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.

 

From the very end of the 19^th century to, say, the early 1920s, the quick development of spectrometric technics enabled physicist to test both the corpuscular nature and the wavy nature of atomic matter as well as the electromagnetic interaction, which was the only accessible one at that time in laboratories. One series of experiments enlighted the corpuscular behavior and another series, the wavy behavior. As the two series were about the very same particles (mostly electrons and photons), the conclusion was unavoidable: for some kind of reasons that didn’t clearly show at the macroscopic level, particles behaved BOTH as corpuscles AND waves. This was mathematically formalized by Louis de Broglie in 1924 under the name of “wave-corpuscle duality”. Since then, ALL observations confirmed that duality, which became observable at large scales once lasers and other condensed states went better mastered. Today, every astrophysicist learns, from college, that “dead” stars are made of condensed matter and show quantum properties.

 

The question is still opened for space-time itself, not really because of the complicated Einstein’s model for gravity, but because gravity is an extremely weak forces. But progresses have been made in recent years. Still, we’re not allowed yet to ASSERT that space-time also has that quantum duality, because no experiment revealed it so far. However, what we call “space” around us is that “emptiness of substance”. In physics, it’s nothing but a VACUUM STATE. Classically, that vacuum is zero, implying that space should be plane. Active discussions are still on about this, because Einstein’s CURVED space(-time) also show a vacuum state (outside sources) and this vacuum should no longer be zero. Anyway, we KNOW that the QUANTUM vacuum CANNOT be zero, because this has been observed numerous times. In fact, the quantum vacuum is even COMPRESSIBLE… Now, from the viewpoint of quantum statistics (the distributions of particles), space-time is expected to follow the Bose-Einstein stat, simply expressing the fact that it’s a fundamentally NON-substantial vacuum. If this wasn’t the case, the whole universe surrounding us would be full of matter. And what we observe is exactly the OPPOSITE: the universe is full of VACUUM. We can even explain why: because the vacuum precisely enables THE STABILITY OF MATERIAL ASSEMBLIES… Our solar system, for instance, is stable BECAUSE the sun and planets are separated with large vacuum areas. Most atoms are stable BECAUSE the nucleus is separated from the first layer of electrons with a large vacuum (to that scale). Etc. Vacuum states play a fundamental role, not only because they represent the lowest energy levels, but also because they stabilize material assemblies.

 

So, space-time being after all nothing else but a physical vacuum, its fundamentally non-substantial nature relates it to bosons, not fermions. If it’s to be quantized as all the rest, then its mathematical description must go from real to complex-valued quantities. Real quantities are for the classical description. Complex quantities are for the quantum description. More precisely, in the classical, the use of complex numbers is a mere ABSTRACT tool to make calculations much easier, but what is kept in the end is ALWAYS the real part of the result. In the quantum, on the contrary, we cannot do otherwise than keeping the IMAGINARY parts, ALL ALONG, because the wavy behavior of objects IMPOSES us to maintain the sine components or the results would just don’t match observations. It’s Hamilton’s “optical-mechanical analogy”: the equation for the mechanical (i.e. “corpuscular”) path of a system is in all points analogue to the equation for the OPTICAL path of a signal… That’s what led to the discovery of “matter waves” and to Schrödinger’s “wavefunction”.

 

Thus, in order to describe “quantum space(-time)” in agreement with the wave-corpuscle duality, we have no other choice but to complexify its coordinate systems. This is replacing the classical POINT x with a QUANTUM CIRCLE x(ksi) = xexp(iksi), where the initial x, off its arbitrary sign, now only make the AMPLITUDE of the “quantum coordinate position” x(ksi). That amplitude is known as always being a non-negative quantity, so the sign is now COMPLETELY DEFINED by the angle ksi: when ksi = 2npi, where n is in Z, we find a positive sign; when ksi = (2n+1)pi, a negative sign.

 

Angles in quantum spaces completely define their orientation.

