Let’s now spend some time on DIMENSIONAL COUPLINGS. There basically are two kinds of coupling between spaces, the CARTESIAN PRODUCT and the TENSOR PRODUCT. The Cartesian product is an INTERACTIVE COUPLING, the tensor product is a coupling WITHOUT INTERACTION. If you take two 1-dimensional lines and you associate them under a Cartesian product, you obtain a 2-dimensional plane, within which your two generating lines INTERSECT (I’m really annoyed I can’t use special characters nor draw any figure because they can’t be seen the same way on different processors. Explanations would appear much clearer). Anyway. In Cartesian products, dimensions ADD, whereas they MULTIPLY in tensor products. So, let’s begin with that last product of spaces, for it’s the simplest one.

 

Let {X^[alpha(k)]}_k=1,…,n be a countable and finite set of spaces with quantum dimension:

 

(1)               D_k(alpha_k) = D_kexp(2ipialpha_k)  ,  1 =< k =< n < +oo , n and D_k in N*

 

The tensor product of these spaces gives a space X^(alpha) with dimension:

 

(2)               D(alpha) = Dexp(2ipialpha) = D₁(alpha₁)…D_n(alpha_n)

 

so that,

 

(3)               D = D₁…D_n > 0

(4)               alpha = S_k=1ⁿ alpha_k

 

IN NON-INTERACTIVE COUPLINGS, THE CLASSICAL DIMENSION IS THE PRODUCT OF THE CLASSICAL DIMENSIONS OF ALL FACTORS AND THE RESULTING SCALE FACTOR, THE SUM OF ALL SCALE FACTORS.

 

It immediately follows that:

 

(5)               alpha in Z  =>  D(alpha) = D > 0

(6)               alpha = p + ½ , p in Z  => D(alpha) = -D < 0

 

In particular, for n = 2:

 

(7)               D = D₁D₂ , alpha = alpha₁ + alpha₂

(8)               alpha₁ = alpha₂ = ½ => alpha = 1 => D(1) = D

 

THE NON-INTERACTIVE COUPLING OF TWO ANTI-UNIVERSES IN STATE 1 IS A UNIVERS IN STATE 1.

 

whereas,

 

(9)               alpha₁ = alpha₂ = ¼ => alpha = ½ => D(1/2) = -D

 

THE NON-INTERACTIVE COUPLING OF TWO UNIVERSES IN STATE 2 IS AN ANTI-UNIVERS IN STATE 1.

 

This is absolutely similar to the complex i² = -1. As expected, the rule of signs favors the negative over the positive.

 

Let’s now turn to the Cartesian product. This time, the X^[alpha(k)]s interact, so that quantum dimensions ADD and there appears INTERFERENCES. We don’t see such interferences between RIGID dimensions. The result is now a space X^(alpha) with quantum dimension:

 

(10)           D(alpha) = S_k=1ⁿ D_k(alpha_k) = S_k=1ⁿ D_kexp(2ipialpha_k)

 

As a result, the classical dimension D EXPLICITLY DEPENDS ON ALL SCALE FACTORS:

 

(11)           D² = S_k=1ⁿ D_k² + 2S_k=1^n-1S_l=k+1ⁿ D_kD_lcos(alpha_k - alpha_l)

     = [S_k=1ⁿ D_kcos(alpha_k)]² + [S_k=1ⁿ D_ksin(alpha_k)]²

 

and the resulting scale factor EXPLICITLY DEPENDS ON ALL CLASSICAL DIMENSIONS:

 

(12)           alpha = Arctan{[S_k=1ⁿ D_ksin(alpha_k)]/[S_k=1ⁿ D_kcos(alpha_k)]} mod pi

 

It follows that, for alpha_k+1 = alpha₁ + ½ k:

 

cos(alpha_k+1) = (-1)^kcos(alpha₁) , sin(alpha_k+1) = (-1)^ksin(alpha₁)

 

implying,

 

