 1/2 Energy

Φ2 * (K / Q)1/2 [ l -2 t 2 u ] [l -1 t 2 u] [ l ] -1 Mass (Q/K)1/2 * (H/Φ) [ l t -2] [ l ] [ t ] -2 Acceleration

Q [ l u ] 1/2 [l] [l -1 u] 1/2 Length * Force1/2 K [ l -3 u ] 1/2 [l] -1 [l -1 u] 1/2 Length-1* Force1/2 Φ [ l -1 t 2 u ] 1/2 [t] [l -1 u] 1/2 Time * Force1/2 H [ l -1 t -2 u] 1/2 [t] -1 [l -1 u] 1/2 Time-1* Force1/2

Q2 (= Q*Q) [ l u ] [l u] 1/2 [l u] 1/2 Energy * Length Q * Φ [ t u ] [l u] 1/2 [l -1 t 2 u] 1/2 Time * Energy = h Q / H [ l t ] [l u] 1/2 [l -1 t -2 u] -1/2 Length * Time Φ / K [ l t ] [l -1 t 2 u] 1/2 [l -3 u] -1/2 Length * Time

Φ * H [ l -1 u ] [l -1 t 2 u] 1/2 [l -1 t -2 u] 1/2 Energy / Length = F Q * K [ l -1 u ] [l u] 1/2 [l -3 u] 1/2 Energy / Length = F Q / Φ [ l t -1] [l u] 1/2 [l -1 t 2 u] -1/2 Length / Time = v Q * H [ t -1 u ] [l u] 1/2 [l -1 t -2 u] 1/2 Energy / Time = P

You note that:

α-14) Length = (Q / K)1/2 Purely electric nature α-15) Time = (Φ / H)1/2 Purely magnetic nature α-16) Mass = Φ2 * (K/Q)1/2 Elettromagnetic nature

While energy presents itself in three forms (a is the acceleration):

α-17) Energy = Q * (Q * K)1/2 Electric nature α-18) Energy = Q * (Φ * H)1/2 Elettromagnetic nature α-19) Energy = Φ2 * a Magneto-mechanical nature

From α-19) is deduced that it is possible to produce energy accelerating a magnetic flux: to make a practical example, electric energy can be extracted making a permanent magnet rotate on its own axle in form of an axially magnetized disk (the classical experience of the so-called Faraday Disk, in which the rotation submits to radial acceleration a conductive disk-shaped permanent magnet and the electric energy is withdrawn between the axle and the circumference of the same disk). IT IS TO REMEMBER WELL, FINALLY, THAT THE PRODUCT Q * Φ HAS THE SAME DIMENSIONS [ t u ] OF THE INTRINSIC ANGULAR MOMENT OF A PARTICLE, WHOSE UNIT IS h/(2*π). By such unit we measure the SPIN (that can assume values equal to 0 ± ½, ± 1, ± 2, and so on).


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β) THE MP UNCERTAINITY [in the text this character is reserved to the dimensional analysis and to the

concerning comments] N.B.: All the ĉ used in the text are adimensional constants, that don't influence the

qualitative behaviour of the formulas, but only serve to keep into account the unities of measure adopted.

Let’s take back TABLE α-a of the preceding paragraph:

TABLE α-a

MODIFIED INTERNATIONAL SYSTEM

Physical quantity Dimensions

l = length [ l ] t = time [ t ] m = mass [ m ] f = frequency [ t -1] v = speed [ l t -1] a = acceleration [ l t -2] F = force = m*a [ l m t -2] U = energy [ l 2 m t –2] P = power [ l 2 m t -3] i = electric current (IS)

i = electr. curr. (from CGS)[ i ]

[l 3 m t -4] 1/2 ε0 = dielectric constant [ l -3 m -1 t 4 i 2]

1 (CGS typ. value) µ0 = absolute permeability

µ0 = 1/v2 [ l m t -2 i -2]

[ l t -1] –2 G = gravitation constant [ l 3 m -1 t -2] h = Planck’s constant

H = Q 2 / v = Φ 2 * v [ l 2 m t -1]

K = electric field strength [ l m t -3 i -1]

[ l -1 m t -2] 1/2 H = magnetic field strength [ l -1 i ]

[ l m t -4] 1/2 Q = electric flux (electric charge)

Q2 = Energy * Length

[ t i ] [ l 3 m t -2] 1/2

[ l 3 m t -2] Φ = magnetic flux

Φ = Q/v

Φ2 = Space * Mass

[ l 2 m t -2 i -1] [ l m ] 1/2

[ l m]