 

There’s no arbitrary anymore. The quantization of x reveals the existence of a SECOND SET OF VARIABLES, ksi, and ksi has NO PHYSICAL UNIT. This is important, because it makes them UNIVERSAL, which is not the case for classical variables. What now enables us to distinguish them is their AFFILIATION with a dimensioned classical quantity: here, ksi is affiliated with x, measured in meters. So, we now have two kinds of COMPLETELY INDEPENDENT variables: x is an “EXTERNAL” variable, ksi is an “INTERNAL” variable. Together, they make a “QUANTUM” variable. The real dimension is doubled, but the COMPLEX dimension remains the same. It follows that the picture of the world is not that of “additional dimensions” but, instead, that of dimensions TOGETHER WITH THEIR STATES: the set (x,ksi) means we have one physical dimension IN THE PHYSICAL STATE ksi. Classically, ksi = 0 or pi. These are the only classically-allowed values. In the quantum, ksi needs not be a limited variable, because both cos(.) and sin(.) are bounded functions of their argument. So, ksi can take any value along the real line, cos(ksi) and sin(ksi) will ALWAYS remain between -1 and +1. This is very important for what will follow. Usually, we take ksi in [0,2pi[. Actually, ksi does not require to be bounded.

 

What happens when we complexify variables, parameters and functions and still want to apply the Newton laws of motion?

 

First, let’s introduce that “quantum differential” d(delta), which is nothing else but the complexified differential. When applied to a “quantum time” t(tau), it must give the same result as in traditional complex calculus, that is:

 

(1)               d(delta)t(tau) = d[texp(itau)] = exp(itau)(dt + itdtau) = dtexp(ideltatau)

 

However, the second writing is improper, because d was first introduced in the frame of REAL analysis. The polar expression for the result is:

 

(2)               d(delta)t(tau) = (dt² + t²dtau²)^1/2exp{i[tau + Arctan(dtau/dt)]}

 

so that,

 

(3)               dt = (dt² + t²dtau²)^1/2 >= 0

(4)               deltatau = tau + Arctan(dtau/dt)

 

Then, let’s write Newton’s equations of motion for a constant quantum mass m(mu):

 

(5)               m(mu)a(alpha)[t(tau)] = F(PHI)[t(tau)]

 

We do not consider a FIELD force F(PHI) for the time being, not to complicate the debate from the start. The acceleration of the mass is:

 

(6)               a(alpha)[t(tau)] = [d²(delta)/d(delta)t(tau)²]x(ksi)[t(tau)]

 

Well, surprisingly enough, it happens that it is more convenient to work from the INTEGRAL version of Newton’s law rather than with the usual second-degree ODE. The velocity of the mass is:

 

(7)               v(stigma)[t(tau)] = [m(mu)]^-1S₀^t(tau) F(PHI)[t’(tau’)]d(delta)t’(tau’) + cte

 

The position of that mass will therefore be:

 

(8)               x(ksi)[t(tau)] = S₀^t(tau) v(stigma)[t’(tau’)]d(delta)t’(tau’) + cte

= [m(mu)]^-1S₀^t(tau){S₀^t’(tau’) F(PHI)[t”(tau”)]d(delta)t”(tau”)}d(delta)t’(tau’) +

U.M.

 

where “U.M.” stands for Uniform Motion. It’s clear from (2) and:

 

F(PHI)[t(tau)] = F(t,tau)exp[iPHI(t,tau)]

 

that, in general, the motion x(t,tau) and the motion ksi(t,tau) will INTIMATELY BE INTRICATED. x(t,tau) expresses the move through EXTERNAL space (variable x), while ksi(t,tau) expresses the move through INTERNAL space (variable ksi), that is, FROM ONE SPACE STATE TO ANOTHER. Both depend on external time t and its state tau, in the general case.

 

Let’s look at what happens when we set:

 

(9)               t = t₀ = cte  ,  x(t,tau) = x₀ = cte

 

Then, x(ksi) = x₀exp(iksi) and x(ksi)[t(tau)] = x₀exp[iksi(t₀,tau)] = x₀exp[iksi₀(tau)]:

 

EXTERNALLY, NOTHING HAPPENS, THE SYSTEM LOOKS STEADY AND TIME IS FROZEN.

