Pas besoin de chercher bien loin: il n'y a, à ce jour, QU'UN SEUL pays du continent Américain qui soit dirigé par un membre tribal. Afin d'éviter le genre de désagrément trouvé, par exemple, dans:
https://newfoodeconomy.org/tanka-bar-general-mills-epic-provisions-bison-bars/?fbclid=IwAR1_jFjSZwSVOXrrp9aAdFHrFX0IbJpEcW8c8_R1W1fahIes9h1f2whfyT0

voici mes suggestions:
- vous avez un produit INNOVANT à lancer, basé sur une science tribale;
- vous DELOCALISEZ en Bolivie et vous ENREGISTREZ là;
- vous PROTEGEZ IMMEDIATEMENT le produit aux Instituts de la Propriété Industrielle, afin de déposer marques et brevets;
- vous démarrez l'activité, phase de recherche si préalable, puis commercialisation;
- vous focalisez les ventes AU SEIN DES TRIBUS AMERICAINES ET A L'ETRANGER (export);
- vous payez vos taxes et impôts société et individuels au pays hôte.

De ce fait, vous aurez BEAUCOUP MOINS DE RISQUES d'être copiés "à la sauvage" et de vous voir détourner votre chiffre d'affaires. Le dépôt de marques et concepts vous ouvrira la PROTECTION JURIDIQUE et SECURISERA VOTRE MARCHé, gardant en tête que les situations de monopole restent quand même intenables à moyen terme. Il faudra donc, par la suite, faire EVOLUER VOS PRODUITS, REDEPOSER DE NOUVEAUX BREVETS ET MARQUES,...

C'est comme ça qu'on procède, en Occident... :) ça vous évite de plonger et d'aller au tribunal pour rien...

Bonjour à tous. A compter d'aujourd'hui, des posts en français sur le même blog que mes travaux théoriques, mais qui n'ont rien à voir avec ceux-ci et portent surtout sur DIPLOMATIE ET FINANCE. Aujourd'hui, à votre connaissance:

https://newsmaven.io/indiancountrytoday/news/cheyenne-river-citizens-plead-against-water-permit-for-pe-sla-gold-prospecting-H0oGwlzEa0K_92ujJW_LFQ/
https://www.navajotimes.com/wires-wp/index.php?id=1825030680&kid=IZXbj2m0mvvKAgbc#apfeed

(normalement, les liens sont actifs). Voici le pb. Le sort des tribus locales peut ne pas entrez dans vos intérêts c'est une toute autre question. EN REVANCHE, si les affirmations de la Cie canadienne rapportée dans l'article de ICT s'avère correctes, tenant compte de toutes les AUTRES zones d'extraction minière en plein (re)développement, et dans la perspective logique que le minerai suive les lois du marché, alors il faut s'attendre à ce que les cours de l'or S'EFFONDRENT DANS UN AVENIR TOUT PROCHE. Or, les devises "papier-monnaie" restent indexées sur le cours de l'or. Une dépréciation considérable de celui-ci, lié à une surabondance brutale sur le marché des devises, entrainerait des dépréciations équivalentes des monnaies.
POUR CETTE RAISON, je pense que la question devrait être examinée, indépendamment du sort réservé aux tribus locales. Mais, le soutien à leur opposition pourrait faire l'objet d'une activité DIPLOMATIQUE complémentaire.

Vous devriez être informés par ce site des nouveaux posts à venir. Alors, il y aura sans doute un mélange d'articles techniques et non-techniques, mais j'ai préféré conserver le même blog et même fournisseur de contenu.

Mes amitiés à tous et mon bon souvenir.

Quantum physics is essentially replacing the real-valued classical frame with a complex-valued one. It may look simplistic, but that’s what the wave-corpuscle duality says, nothing more. So, we give ourselves a physical frame X with COMPLEX dimension D, that’s REAL dimension 2D, and we realize X as the direct sum X = X₁ + iX₂ of two “projective spaces”, real, with same dimension D. This is the planar representation: each point x of X with coordinates x^a (a = 1,…,D) has “projections” x₁ in X₁ with coordinates x₁^a and x₂ in X₂ with coordinates x₂^a. Saying that a quantum object is “located at point x in quantum space X” becomes formally equivalent to saying that the PROJECTIONS of this location are x₁ in X₁ and x₂ in X₂. And it goes the same with every other physical quantity: parameters, variables, functions. There’s therefore absolutely no mathematical difficulty whatsoever, nothing new, the main difficulty stands on how to INTERPRET RESULTS, their PHYSICAL CONTENT.