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Heisemberg, with his uncertainty principle, affirms that:

β-01) ∆T*∆U ≥ h/(4*π) where h is the Planck’s constant. This principle establishes the uncertainty value h/(4*π) in the simultaneous determination of the energy that a particle owns and of the temporal instant in which it does it: if the temporal uncertainty is reduced to zero, the other indetermination becomes of endless amplitude, making impossible the determination of energy owned by the particle in the selected instant. Naturally this principle allows to reverse the situation too, making impossible the determination of the instant in which the particle owns the exact energy that has been "preselected." The ∆T*∆U = h/(4*π) is often written in another way:

β-02) ∆x*∆p ≥ h/(4*π) = ħ/2 with x = position, p = momentum (m*v) and ħ = h/(2*π). In this form, the uncertainity ∆X*∆p represents the indetermination in the contemporary definition of the position that a particle has and of the momentum (the product between mass and speed of the same particle) that it has in that position. Instead of:

β-03) ∆X*∆p ≥ h/(4*π) it could be written:

β-04) ∆x*∆m*∆v ≥ h/(4*π) expression that implicates the uncertainity in the contemporary definition of position x, mass m and speed v of the particle. The presence or the absence of (4*π) in the denominator of the second term of the preceding disequations depends on the conventions related to the system of measure adopted, but for our purpose it is not meaningful, since we are exclusively interested in the dimensional meaning of Heisemberg’s uncertainity principle: therefore from now onwards we will write ∆X*∆p ≥ ĉh*h . Note that the equation ∆T*∆U = ĉh*h is of the type x*y = constant, that represents an equilateral hyperbola in a Cartesian plane of which T (Time) and U (Energy) are the coordinated axes. It can be said that the principle implicates the existence of such axes; in fact, having made recourse to them, we obtain a simple graphic representation of Heisemberg’s uncertainity principle, as the place of points beyond a limit curve constituted by the same hyperbola. The equation β-04) shows, nevertheless, the importance of position (length, it is Space), mass and speed (length/time). Altogether: Space, Time and mass. Let’s remember, however, that mass and energy, according to Einstein, are tied up by the:

β-05) U = m*c2


Pag. 11 of 11

with U = energy, m = mass, and c = speed of light in vacuum (classical kinetic energy is worth U = ½ m*v2, that is dimensionally equivalent to the β-05). Considering the β-01) and the β-04) we can believe that the involved Cartesian axes are not only two (Time and Energy), but a third exists, that of the Lengths, is of the Space. We introduce, therefore, the hypothesis according to which the Heisemberg’s uncertainity principle only represents the bidimensional version of a more general (three-dimensional) uncertainity principle: accordingly, to the coordinated axes T and U we will add, in a three-dimensional Cartesian system, the spatial axis S. The expressions β-01) and β-04), here repeated with the introduction of ĉh:

β-01) ∆T*∆U ≥ ĉh*h

β-04) ∆x*∆U*∆S/T ≥ ĉh*h dimensionally are worth:

[ l 2m t -1] . In the new system of orthogonal coordinates S, T and U are born, therefore, three PARTICULAR (bidimensional) uncertainity principles, one for every couple of coordinated axes (the first one is the classical Heisemberg’s uncertainity principle). In fact, setting:

β-06) ∆U = ∆Energy [ l 2m t -2] β-07) ∆T = ∆Time [ t ] β-08) ∆S = ∆Space [ l ]

from the dimensional point of view the three aforesaid principles are:

β-09) ∆U*∆T = ∆Energy * ∆Time [ l 2m t -2][ t ] = [ l 2 m t -1] (Heisemberg) β-10) ∆T*∆S = ∆Time * ∆Space [ t ][ l ] = [ l t] β-11) ∆U*∆S = ∆Energy * ∆Space [l 2m t -2][ l ] = [ l 3 m t -2]

But we can also affirm that: β-12) ∆U [ l 2m t -2] = [ l 2m t -1] [ t -1] ≥ ĉU*h*f That is we can say the uncertainty of the Energy is proportional to a frequency (f). Then why ∆T doesn't have to be proportional (according to a constant we will call ĉT) to a period (T), and ∆S (according to a constant we will call ĉ) to a wavelength (λ)? The wavelength is worth speed/frequency. Accordingly: λ = v/f = v*f-1, with v = speed. It draws e set of three equations (the first one is repeated for convenience):