D² = [S_k=1ⁿ (-1)^k-1D_kcos(alpha₁)]² + [S_k=1ⁿ (-1)^k-1D_ksin(alpha₁)]²

    = [S_k=1ⁿ (-1)^k-1D_k]²[cos²(alpha₁) + sin²(alpha₁)]

    = [S_k=1ⁿ (-1)^k-1D_k]² = [S_k=1ⁿ (-1)^kD_k]²

 

and thus,

 

(13)           alpha_k+1 = alpha₁ + ½ k  =>  D = |S_k=1ⁿ (-1)^kD_k|

 

When alpha_k+1 = alpha₁ + ½ (k + ½):

 

cos(alpha_k+1) = (-1)^k+1sin(alpha₁) , sin(alpha_k+1) = (-1)^kcos(alpha₁)

 

leading to the same result,

 

(14)           alpha_k+1 = alpha₁ + ½ (k + ½)  =>  D = |S_k=1ⁿ (-1)^kD_k|

 

As the sum of classical dimensions is an ALTERNATING one, the resulting classical dimension D can be found ZERO:

 

(15)           S_k=1ⁿ (-1)^kD_k = 0  <=> D = 0

 

For instance, when n = 2, we find:

 

D = |-D₁ + D₂| = |D₁ - D₂|

 

so that, when D₂ = D₁, the two dimensions CANCEL ONE ANOTHER. This is because the interference between them is DESTRUCTIVE. When interferences are FULLY CONSTRUCTIVE:

 

(16)           alpha_k+1 = alpha₁ + k  =>  D = S_k=1ⁿ D_k

 

and we simply retrieve the classical result.

 

It could be worth noticing that the Cartesian QUOTIENT of two spaces with POSITIVE dimensions is formally equivalent to the Cartesian PRODUCT of one space with a POSITIVE dimension and one space with a NEGATIVE dimension. Reasoning this way, we would even be able to define a TENSOR quotient of spaces, something that is ill-defined in the classical, as the concept of “negative dimensions” there does not exist (and is not even constructible, since one argues that “one can’t be smaller than a point”).

 

 

 

 

 

First and foremost, let me explain once again how I consider the overall context: to me, the “way into” paranormal phenomena is shown by Near Death Experiments (NDEs). In order to understand them, it’s just impossible to bet on a SINGLE body. We need TWO bodies, as the biological one stands INERT and UNCONSCIOUS. And, anyway, experiencers all reported they felt they still had a “body”, but “of a different kind”.

 

It happens that physics offers that possibility. The developments of quantum physics showed that space and time themselves need be DOUBLED in order to account for the unification of the physical world as we observed it so far. I want to insist once again on this point: this is DICTATED to us by OBSERVATIONS and FACTS. We do not try to “force the laws of the world inside our understanding” into frames “suitable to us”. That’s why it took nearly a century to get to these results: because we had to adapt OUR well-anchored conceptions to the facts. The difference with supersymmetric theories here is that I object the recourse to a mirror geometry because it hardly matches with the fact that both space and time are typically NOT material: they have nothing “substantial”. I don’t discuss the fact that mirror symmetry is the proper frame that groups fermions and bosons together but, to me, that frame is more MATHEMATICAL, i.e. ABSTRACT, than PHYSICAL: it nicely applies to physical OBJECTS but, once applied to the frame ITSELF (which is ALSO part of quantization), it no longer matches. Now, we can’t pretend to have a COMPREHENSIVE theory of the quantum world if we keep on EXCLUDING the frame from the rest. And the fundamentally NON material aspect of space and time tells us that they rather obey the Bose-Einstein quantum statistics. Since a DIRECT link was established, not only between spin and statistics, but also between spin and GEOMETRY, the Bose-Einstein stat is related with a RIEMANNIAN geometry, while the Fermi-Dirac stat is related with a GRASSMANNIAN geometry. The “choice” is therefore imposed: we need keep a Riemannian geometry, while we don’t exclude for as much that REPRESENTATIONS of reality can be ABSTRACTLY MODELED calling for supersymmetric extensions using BOTH kinds of geometries (just like the skew-symmetric topology of the phase space was used in “classical physics” to model motions inside the symmetric topology of the configuration space).