 

As we will work with projective states all along, we will introduce the s-MATRICES we will widely use in our calculations. These are the real-valued pendings of the sigma-matrices of spin theory. Basically, each time we have a PRODUCT of complex-valued quantities, the real-valued components of the result can be expressed in terms of these matrices, which are all INVERTIBLE. Let’s then consider two complex quantities x = x₁ + ix₂ and y = y₁ + iy₂. We have:

 

(1)               u = xy = x₁y₁ - x₂y₂ + i(x₁y₂ + x₂y₁) = (s₁^AB + is₂^AB)x_Ay_B

(2)               v = x*y = x₁y₁ + x₂y₂ + i(x₁y₂ - x₂y₁) = (s₃^AB + is₄^AB)x_Ay_B

 

Capital Latin indices run from 1 to 2 and label projective states. Thus:

 

(3)               u_C = s_C^ABx_Ay_B  ,  v_C = s_C+2^ABx_Ay_B

 

These s-matrices have components:

 

(4)               s₁¹¹ = -s₁²² = +1  ,  s₁¹² = s₁²¹ = 0

(5)               s₂¹¹ = s₂²² = 0  ,  s₂¹² = s₂²¹ = +1

(6)               s₃¹¹ = s₃²² = +1  ,  s₃¹² = s₃²¹ = 0  =>  s₃ = Id

(7)               s₄¹¹ = s₄²² = 0  ,  s₄¹² = -s₄²¹ = +1 =>  s₄ = J

 

The “identity matrix” is s₃. s₁ and s₃ are diagonal; s₂ and s₄, “anti-” or “off-diagonal”. s₁, s₂ and s₃ are symmetric matrices, s₄ is skew-symmetric. Their genuine properties are:

 

(8)               Tr(s₁,s₂,s₄) = 0  ,  Tr(s₃) = 2

(9)               Det(s₁,s₂) = -1  ,  Det(s₃,s₄) = +1

(10)           (s₁)² = (s₂)² = (s₃)² = -(s₄)² = s₃

(11)           (s₁)^-1 = s₁  ,  (s₂)^-1 = s₂  ,  (s₃)^-1 = s₃  ,  (s₄)^-1 = -s₄

(12)           s₁s₂ = -s₂s₁ = s₄  ,  s₁s₄ = -s₄s₁ = s₂  ,  s₄s₂ = -s₂s₄ = s₁

 

while s₁, s₂ and s₄ obviously commute with s₃. These properties linearize the products of s-matrices. For instance,

 

s₁s₂s₄ = s₄² = -s₁² = -s₃

 

etc., so that we find the CYCLIC properties of the integer powers of i (i² = -1, i³ = -i, i⁴ = +1) REPORTED to the real-valued s-matrices, which make a BASE of M₂(R).

 

Let’s now consider a quantum point x^a = x₁^a + ix₂^a, a = 1,…,D. Both projections x₁^a and x₂^a are a priori SIGNED quantities. If X₁ and X₂ are both Euclidian, they both have SO(D) as EXTERNAL rotation group; if they are pseudo-Euclidian with signature (D₁,D₂), D₁ + D₂ = D, they both have SO(D₁,D₂) as external rotation group. This group defines ORIENTATION in both projective spaces. What about X = X₁ + iX₂? It inherits the EXTERNAL symmetries of its projections, plus an INTERNAL symmetry introduced by the presence of the imaginary unit i. This is a U(1) symmetry group. It defines ANOTHER ORIENTATION, now dealing with PHYSICAL STATES. If we extend (1) and (2) to the tensor product of two quantum vectors x and y of X, we find:

 