Pag. 12 of 12

β-12) ∆U [ l 2m t -2] h * frequency ≥ ĉU*h*f = ĉU*h*f β-13) ∆T [ t ] period ≥ ĉ T*T = ĉ T*f-1 β-14) ∆S [ l ] wavelength ≥ ĉ S*λ = ĉ S*v*f-1

The β-12) has the dimensions of the classical relationship U = h*f , that expresses the energy of the photon, but it is also worth [ l 2m t -2] = [ m ] [ l t -1] 2 , with the dimensions of the as much known U = m*c2, being just [ l t -1] a speed.

NOTE β-α Let’s admit that, for the same particle (PHOTON), are worth both the:

E = m*c2 and E = h*f

Making equal them, we get:

m*c2= h*f

from which we obtain:

m = ( h / c2 ) * f [ l 2 m t -1] [ l t -1] -2 [ t -1 ] = [ m ]

We deduce that the mass of a photon is proportional to its frequency. In fact: f * 6.626 *10-34 / 9 * 1016 = f * 0,7362 * 10-50 Kg For example, at 1 GHz, the mass is 0.7362 * 10-41 Kg

∆S , ∆T and ∆U can be interpreted as THE THREE QUANTITIES THAT DEFINE A PARTICLE IN THE S-T-U DOMAIN (Space-Time-Energy) and we can affirm that:

∆U is proportional to a frequency, ∆T is proportional to a period, ∆S is proportional to a wavelength

Making the product of the dimensions of ∆S , ∆T and ∆U , it is gotten:

β-15) ∆S*∆T*∆U [ l ] [ t ][ l 2m t -2] = [ l 3m t -1] From β-09), β-10) and β-11) we then draw these relationships, characteristic of S-T-U domain:

β-16) ∆S/∆T speed [ l t -1] β-17) ∆U/∆S force [ l m t -2] β-18) ∆U/∆T power [ l 2m t -3]

Adding the β-08), β-09) and β-10) we have all the relationships typical of the S-T-U domain:

β-06) ∆U*∆T h [ l 2 m t -1] (intrinsic angular momentum) β-07) ∆T*∆S [ l t] β-08) ∆U*∆S Q2 [ l 3 m t -2] (electric charge)2


Pag. 13 of 13

The products ∆U*∆S , ∆U*∆T , ∆T*∆S , as pointed out by their pedixes, concern the "planes" U-S, U-T and T-S respectively and, over them, they define real surfaces, whose "area" is defined by the product of two ∆. Their square roots are proportional to "radii" of such areas, if these are thought as circular, or to "sides", if they are supposed square. The complete picture of the relationships resultant from what till now exposed in paragraphs α) and β) is the following:

β-19) ∆S*∆T*∆U [ l t u ] = [ l 3m t -1] = Q2*(Φ/H)1/2

β-20) ∆U*∆T [ t u ] = [ l 2 m t -1] = Q*Φ h (intrinsic angular mom.) β-21) ∆T*∆S [ l t] = [ l t ] = Φ/K = Q/H β-22) ∆U*∆S [ l u ] = [ l 3 m t -2] = Q2 (electic charge)2

β-23) ∆S/∆T [ l t -1] = [ l t -1] = Q/Φ v (speed) β-24) ∆U/∆S [ l -1 u ] = [ l m t -2] = Q*K = Φ*H F (force) β-25) ∆U/∆T [ t -1 u ] = [ l 2m t -3] = Q*H P (power) As a result it can be defined the:

MP GENERAL UNCERTAINITY PRINCIPLE expressed as:

β-26) ∆S*∆T*∆U ≥ z = constant The β-26 ) is a product of three ∆, a kind of "volume" comparable to that of a sphere and, therefore, proportional to an opportune "radius" raised to the third power, or to a "cubic volume", of which it is proportional to the "side", raised to the third power too. It is already seen that ∆U*∆S has the dimensions of an electric squared charge, therefore it can be assumed that it is proportional to e2 (e = electron charge). Instead ∆U*∆T , as noticed, has the dimensions of the Planck’s constant (h). From:

β-12) ∆U [ l 2m t -2] h * frequency ≥ ĉU*h*f = ĉU*h*f β-13) ∆T [ t ] period ≥ ĉT*T = ĉT*f-1 β-14) ∆S [ l ] wavelength ≥ ĉS*λ = ĉS*v*f-1

if, in place of v, we write c (light speed in vacuum) and we adopt, as value of Q (in β-22), the charge e of the electron, we can write:

β-27) ∆U*∆S ≥ ĉS*h*v = ĉ0*h*c = Q2 = ĉ1*e2 β-28) ∆U*∆T ≥ ĉT*h = ĉ1*ĉT*e2/(ĉ0*c) (h from β-27) = ĉ2*e2*c-1 β-29) ∆T*∆S ≥ ĉT*ĉs*v*f-2 = ĉ3*c*f-2 (c from β-27) = ĉ4*e2*f-2*h-1

The products ∆U*∆S , ∆U*∆T , ∆T*∆S , as their pedixes point out, respectively concern the "planes" U-S, U-T and T-S, and define "areas" that can be compared to those of circles, whose radii are:


Pag. 14 of 14

β-30) ∆U*∆S → area ≥ ĉ0*h*c = ĉ1*e2 → radius ≥ e*(ĉ1*π-1)1/2 β-31) ∆U*∆T → area ≥ ĉT*h = ĉ2*e2*c-1 → radius ≥ e*(ĉ2*π-1*c-1)1/2 β-32) ∆T*∆S → area ≥ ĉ3*c*f-2 = ĉ4*e2*f-2*h-1 → radius ≥ e*f-1*(ĉ4*π-1*h-1)1/2 MP GENERAL UNCERTAINITY PRINCIPLE becomes, this way:

β-33) ∆S*∆T*∆U ≥ ĉ0*ĉT*h*c*f-1 = ĉ6*e2*f-1 (being: h*c = e2*ĉ1*ĉ0-1) The “radius” ∆STU of the “volume”, considered spherical, therefore is:

β-34) ∆STU ≥ (¾ ĉ6*e2*f-1*π-1)1/3 = (3/2*ĉ6*e2*1/2*π-1*f-1)1/3 = ĉ7*( e2*ω-1)1/3

or, being e2 = h*c*ĉ0*ĉ1-1:

β-35) ∆STU ≥ ĉ8*(h*c*ω-1)1/3 with ω (pulsation or angular velocity), equal to 2*π*f. It is important to notice that: ω IS A CHARACTERISTIC PARAMETER OF ROTATION. Besides: According to (β-33) the MP GENERAL UNCERTAINITY PRINCIPLE, the product of the uncertainties of Space, Time and Energy is at least equal to A (dimensional) CONSTANT DIVIDED BY A FREQUENCY. A particle subjected to such principle would practically behave like a ball (with dimensions depending from frequency) made of very thin and extremely elastic rubber, full with water and suspended at middle height in a water tub. Crushing it, the ball deforms itself and widens more the more it is crushed. Since the quantity of water contained in it is always the same one, its volume is constant, but its aspect will change. CONCLUSIONS • The Heisemberg’s uncertainity principle validity has been proven along every one of the

three classical axes of the Space. In fact three components of Space: Sx, Sy and Sz (usually simply called: x, y and z) exist, for everyone of which the aforesaid principle is worth. Nevertheless, we can suppose that, for Time too, three components exist: Tx, Ty and Tz. Accordingly, for Energy three other components will exist, Ux, Uy and Uz. Altogether nine dimensional components: 3 for Space, 3 for Time and 3 for Energy.

• Since ω is a characteristic parameter of rotation and in the β-12), β-13) and β-14) frequency, period and wavelength appear, it is spontaneously born the hypothesis that f, T and λ can refer to the same phenomenon: a rotation with angular velocity ω.


Pag. 15 of 15

γ) THE NINE DIMENSIONS DOMAIN In the preceding paragraph we have talked of an S-T-U domain with 9 dimensions; now let’s see what characteristics it possesses. To begin we take in examination a system of orthogonal coordinates, that we will call, respectively, S, T and U.

Let’s consider, then, a vector R, departing from the origin O’ of those coordinates.

The vector R is projected over everyone of the three coordinated planes (S-T, T-U and U-S) and each of the three projection-vectors (RST, RTU ed RUS) is projected, in turn, over two coordinated axes, giving rise to three resultant vectors, which represent the decomposition of the vector R over the three coordinated axes S, T and U; we will respectively call them ∆S, ∆T and ∆U.