 

Well, it so happens that the quantization of the physical frame actually does NOT concern variables, but THE PHYSICAL DIMENSIONS THEMSELVES. It may, at first glance, look a rather surprising result, but it soon appears natural, after all: if dimensions are quantized, so are both space and time, automatically, as everything inside that frame. :)

 

In classical physics, the physical space is 3-dimensional: D = 3 and we have three components x^a for each point of space, a = 1,2,3. However, x^a can mathematically be viewed also as an APPLICATION from the FINITE set {1,…,D}, D in N*, D < oo, into the real line:

 

(1)               x: {1,…,D} -> R  ,  a -> x(a) = x^a

 

In plane space endowed with a globally-defined metrical 2-tensor g_ab, the dual of x^a is:

 

(2)               x_a = S_b=1^D g_abx^b = g_abx^b                   (Einstein’s convention)

 

and the contracted product of x^a with x_a is a scalar,

 

(3)               s² = x^ax_a

 

that NO LONGER DEPENDS ON THE INDEX a.

 

It should be quite obvious that the rules of tensor calculus can be extended, not only to the infinite dimension, but to the continuum. We then get an application:

 

(4)               x: R -> R  ,  a -> x(a) = x^a

 

The only possible “obstruction” is purely CONCEPTUAL: when a is negative, what would “negative dimensions” represent? That question will be solved in the quantum. For the rest, only the departure set differs from (1) to (4). Summations are then replaced with integrals over indices. Thus, (2) becomes:

 

(5)               x_a = S_R g_abx^bdb = g_abx^b

 

extending, for convenience, the Einstein summation convention to the continuum.

 

When going from the classical to the quantum, the COMPONENT INDEX IS QUANTIZED, becoming a COMPLEX-VALUED QUANTITY:

 

(6)               a(alpha) = aexp(2ipialpha)

 

with a in {1,…,D} as before, but now alpha in R, rendering the frame both complex-valued and infinite-dimensional at the same time. Since a runs from 1 to D, the first continuous series of components will be represented by 1(alpha) = cos(2pialpha) + isin(2pialpha) and the last continuous series of components, by D(alpha) = Dcos(2pialpha) + iDsin(2pialpha), so that, at EACH scale alpha, we’ll find TWO generally NON-INTEGER dimensions, D₁(alpha) = Dcos(2pialpha) and D₂(alpha) = Dsin(2pialpha), except at the four “poles”:

 

alpha = 0 (“east pole”), D₁(0) = D, D₂(0) = 0, “classical, state 1”;

alpha = ¼ (“north pole”), D₁(1/4) = 0 , D₂(1/4) = D, “classical, state 2”;

alpha = ½ (“west pole”), D₁(1/2) = -D, D₂(1/2) = 0, “classical, ANTI-state 1”;

alpha = ¾ (“south pole”), D₁(3/4) = 0 , D₂(3/4) = -D, “classical, ANTI-state 2”;

 

(modulo ¼). For instance, at scale alpha = 1/6, one finds D₁(1/6) = ½ D and D₂(1/6) = 3^1/2D/2, so that CLASSICAL observers in both states will observe the effects of SCALINGS over what they usually perceive as “THE classical dimension D”: in state 1, D will seem to be reduced a factor ½, while in state 2, it will seem to be reduced a factor 3^1/2/2. A QUANTUM observer won’t observe this: for such an observer, there will be TWO DIMENSIONAL DATAS, D (a non-negative integer) and alpha (a continuous number), so that all dimensions will OSCILLATE a continuous way between the extremal values -D and +D, back and forth all the time. It now becomes obvious that, if dimensions are the first ones to oscillate, than everything inside them does…

 

Application (1) becomes an application:

 

(7)               x : C -> C , a(alpha) -> x[a(alpha)] = x^a(alpha) = [x^{acos(2pialpha)} , x^{asin(2pialpha)}]

 

generating, point to point, a complex space X^(alpha) with complex dimension D(alpha), canonically decomposing into two REAL spaces: X₁^(alpha) with (local) coordinates x^acos(alpha) and X₂^(alpha) with (local) coordinates x^asin(alpha). X₁^(alpha) has dimension D₁(alpha) and X₂^(alpha) has dimension D₂(alpha). The continuous set of ALL X^(alpha)s when alpha runs R make that “quantum space” X = [X^(alpha)]_{alpha in R}. It’s actually a SINGLE space, because alpha is a SCALE FACTOR. So, different alphas don’t give distinct “space folds” as could be thought, leading to no “multiverse picture”. What (7) says is that, instead of having RIGID (and strictly positive!) dimensions as in the classical, we have PULSING dimensions with the possibility of taking NEGATIVE values.

 

No quantum dimension D(alpha) can vanish unless its classical part D does. But both D₁(alpha) and D₂(alpha) can, at poles. So, when one dimension reaches zero, we set the rather obvious convention:

 

(8)               x⁰ = 0

 

(no dimension => no space).

 

You’ll also notice that we can now use THE SAME VARIABLE x IN BOTH STATES, without any risk of confusion, since COMPONENTS ARE DIFFERENT. :) Even for alpha = 1/8, we find D₁(1/8) = D₂(1/8) = 2^-1/2, only indicating that X₁^(1/8) and X₂^(1/8) have same dimension 2^-1/2D, but keep distinct coordinate sets.

 

As,

 

(9)               x^{a(alpha + pi)} = x^-a(alpha)

 

we have an interpretation that was hard to find in the real-valued context:

 

A “NEGATIVE DIMENSION” IS A POSITIVE ONE, BUT IN PHASE OPPOSITION.

 

We said that the phase angle alpha also plays the role of a (unit-free) SCALE FACTOR. Let’s place ourselves in state 1 and introduce a CLASSICAL observer. When alpha = 0, our observer sees no “contraction”, no “scale (or ‘lens’) effect”, so that distances and lengths match with their classical values, while state 2 simply does NOT exist for this observer (it’s reduced to a mere point, so it does not exist). Let’s do the same in state 2, now (alpha = ¼). Similar conclusion for an observer in that state: this time, state 1 does not exist, while state 2 looks classical. When alpha = ½, we find an “anti-state 1” with an “anti-observer” and, when alpha = ¾, an “anti-state 2” with an “anti-observer”.

 

EVERYWHERE ELSE, we find NON-INTEGER DIMENSIONS IN BOTH STATE 1 AND STATE 2, as a result of SCALING. It remains that the CLASSICAL dimension is everywhere FINITE and equal to D.

 

The algebra of tensor calculus easily extends to the quantum dimension. The “plane” situation is described with a globally-defined still symmetric 2-tensor g_{a(alpha)b(beta)} and the dual of x^a(alpha) is:

 

(10)           x_a(alpha) = S_b(beta) g_{a(alpha)b(beta)}x^b(beta) = S_b=1^DS_R g_{a(alpha)b(beta)}x^b(beta)dbeta

 

what we’ll simply write,

 

(11)           x_a(alpha) = g_{a(alpha)b(beta)}x^b(beta)

 

You can see the need for introducing a DIFFERENT phase to EACH index: using the same phase alpha for all indices wouldn’t match integration in (16). And this is logical: in the classical, we had two different indices a and b; in the quantum, we have to different indices a(alpha) and b(beta), with:

 

(12)           a(alpha) = b(beta)  <=>  a = b  AND  alpha = beta (mod 2pi)

 

The scalar:

 

(13)           s² = x^a(alpha)x_a(alpha) = g_{a(alpha)b(beta)}x^a(alpha)x^b(beta)

 

depends on NO INDEX.