(13)           u^ab = x^ay^b = x₁^ay₁^b - x₂^ay₂^b + i(x₁^ay₂^b + x₂^ay₁^b) = (s₁^AB + is₂^AB)x_A^ay_B^b

(14)           v^ab = x*^ay^b = x₁^ay₁^b + x₂^ay₂^b + i(x₁^ay₂^b - x₂^ay₁^b) = (s₃^AB + is₄^AB)x_A^ay_B^b

(15)           u_C^ab = s_C^ABx_A^ay_B^b  ,  v_C^ab = s_C+2^ABx_A^ay_B^b

 

It is clear the small Latin indices representing the EXTERNAL components aren’t affected at all by the INTERNAL transformation involving only capital Latin indices. What we find instead is that the state-1 projection of u^ab is u₁^ab = x₁^ay₁^b - x₂^ay₂^b, its state-2 projection is u₂^ab = x₁^ay₂^b + x₂^ay₁^b; the state-1 projection of v^ab is v₁^ab = x₁^ay₁^b + x₂^ay₂^b and its state-2 projection, v₂^ab = x₁^ay₂^b - x₂^ay₁^b. Each time, it involves ALL FOUR INITIAL VECTORS, x₁, y₁, x₂ and y₂ and this is what the INTERNAL SCALAR PRODUCTS express in (15). Notice that:

 

x₁^ay₁^b = ½ (u₁ + v₁) , x₁^ay₂^b = ½ (u₂ - v₂)

x₂^ay₁^b = ½ (u₂ + v₂) , x₂^ay₂^b = -½ (u₁ - v₁)

 

so that,

 

(16)           x_A^ay_B^b = ½ (u_Cs^C_AB + v_Cs^C+2_AB)

 

In the special case y = x,

 

(17)           u^ab = x^ax^b = x₁^ax₁^b - x₂^ax₂^b + i(x₁^ax₂^b + x₂^ax₁^b) = (s₁^AB + is₂^AB)x_A^ax_B^b

(18)           v^ab = x*^ax^b = x₁^ax₁^b + x₂^ax₂^b + i(x₁^ax₂^b - x₂^ax₁^b) = (s₃^AB + is₄^AB)x_A^ax_B^b

(19)           u_C^ab = s_C^ABx_A^ax_B^b  ,  v_C^ab = s_C+2^ABx_A^ax_B^b

(20)           x_A^ax_B^b = ½ (u_Cs^C_AB + v_Cs^C+2_AB)

 

and external traces give:

 

(21)           u = S_b=a=1^D u^ab = S_a=1^D [(x₁^a)² - (x₂^a)² + 2ix₁^ax₂^a]

(22)           v = S_b=a=1^D v^ab = S_a=1^D [(x₁^a)² + (x₂^a)²]

 

While v is always non-negative and real-valued in an Euclidian geometry, both u₁ and u₂ can either be positive, negative or zero, EVEN IN AN EUCLIDIAN GEOMETRY. Here is the effect of the INTERNAL SYMMETRY: the EXTERNAL geometry remain unchanged, but PROJECTIVE SQUARES are affected by the INTERNAL geometry (in circle). This is the great difference with the classical situation. Here, we have:

 

(23)           u₁ = S_a=1^D [(x₁^a)² - (x₂^a)²]

(24)           u₂ = 2S_a=1^D x₁^ax₂^a

 

Both are HYPERBOLIC squares, whereas v is an ELLIPTIC square. It follows that, independent of the external geometry, we have to define INTERNAL GENUS. If u₁ is found > 0, the state-1 contribution is HIGHER than the state-2 contribution and we’ll say that u₁ is “(state)1-like”. If u₁ < 0, it will be “(state)2-like” and if u₁ = 0, we’ll keep the name “isotropic” used in the mathematics of space-time relativity. It means that, if the Euclidian areas measured in both states are EQUAL, than the resulting area, OBSERVED IN STATE 1, will be ZERO… otherwise said, it will reduce to a POINT. If u₂ now is equal to zero, the two projected vectors x₁ and x₂ appear ORTHOGONAL to a state-2 observer. In the POLAR representation now:

 

(25)           x₁^a = r^acos(ksi) , x₂^a = r^asin(ksi)