RUS

RST RUT

∆S

∆T

∆U R

O’

S

T

U

O’

R

S

T

U

O’

Fig. γ-F02

Fig. γ-F01

Fig. γ-F03


Pag. 16 of 16

It is to notice that the projection-vectors RST, RTU and RUS contain, everyone, informations related to two of the vectors resulting from the decomposition of R over the three main axes (S, T and U). The whole S-T-U reference system is supposed, in turn, inserted, with generic orientation, in another orthogonal reference system, whose coordinated axes we will call, respectively, x, y and z (Fig. γ-F04).

The origins of the two reference systems can be considered coincident, as in Fig. γ-F04, but let’s admit, for greater graphic clearness, that they are non-coincident (Fig. γ-F05).

∆S

∆T

∆U

O’

R

x

y

z

∆S

∆T

∆U R

x

y

z

Fig. γ-F04

Fig. γ-F05


Pag. 17 of 17

The three orthogonal axes x, y and z are traditionally used to define spatial coordinates; in our case, instead, they are simply defined as "secondary axes", while the role of "main axes" is assumed by S (Space), T (Time) and U (Energy). ∆S , ∆T and ∆U represent differences of main coordinates:

γ-01) ∆S = S1 – S0

γ-02) ∆T = T1 – T0

γ-03) ∆U = U1 – U0 Likewise to what has been exposed with respect to the vector R in the S-T-U system of coordinates, in the new x, y, z system of coordinates every difference of main coordinates can be, in turn, decomposed over the secondary axes, resulting in three new vectors, that we will respectively call ∆Sx, ∆Sy, ∆Sz, ∆Tx, ∆Ty, ∆Tz, ∆Ux, ∆Uy, ∆Uz: altogether 9 vectors (Fig. γ-F06, Fig. γ-F07 e Fig. γ-F08).

∆S

∆T

∆U R

O’

∆Sz

x

y

z

∆Sy

∆Sx

∆Szx

∆Szy

∆Sxy

O

Fig. γ-F06


Pag. 18 of 18

0

∆T

∆U R

O’

y

z

O

∆Tzx

∆Tz ∆Tzy

∆Txy

∆Ty

∆Tx

∆S

x Fig. γ-F07

∆T

∆U R

O’

x

y

z

O

∆S∆Uzx

∆Uzy

∆Uxy

∆Uz

∆Ux

∆Uy

Fig. γ-F08


Pag. 19 of 19

Over everyone of axes x, y and z three of such vectors will be added, giving respectively place to:

γ-04) ∆Sx + ∆Tx + ∆Ux = ∆x

γ-05) ∆Sy + ∆Ty + ∆Uy = ∆y

γ-06) ∆Sz + ∆Tz + ∆Uz = ∆z

To satisfy, then, the condition according to which ∆S, ∆T and ∆P (Fig. γ-F02, under transcribed for convenience) are reciprocally orthogonal, it has to be worth the:

γ-07) R2 = ∆S2 + ∆T2 + ∆U2

∆S

∆T

∆U R

R O’

x

y

z

O

∆zx

∆zy

∆xy ∆x

∆y

∆z

∆Sy

∆Ux

∆Ty

∆Uy

∆Tx

∆Sx

∆Uz

∆Sz

∆Tz

Fig. γ-F09

Fig. γ-F02


Pag. 20 of 20

or, for wide, the:

γ-08) R2 = (S1 – S0)2 + (T1 – T0)2 + (U1 – U0)2 Every difference of main coordinates, as it has been said, is decomposed over the secondary axes, resulting in 3 differences of secondary coordinates: γ-09) ∆Sx = Sx1 – Sx0 γ-12) ∆Tx = Tx1 – Tx0 γ-15) ∆Ux = Ux1 – Ux0 γ-10) ∆Sy = Sy1 – Sy0 γ-13) ∆Ty = Ty1 – Ty0 γ-16) ∆Uy = Uy1 – Uy0 γ-11) ∆Sz = Sz1 – Sz0 γ-14) ∆Tz = Tz1 – Tz0 γ-17) ∆Uz = Uz1 – Uz0 For these, being x, y and z orthogonal axes, they are worth the: γ-18) ∆Sx2 + ∆Sy2 + ∆Sz2 = ∆S2 γ-19) ∆Tx2 + ∆Ty2 + ∆Tz2 = ∆T2

γ-20) ∆Ux2 + ∆Uy2 + ∆Uz2 = ∆U2 Then:

γ-21) R2 = ∆S2+∆T2+∆U2 = ∆Sx2+∆Sy2+∆Sz2+∆Tx2+∆Ty2+∆Tz2+∆Ux2+∆Uy2+∆Uz2 Besides, as we have said, the secondary components too of ∆S, ∆T and ∆U are added over everyone of axes x, y and z, giving origin to the:

γ-22) R2 = (∆Sx + ∆Tx + ∆Ux)2 + (∆Sy + ∆Ty + ∆Uy)2 + (∆Sz + ∆Tz + ∆Uz)2 or, written for wide, to the:

γ-23) R2 = [(Sx1 – Sx0) + (Tx1 – Tx0) + (Ux1 – Ux0)]2 + + [(Sy1 – Sy0) + (Ty1 – Ty0) + (Uy1 – Uy0)]2 +

+ [(Sz1 – Sz0) + (Tz1 – Tz0) + (Uz1 – Uz0)]2 Accordingly they will have to be worth, contemporarily, both the:

γ-21) R2 = ∆Sx2 + ∆Sy2 + ∆Sz2 + ∆Tx2 + ∆Ty2 + ∆Tz2 + ∆Ux2 + ∆Uy2 + ∆Uz2

γ-22) R2 = (∆Sx + ∆Tx + ∆Ux)2 + (∆Sy + ∆Ty + ∆Uy)2 + (∆Sz + ∆Tz + ∆Uz)2

of which the second, written for wide, becomes:

γ-24) R2 = ∆Sx2 + ∆Tx2 + ∆Ux2 + 2*∆Sx*∆Tx + 2*∆Sx*∆Ux + 2*∆Tx*∆Ux + + ∆Sy2 + ∆Ty2 + ∆Uy2 + 2*∆Sy*∆Ty + 2*∆Sy*∆Uy + 2*∆Ty*∆Uy +

+ ∆Sz2 + ∆Tz2 + ∆Uz2 + 2*∆Sz*∆Tz + 2*∆Sz*∆Uz + 2*∆Tz*∆Uz and, combined with the first one, gives:

γ-25) ∆Sx2 + ∆Sy2 + ∆Sz2 + ∆Tx2 + ∆Ty2 + ∆Tz2 + ∆Ux2 + ∆Uy2 + ∆Uz2 = = ∆Sx2 + ∆Tx2 + ∆Ux2 + 2*∆Sx*∆Tx + 2*∆Sx*∆Ux + 2*∆Tx*∆Ux +

+ ∆Sy2 + ∆Ty2 + ∆Uy2 + 2*∆Sy*∆Ty + 2*∆Sy*∆Uy + 2*∆Ty*∆Uy + + ∆Sz2 + ∆Tz2 + ∆Uz2 + 2*∆Sz*∆Tz + 2*∆Sz*∆Uz + 2*∆Tz*∆Uz


Pag. 21 of 21

from which we deduce that the mixed products sum is zero:

γ-26) ∆Sx*∆Tx+∆Sx*∆Ux+∆Tx*∆Ux+∆Sy*∆Ty+∆Sy*∆Uy+ +∆Ty*∆Uy+∆Sz*∆Tz+∆Sz*∆Uz+∆Tz*∆Uz = 0

This last equation, together, for instance, with the first of two, composes the system of two equations that have contemporarily to be satisfied in every point of the domain:

γ-22) R2 = (∆Sx + ∆Tx + ∆Ux)2 + (∆Sy + ∆Ty + ∆Uy)2 + (∆Sz + ∆Tz + ∆Uz)2

γ-26) 0 = ∆Sx*∆Tx + ∆Sx*∆Ux + ∆Tx*∆Ux + ∆Sy*∆Ty + ∆Sy*∆Uy + + ∆Ty*∆Uy + ∆Sz*∆Tz + ∆Sz*∆Uz + ∆Tz* ∆Uz Because both are verified, it needs, in conclusion, that:

γ-27) ∆Sx2 + ∆Tx2 + ∆Ux2 + ∆Sy2 + ∆Ty2 + ∆Uy2 + ∆Sz2 + ∆Tz2 + ∆Uz2 = R2 which guarantees the orthogonality both of the main and secondary axes, any is their mutual orientation, and it may be written as follows (γ-21): γ-21) (∆Sx2+∆Tx2+∆Ux2)+(∆Sy2+∆Ty2+∆Uy2)+(∆Sz2+∆Tz2+∆Uz2) = ∆S2+∆T2+∆U2 = R2

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