 

SCALARS REMAIN THE SAME AT ALL SCALES.

 

Another way of putting this is to say that:

 

SCALARS ARE SCALE-INVARIANT.

 

Indeed, following (10), we have:

 

s² = S_a=1^DS_b=1^DS_RS_R g_{a(alpha)b(beta)}x^a(alpha)x^b(beta)dalphadbeta

 

so that s² depends on neither a nor b, alpha or beta.

 

The INVERSE of g_{a(alpha)b(beta)} is g^{a(alpha)b(beta)}. When applied to x_a(alpha), it must give x^a(alpha) back:

 

x^a(alpha) = g^{a(alpha)b(beta)}x_b(beta) = S_b=1^DS_R g^{a(alpha)b(beta)}x_b(beta)dbeta

 

Applying (10):

 

x^a(alpha) = S_b=1^DS_R g^{a(alpha)b(beta)}dbeta{S_c=1^DS_R g_{b(beta)c(khi)}x^c(khi)dkhi}

= S_b=1^DS_c=1^DS_RS_R g^{a(alpha)b(beta)}g_{b(beta)c(khi)}x^c(khi)dbetadkhi

 

Now, x^a(alpha) must also be equal to:

 

x^a(alpha) = d^a(alpha)_b(beta)x^b(beta) = S_b=1^DS_R d^a(alpha)_b(beta)x^b(beta)dbeta

 

and considering d^a(alpha)_b(beta) as an application d[a(alpha),b(beta)], we meet the requirement for,

 

(14)           d^a(alpha)_b(beta) = d^a_bdelta(alpha - beta)

 

where d^a_b is the discrete and finite Kronecker delta and delta(.) is Dirac’s delta function. Indeed,

 

x^a(alpha) = S_b=1^DS_R d^a_bdelta(alpha - beta)x^b(beta)dbeta

= S_b=1^D d^a_b{S_R delta(alpha - beta)x^b(beta)dbeta}

= S_b=1^D d^a_bx^b(alpha) = S_R delta(alpha - beta)x^a(beta)dbeta

 

It follows that:

 

S_b=1^DS_c=1^DS_RS_R g^{a(alpha)b(beta)}g_{b(beta)c(khi)}x^c(khi)dbetadkhi = S_b=1^DS_R d^a(alpha)_b(beta)x^b(beta)dbeta

  = S_c=1^DS_R d^a(alpha)_c(khi)x^c(khi)dkhi

 

and thus, that:

 

S_b=1^DS_R g^{a(alpha)b(beta)}g_{b(beta)c(khi)}dbeta = d^a(alpha)_c(khi)

 

or else,

 

(15)           g^{a(alpha)b(beta)}g_{b(beta)c(khi)} = d^a(alpha)_c(khi) = d^a_cdelta(alpha - khi)

 

We observe that, when alpha = khi (mod 2pi), we have a “pulse” driven by Dirac’s function, that is, we have a “signal response” to the synchronization of alpha and khi. This is the continuous version of d^a_c = 1 for c = a. Besides:

 

g^{a(alpha)b(beta)}g_{b(beta)a(alpha)} = d^a(alpha)_a(alpha) = S_a=c=1^DS_R d^a_cdelta(alpha - khi)d(alpha - khi)

  = (S_a=c=1^D d^a_c)S_R delta(alpha - khi)d(alpha - khi)

 

so that,

 

(16)           g^{a(alpha)b(beta)}g_{a(alpha)b(beta)} = d^a(alpha)_a(alpha) = d^a_a = D

 

since S_R delta(alpha - khi)d(alpha - khi) = +1. We only retrieve the fact that the CLASSICAL dimension, D, is a SCALAR and, as such, must be both index- and scale-invariant.

 

ALL THE REST OF TENSOR CALCULUS PROCEEDS THE SAME AS USUAL, PROVIDED WE USE THE EXTENDED SUMMATIONS.