(26)           u₁ = [S_a=1^D (r^a)²]cos(2ksi)

(27)           u₂ = [S_a=1^D (r^a)²]sin(2ksi)

 

and, whether S_a=1^D (r^a)² = 0 in which case x₁ = x₂ = 0 and  u₁ = u₂ = 0, or S_a=1^D (r^a)² <> 0 and then u₁ = 0 for ksi = (2k+1)pi/4, k in Z, giving u₂ = (-1)^kS_a=1^D (r^a)² or u₂ = 0 for ksi = kpi/2, giving u₁ = (-1)^kS_a=1^D (r^a)². We can see that, in both situations, we alternate between a positively-counted and a negatively-counted area and that, more importantly:

 

A NEGATIVELY-COUNTED area is a POSITIVELY-COUNTED one in PHASE OPPOSITION.

 

This enables us to talk of “negatively-counted distances”. What happens is that we have a state-1⁺ and a state-2⁺, where distances are positively counted, plus a state-1^- and a state-2^-, where distances are negatively counted, because 1^- is opposite in phase to 1⁺ and 2^- to 2⁺.

 

We don’t encounter this in the classical, since ksi is set to zero there, giving x₁^a = x^a, x₂^a = 0 and u₁ = S_a=1^D (r^a)² >= 0 while u₂ = 0: the component X₂ is reduced to a point (as to know, {0}) and X₁ identifies with classical D-space.

 

Notice that there’s no contradiction of principle in finding negatively-counted areas, because these are PROJECTIVE EFFECTS: the INVARIANT area remains S_a=1^D (r^a)² IN ALL CASES [however, each r^a in (25) is SIGNED, since it remains subjected to EXTERNAL orientation].

 

Much more generally, when X is CURVED but RIEMANNIAN, its elementary area is given by the second quadratic form:

 

(28)           dl² = g_abdx^adx^b

 

where ALL quantities are complex-valued. Developing in real-valued components, it’s not difficult to show that:

 

(29)           dl² = (dl²)₁ + i(dl²)₂ = (s₁^AB + is₂^AB)dl_AB²

(30)           (dl²)_C = s_C^ABdl_AB²

(31)           dl_AB² = g_Aabs_B^CDdx_C^adx_D^b

 

Explicitly:

 

(32)           (dl²)₁ = g_1ab(dx₁^adx₁^b - dx₂^adx₂^b) - g_2ab(dx₁^adx₂^b + dx₁^bdx₂^a)

(33)           (dl²)₂ = g_1ab(dx₁^adx₂^b + dx₁^bdx₂^a) + g_2ab(dx₁^adx₁^b - dx₂^adx₂^b)

 

Also notice the linear form:

 

(34)           dl = u_adx^a  =>  dl_C = s_C^ABu_Adx_B^a

 

If we square dl = dl₁ + idl₂, we find dl² = dl₁² - dl₂² + 2idl₁dl₂. Comparing with (29) gives:

 

(35)           (dl²)₁ = dl₁² - dl₂² = s₁^ABdl_Adl_B

(36)           (dl²)₂ = 2dl₁dl₂ = s₂^ABdl_Adl_B

 

So, we should not confuse (dl²)_C, which are the projections of dl², with (dl_C)², which are the squares of the projections of dl.

 

First of all, I’d like to apologize near the regular reader for the permanent updates on this blog. The reason is: I have few time to allow it. So, it’s calculations and reasonings “when I can”, I prepare a lot of things but I’m still not in a REGULAR ENVIRONMENT where work can be properly done.

 

Usually, I don’t keep versions I’m not satisfied with nor articles that went the wrong way. However, as it begins to accumulate, what I decided to do this time is: make a break with that B149 and starts it all over again from B150.

 

It’s essential that we understand how the PHYSICAL FRAME itself work, because it contains everything else. So, it has INCIDENCES on the rest. As a consequence, if we do understand the way it articulates, we will understand how objects inside it articulate.

 

The work to come is simplified and, overall, CORRECTED. It does bring interesting informations but I can’t tell by now if this will be enough to answer the main questions about “paranormal behaviors”. What I can say is that it sticks to the knowledge and understanding of physical laws we have today.