 

We described earlier the quantum universe as a naturally pulsing one, where dimensions are allowed to contract or dilate a cyclic way and “anti-dimensions” appear opposite in phase to “positive” ones. ELASTICITY, i.e. the property that the physical universe can “stretch”, is something else, it’s a characteristic of CURVED spaces. I want to emphasize that, even in a PLANE situation, the quantum universe still “pulses”, because its dimensions are subject to a scale factor.

 

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.

 

The reader understood it: after several attempts to get rid of i, I finally reintroduced it, because I found it a physical content.

 

In this article, we’re going to talk about MASS COUPLING, because there are important consequences.

 

As any other physical quantity, a QUANTUM MASS is a complex-valued quantity:

 

(1)               m(mu) = mexp(imu)

 

It measures the quantity of quantum substance within a delimited quantum volume. As an amplitude, the external mass m is ALWAYS a non-negative quantity. This means that, in the quantum, we will ALWAYS deal with SUBSTANCE:

 

THERE’S NO “ANTI-SUBSTANCE” IN THE QUANTUM.

 

Wow… will immediately hurt the quantum physicist… K Not that much actually. We now know that the SIGN of a quantity all depends on the INTERNAL factor. The notion of “anti-substance with a positive energy” arose from the classical vision, where signs are arbitrary. People preferred to talk about this rather than about “substance with negative energy (or mass)”, because such substance was not observed at levels immediately higher than that of “elementary” particles (today, it’s possible to create “anti-atoms”, but they remain highly unstable and must be kept inside magnetic fields so as to avoid interacting with any atoms). The internal mass mu of a quantum body is perceived as a “mass state” by a CLASSICAL observer. In other words, in some 1-dimensional ISO-space, we have that mass representation made of a pair (m,mu) of masses where m is classically perceived as being “the” mass of a quantum object, while mu is classically perceived as representing the STATE in which m is. As a result, when mu = 0, m(0) = m > 0 appears perfectly logical, whereas mu = pi gives m(pi) = -m < 0 or “anti-substance”. What we instead have is actually some substance (m > 0), but in a phase mu = pi OPPOSITE to the phase mu = 0. We also find much more: we find that m(pi/2) = im and m(3pi/2) = -im are no longer “abstract”, but represent PURELY QUANTUM MASSES, opposite in phase. All other values of mu are matters of PROJECTIONS. We have a first projection m₁(mu) = mcos(mu): that’s what was previously assumed to be “classical”. We have a second projection m₂(mu) = msin(mu) that was REJECTED in the classical, but INCLUDED in the quantum. Thus, m₁(mu) is the quantity a classical observer will perceive, while im₂(mu) is the quantity a PURELY QUANTUM OBSERVER will perceive. The presence of i is important here: again, as a REAL-valued quantity, m₂(mu) is CLASSICAL, whereas im₂(mu) is purely quantum, because i is.

 

Consequently, as mu changes, so do m₁(mu) and m₂(mu), which are SIGNED quantities, and this is absolutely normal, since m(mu) CHANGES STATE. The important thing is that m does NOT change, so that:

 

EXTERNALLY, we keep THE SAME AMOUNT OF SUBSTANCE.

What is likely to change is the INTERNAL amount of substance.

 

It would be perfectly possible to attribute an object to EACH value (m,mu), but this would lead to a plethora of particles. We can drastically reduce that number considering that a given object has CONSTANT external mass in VARIABLE STATES. This also has the advantage of unifying the former concepts of “substance” and “anti-substance”: it’s well-known indeed that a particle and its “anti-partner” have SAME MASS AT REST. Yes, indeed: same EXTERNAL mass at rest… :) but DIFFERENT INTERNAL MASSES. And that’s what enables us to distinguish them, at least on the mass level.