 

So, let’s go again, hoping this time we won’t have to go back.

 

Because I’m stupidly loosing time, the reader may begin to think this is all very confuse and loose patience because of this. And, to me, it’s frustrating anyway, because it stagnates…

 

Before continuing on anything else, I’ll like to examine that question of time in the quantum, because it’s a central point that extends to energy, since they are dual quantities.

 

Classically, the distinction between “space” and “time” is known since Einstein’s relativity as being linked to the SIGN of the diagonal components of the metrical 2-tensor of the physical frame. In plane Minkowski space-time, we have g_aa = -1 (a = 1,2,3) and g₄₄ = +1 for a “time-like interval” and g_aa = +1, g₄₄ = -1 for a “space-like interval”. Physically, this is required for the velocity of light in the vacuum to be guaranteed “invariant”, i.e. with the same value in all coordinate systems. It appears that, along the four main directions of space-time, the requested “surface element” must be of the form s² = (x⁴)² - S_a=1³ (x^a)² for the time-like formulation, or s² = S_a=1³ (x^a)² - (x⁴)² for the space-like one.

 

This all changes when we go to the quantum. In place of the four classical coordinates x^a (a = 1,2,3,4), we find for quantum coordinates:

 

(1)               x^a(ksi^a) = x^aexp(iksi^a)                                 (a = 1,2,3,4)

 

that’s four external x^as together with four internal ksi^as. So, even if the topology of the quantum space remains Riemannian because we need the physical vacuum to remain boson-like, the classical g_ab turns into the quantum:

 

(2)               g_ab(gam_ab) = g_abexp(igam_ab)            (a,b = 1,2,3,4)

 

and the symmetry of the quantized 2-tensor,

 

(3)               g_ba(gam_ba) = g_ab(gam_ab)

 

imposes that,

 

(4)               g_ba = g_ab  ,  gam_ba = gam_ab

 

so that BOTH the external AND the internal topologies are Riemannian.

 

The great advantage of (2) versus the classical metrical tensor is that component signs are now determined by the values taken by the INTERNAL metrical tensor, so that, in EACH direction a, g_aa(gam_aa) can give projections with either a positive or negative sign. As a direct consequence of this, we no longer require that the external tensor be of ALTERNATED signature, like at Minkowski. Right on the contrary, we have physical interest in having it FULLY EUCLIDIAN. So, let’s place ourselves in plane 4-space and set g_ab as the Kronecker delta:

 

(5)               g_aa = +1  ,  g_ab = 0                                     (a,b = 1,2,3,4; a <> b)

 

The off-diagonal components of gam_ab are out of the game, so that we can restrict to the four gam_aa. This gives a:

 

(6)               g_aa(gam_aa) = exp(igam_aa)                             (a = 1,2,3,4)

 

and is far enough to attribute various signs to the projection spaces. In particular:

 

(7)               gam_aa = pi (a = 1,2,3)  ,  gam₄₄ = 0  ->  time-like Minkowski metric

(8)               gam_aa = 0 (a = 1,2,3)  ,  gam₄₄ = pi  ->  space-like Minkowski metric

 

and many other possibilities, since we have a QUADRUPLE CONTINUOUS INFINITY OF CHOICES. So, when the metrical tensor explicitly depends on the coordinates, as in curved quantum space, we can even witness a CHANGE IN THE SIGN OF THE DIAGONAL COMPONENTS FROM ONE POINT TO ANOTHER. It follows that:

 

The notion of “time” LOOSES ALL PHYSICAL SIGNIFICANCE IN THE QUANTUM.

And, with it, the dual notion of ENERGY.

 

In quantum space, we can only talk about space and momentum.

 

This has interesting consequences on the behavior of SIGNALS. In the classical, we had a delay of order ct, due to the FINITE velocity at which a signal propagated. In the quantum, that term turns into c(khi)t(tau) = ctexp[i(khi + tau)]. c and t are always non-negative quantities. But khi and tau are SIGNED ones. So, when khi + tau = 2npi, n in Z, we find a delay of order ct; but when khi + tau = (2n+1)pi, we find a delay -ct, that is, an ADVANCE of order ct. And this is perfectly consistent. It has many possible interpretations: whether signal propagation is reversed (c -> -c), or time is (t -> -t), or many others as long as khi and tau satisfy khi + tau = (2n+1)pi. Remember that, in the classical, only x - ct spatial dependence were kept, because x + ct represented an INCIDENT signal hitting a given system of bodies. In the quantum, the interpretation is radically different. It all becomes a question of STATES. In x⁴(ksi⁴) = c(khi)t(tau), the INTERNAL position is ksi⁴ = khi + tau. It also defines the STATE associated with x⁴ = ct >= 0. An “advance” then becomes OPPOSITE IN PHASE TO A DELAY. There’s no “space or time reversal” any longer: these are all EXTERNAL PERCEPTIONS. In the classical solution to the 1-dimensional wave equation:

 

f(x - ct) + f(x + ct)

 

x, c and t were SIGNED quantities, needing f(x + ct) to be REJECTED because it didn’t represent a signal PRODUCED by the system. Signals propagating in the vacuum at a finite velocity c, there was NECESSARILY a delay between their emission and their observation and that delay was modeled by x - ct. So, “advanced signals” were rejected.

 

In the quantum,

 

f(phi)[x(ksi) - c(khi)t(tau)] + f(phi)[x(ksi) + c(khi)t(tau)]

 

nearly make a “REDUNDANCY” so that we can keep f(phi)[x(ksi) - c(khi)t(tau)] alone and get the same results, since signs depend on the INTERNAL. Thus, for:

 

(9)               (phi, ksi, khi, tau) = (0,0,0,0)  =>  f(x - ct)

 

we retrieve the classical delayed signal, but with

 

(10)           (phi, ksi, khi + tau) = (0,0,pi)  =>  f(x + ct)

 

we find the ADVANCED signal. Again, WITHOUT PERFORMING ANY EXTERNAL REVERSAL. This last signal is still PRODUCED by the quantum system, but externally observed, it LOOKS LIKE an “incident wave” coming from outside to hit the system… :)

 

The great lesson of complexification is to learn us that:

 

“POLARITIES” OR THE NOTION OF SIGNS IS AN INTERNAL MATTER.

 

It’s INTERNAL, it does NOT belong to the external…

 

Want a blatant example? The Newtonian electrostatic potential between two electric charges q and q’ is:

 

(11)           U(r) = qq’/(4pi)e₀r

 

Where e₀ (actually, epsilon₀) is the electrical “permittivity” (= conductivity) of the “classical vacuum” (i.e. the medium OUTSIDE the system of charges). Quantize q and q’ through complexification and write them in the planar representation:

 

(12)           q(theta)q’(theta’) = q₁(theta)q’₁(theta’) - q₂(theta)q’₂(theta’) +

+ i[q₁(theta)q’₂(theta’) + q₂(theta)q’₁(theta’)]

 

Now, set:

 

(13)           q(theta) = q + i(4pie₀k)^1/2m

 

where q and m are CLASSICAL charge and mass, respectively. What do you get?

 

(14)           tan(theta) = (4pie₀k)^1/2m/q

 

valid for ANY value of q and m… And what do you get in (12)? THE MINUS SIGN OF GRAVITATION IN THE REAL COMPONENT…:

 

(15)           q(theta)q’(theta’) = qq’ - (4pie₀k)mm’ + i(4pie₀k)^1/2(mq’ + m’q)

 

plus two charge-mass couplings in the imaginary part. Theta is zero OR PI for m = 0, and pi/2 OR 3PI/2 for q = 0… So, the minus sign in (12) DOES arise from i² = -1, which is a PURELY QUANTUM AND THEREFORE INTERNAL PROCESS… hence the presence of pi in the INTERNAL potential of gravity in the previous article.

 

If you stick to the classical, you find NO CLEAR EXPLANATION to the reversal of the sign between electrostatic and gravitostatic interactions.