 

Before quantizing, we now have to ask ourselves why the gravitational interaction that couples masses has a classical potential OPPOSITE IN SIGN with the electromagnetic interaction that couples electric charges. Potentials are SCALAR quantities, so we cannot invoke any space orientation. There’s another reason. Of course, this was set so because it was observed that two electric charges with same sign REPULSE, whereas two masses with same sign ATTRACT. We now need to understand why that reversal. The Newtonian G-potential between a mass m’, acting like the source, and an incident mass m, is given by:

 

(2)               U(r) = -km’m/r

 

where r is the distance between the two masses. Such a potential immediately eliminates all possibility of self-interaction, since it diverges near r = 0. Now, r is assumed to be a NON-NEGATIVE quantity, since it represents a RADIAL distance. However, Penrose admitted the possibility of NEGATIVE values for r, synonymous of a repulsion “beyond a central singularity” (crazy how the fact of ARBITRARILY defining sign can lead to multiple interpretation attempts…). It occurs that we can rewrite U(r) this way:

 

(3)               U(r,pi) = (km’m/r)exp(ipi)

 

We didn’t change anything. But we enlighted a SPATIAL STATE rho = pi, while the AMPLITUDE of the potential becomes:

 

(4)               U(r,0) = km’m/r > 0

 

a NON-NEGATIVE QUANTITY… In other words, what we just did was to REFORMULATE the CLASSICAL observation (2) a QUANTUM way. Instead of saying “gravity is attractive between two masses with same sign”, we say:

 

EXTERNAL gravity is ALWAYS REPULSIVE and, ACCORDING TO THE STATE THE G-POTENTIAL IS IN, we’ll have attractions or repulsions.

 

The question of mass signs is solved complexifying (2). U(r) turns:

 

(5)               U(UPS)[r(rho)] = -k(kap)m’(mu’)m(mu)/r(rho)

 

giving an external potential,

 

(6)               U(r) = km’m/r

 

just like (4), but with NOTHING BUT NON-NEGATIVE QUANTITIES, and an internal potential

 

(7)               UPS(rho) = kap + mu’ + mu + pi - rho

 

The minus sign in (5) BELONGS TO THE INTERNAL. It’s a pi-shift. kap is an internal parameter required because nothing allows us to assert that the constant of physics we, as classical observers, measure in the vacuum, are truly “universal”, i.e. the same in all quantum states. CLASSICALLY universal, they are; QUANTUM universal is nothing for granted at all. Asserting this would be pure speculation for the time being.

 

Look at the form of the internal G-field: it’s GLOBAL (independent of r) and LINEAR in rho.

 

Externally, the Newtonian static G-field is a typically DECONFINED FIELD.

Internally, it is a CONFINING FIELD.

 

At the critical distance:

 

(8)               rho_c = kap + mu’ + mu + pi  =>  UPS(rho_c) = 0

 

the internal field just vanishes. At all other distances, it grows in absolute value with the internal distance.

 

(9)               rho = 0  =>  UPS(0) = rho_c

 

So, even in the worst case where we would fix kap, mu’ and mu to zero, we would still find the non-zero value UPS(0) = pi: this is precisely where the minus sign comes from, in the classical model.

 

The external potential (6) is clearly a potential WALL, that is, schematically, a potential barrier with unlimited height at r = 0. This now represents the shortest external distance one can find. It can no longer be prolonged to negative values.

 

Opposite to U(r), which is now strictly positive and only asymptotically zero, UPS(rho) can be either positive, negative or zero. We’ll have:

 

(10)           rho < rho_c  =>  UPS(rho) > 0  =>  INTERNAL REPULSION

(11)           rho = rho_c  =>  UPS(rho) = 0  => LIBRATION POINT

(12)           rho > rho_c  =>  UPS(rho) < 0  =>  INTERNAL ATTRACTION

 

What’s interesting in the Newtonian static potential is that, even once quantized, the external field only depends on external quantities and the internal field, on internal ones. This is obviously far from being that simple for other field distributions. That kind of field thus gives us a quite good idea of the mechanisms at play. We can see that the functioning of the internal G-field is RADICALLY different from that of the external one, because the correspondence between the two is LOGARITHMIC. So, what were external products turn into internal sums and internal products turn into external POWERS (exponentiation).