 

If, instead, you quantize the electrostatic field IN SUCH A WAY that it unites the concept of charge with that of mass, that sign reversal appears NATURALLY…

 

The physical consequence is dramatically different: while you classically cannot find any accumulation of charges, you do find an accumulation of masses with same sign. But if you shift theta of pi/2, then (13) becomes iq - (4pie₀k)^1/2m and you find an accumulation of CHARGES while no accumulation of masses… This is simply because electric charges and masses are now treated on an equal footing.

 

I noticed a rather astonishing property of complex-valued quantities. Let’s consider again two quantum masses m(mu) and m’(mu’). Coupling them gives a resulting mass:

 

(16)           [m”(mu”)]² = m”²exp(2imu”) = m(mu)m’(mu’) = mm’exp[i(mu + mu’)]

(17)           m”² = mm’

(18)           mu” = ½ (mu + mu’)

 

the external mass m” is the geometric average of m and m’, while the internal mass is the arithmetic average of mu and mu’. This is already well-known. What’s new in terms of physical interpretation is this:

 

(19)           m(mu)m’(mu’) = m(mu’)m’(mu)

 

COUPLING QUANTUM QUANTITIES OF THE SAME KIND ENABLES THE EXCHANGE, WHETHER OF THEIR EXTERNAL COMPONENTS OR OF THEIR INTERNAL ONES (which amounts to the same).

 

We have no classical equivalent, because we lack internal variables. Coupled quantities need be of the same kind because coupling, say a mass with a velocity and exchanging their internal components is meaningless, as you can’t associate an external mass with an internal velocity and an external velocity with an internal mass.

 

This result easily generalizes to n quantum quantities of the same kind. Let’s take, for instance, two quantum lengths x¹(ksi¹) and x²(ksi²). Their coupling gives:

 

(20)           x¹(ksi¹)x²(ksi²) = x¹(ksi²)x²(ksi¹) = x¹x²exp[i(ksi¹ + ksi²)] = s²exp(2isig)

 

This time, exchange of internal variables is possible, because they are two internal lengths. Externally,

 

(21)           s² = x¹x²

 

is an area. Internally,

 

(22)           2sig = ksi¹ + ksi²

 

is HALF A PERIMETER (logarithmic correspondence as always). Take three x^a(ksi^a):

 

(23)           x¹(ksi¹)x²(ksi²)x³(ksi³) = x¹x²x³exp[i(ksi¹ + ksi² + ksi³)] = v³exp(3isti)

 

Externally,

 

(24)           v³ = x¹x²x³

 

is a volume. Internally,

 

(25)           3sti = ksi¹ + ksi² + ksi³

 

is again half the perimeter of a 3D internal volume (take a parallelepiped with sizes ksi^a and check). There are exactly 3! = 6 ways of exchanging the three ksi^as. By recurrence, we immediately see that:

 

There are n! ways of exchanging the ksi^as (a = 1,…,n) in the coupling:

 

(26)           x¹(ksi¹)…xⁿ(ksiⁿ) = v_nⁿexp(insti_n)

 

There’s only one n-dimensional external volume and it’s always NON-NEGATIVE:

 

(27)           v_nⁿ = x¹…xⁿ >= 0

 

and there’s a n-dimensional HALF-PERIMETER:

 

(28)           nsti_n = ksi¹ +…+ ksiⁿ

 

The quantity v_n schematically represents the size of a n-dimensional hypercube with a hyper-volume equivalent to x¹…xⁿ. The quantity sti_n schematically represents half of the perimeter of a n-dimensional parallelepiped, divided by the total number of its sides.

 

What becomes properly amazing once given a physical content is that internal substances of the same kind can exchange external substances and external substances of the same kind can exchange internal ones:

 

External or internal substances can be TRANSFERRED to another physical object of the same kind.

 

Now, whereas external amounts are never negative, internal ones can either be positive, null or negative. So, when they’re transferred, they are WITH THEIR SIGNS. When it comes to quantum fields, we have couplings like F₁(PHI₁)…F_n(PHI_n), each field depending on the four space variables x^a(ksi^a). Independent on those variables, the n fields can exchange their external or internal components. This is normal, after all, since a coupling means an INTERACTION between the fields… and, when two physical objects or more interact, they EXCHANGE THEIR INFORMATIONS, i.e. their CONTENTS.