 

Let set again kap, mu’ and mu to zero, as in the classical. We’ll find internal attraction at internal distances rho GREATER than (2n+1)pi, where n is a non-negative integer. It means that, for rho = 0, we have NO CHANCE to find assemblies of internal substance in this model. At the libration points rho = (2n+1)pi, we have those assemblies of external substances. These are the only situations where substantial assemblies are possible. It shows that, even in such a simple model, substantial assemblies are everything but granted. This is because the notions of “attraction” and “repulsion” relate to a SIGN and become meaningless in the quantum, where complex-valued quantities have NO DEFINITE SIGN.

 

Aside of the potential energy, let’s now introduce the kinetic one, K(KAP). The total energy of the system will be:

 

(13)           H(ETA) = K(KAP) + U(UPS)

 

Externally:

 

(14)           H² = K² + U² + 2KUcos(KAP - UPS)

 

Because of interferences, external energies are generally NOT additive quantities.

 

They turn to be only for KAP - UPS = 2npi, n in N, in which case H = |K + U| and KAP - UPS = (2n+1)pi, in which case, H = |K - U|. Everywhere else:

 

(15)           0 =< |K - U| =< H =< |K + U|

 

and, as K and U are >= 0, we have K + U = 0 only for K = U = 0. When K = U, H can reach zero. But:

 

When K is different from U, the LOWEST accessible energy threshold is |K - U| > 0.

 

Compare with the classical, where H = K + U, with K and U SIGNED quantities…

 

The total INTERNAL energy now:

 

(16)           ETA = Arctan{[Ksin(KAP) + Usin(UPS)]/[Kcos(KAP) + Ucos(UPS)]}     (mod pi)

 

It will vanish along:

 

(17)           Ksin(KAP) + Usin(UPS) = 0

 

while reaching pi/2 for:

 

(18)           Kcos(KAP) + Ucos(UPS) = 0

 

These are two Fresnel-like equations. Look at (17): it doesn’t require any of the four variables involved to be zero for ETA to reach zero. On the contrary, if, say KAP = 0 (mod pi), then we need U = 0 or UPS = 0 (mod pi); if K = 0, same. And similar results for (18). Conclusion:

 

One can have a QUANTUM kinetic energy and a QUANTUM potential energy and still find a CLASSICAL total energy or a PURELY QUANTUM one… :|

 

In space relativity, K is classically defined as K = ½ mv², with v the RESULTING velocity. In the quantum,

 

(19)           K(KAP) = ½ m(mu)[v(sti)]²

(20)           K = ½ mv² >= 0

(21)           KAP = mu + 2sti

 

(do NOT confuse with the kap of Newton’s gravitational constant!)

 

We have a LINEAR progression internally vs a PARABOLIC progression externally. The other significant difference is that KAP can turn negative. If we have KAP = UPS, we immediately find ETA = KAP = UPS, but if we have K = U, we only find:

 

Exp(iKAP) + exp(iUPS) = 2exp[i(KAP + UPS)/2]cos[(KAP - UPS)/2]

 

leading to,

 

(22)           H = 2K|cos[½ (KAP - UPS)]|  ,  ETA = ½ (KAP + UPS)                       (K = U)

 

And if we have K = -U (which NO LONGER corresponds to the mechanical equilibrium in the external, because the additive property of energy is lost), from:

 

Exp(iKAP) - exp(iUPS) = 2exp[i(KAP + UPS + pi)/2]sin[(KAP - UPS)/2]

 

we find,

 

(23)           H = 2K|sin[½ (KAP - UPS)]|  ,  ETA = ½ (KAP + UPS + pi)                 (K = -U)

 

Thus, in the two cases K = U and K = -U, the total internal energy is a LINEAR SUPERPOSITION of the internal kinetic energy and the internal potential energy. In all other cases, it’s not.