 

Internally, this is not exactly the same sound. Equation (8) gives:

 

x₀exp[iksi₀(tau)] = m^-1t₀²exp(-imu)S₀^t(tau){S₀^t’(tau’) F₀(tau”)exp[iPHI₀(tau”)]exp(itau”)dtau”}

exp(itau’)dtau’

 

Let’s simply a little bit more, just to make ourselves an idea about the internal move:

 

(10)           F₀(tau) = F₀(0) = cte  ,  PHI₀(tau) = (n - 1)tau + PHI₀(0)  ,  n in Z - {-1,0}

 

Calculation is explicit and easy and gives:

 

(11)           exp[iksi₀(tau)] = K₀exp{i[PHI₀(0) - mu]}{exp[i(n+1)tau] - (n + 1)exp(itau) + n}

(12)           K₀ = F₀(0)t₀²/n(n+1)mx₀

 

leading to,

 

(13)           tan[ksi₀(tau) - PHI₀(0) + mu] =

= {sin[(n+1)tau] - (n+1)sin(tau)}/{cos[(n+1)tau] - (n+1)cos(tau) + n}

 

Particular values are:

 

(14)           ksi₀(0) = PHI₀(0) - mu  ,  ksi₀(pi/2) = ksi₀(0) + Arctan{[(-1)ⁿ⁺¹ - (n+1)]/n}

ksi₀(pi) = ksi₀(0) + pi  ,  ksi₀(3pi/2) = ksi₀(0) + Arctan{[(-1)ⁿ⁺¹ + (n+1)]/n}

 

INTERNALLY, THERE’S AN ACTIVITY AND IT’S NOT LINEAR AT ALL…

 

We do find a MOVE. Now, what is concerned, more precisely? Look at (13): it does NOT involve the EXTERNAL mass m, only the INTERNAL mass mu. Conclusion:

 

The EXTERNAL mass m remains INERT in external space, where time is FROZEN.

The INTERNAL mass mu MOVES, and THROUGH SPACE-TIME STATES.

 

It even follows a rather complicated trajectory, despite the quantum force we chose is externally constant (and global) and internally linear in the time state. For n = 1, the internal force is itself constant and global and (13) gives us:

 

(15)           PHI₀(tau) = PHI₀(0)  => ksi₀(tau) = tau + PHI₀(0) - mu

 

which is still an UNBOUNDED motion…

 

Well, still-to-observed physical reality or not, the “simple” fact of complexifying everything, down to space and time themselves, due to quantization demands, already answers an important question:

 

Is it possible to have TWO bodies, an external one and an internal one, with the internal steady and the internal independently moving?

 

The answer is:

 

YES.

 

So, it’s encouraging for the rest. At least, we found ONE POSSIBLE answer. We have a classical body made of classical substance and we have that second body made of substance STATES: already their constituency IS NOT THE SAME.

 

The classical observer stands at space state ksi = 0 or pi and time state tau = 0 or pi. Other space or time or mass or whatever states BELONG TO OTHER “REALITY LEVELS”. As a result, they’re NOT ACCESSIBLE TO HIS/HER OBSERVATION. But ALL levels are accessible to a QUANTUM observer.

 

And in particular, to an INTERNAL observer, since he precisely moves ALONG STATES…

 

Also notice that we can even set F₀(0) = 0 and still have internal motion. If we had fixed this value from the start, general equation (8), we would have been tempted to deduce that x(ksi)[t(tau)] is zero (up to uniform motion), a correct result, but only implying the EXTERNAL trajectory x(t,tau), IN NO WAY THE INTERNAL ONE (a complex number is zero if and only if its AMPLITUDE is zero…).

 

I wanna take a look at the more-than-well-known quadratic equation in R, because there may be something new about it. Let:

 

(1)               P_2-(x) = ½ x² + bx - ½ c²

 

be the quadratic equation of type I and,

 

(2)               P₂₊(x) = ½ x² + bx + ½ c²

 

the quadratic equation of type II. Let’s first examine (1). We have:

 

P_2-(x) = ½ (x² + 2bx - c²) = ½ [(x + b)² - (b² + c²)] = ½ [(x + b)² - D²]

 

The quantity D² = b² + c² being always non-negative, D is a real quantity and P_2-(x) can be factored in R into:

 

(3)               P_2-(x) = ½ (x + b + D)(x + b - D) = ½ (x - x₁)(x - x₂)

(4)               x₁ = b + D  ,  x₂ = b - D

 

Let’s turn to (2). We now find:

 

P₂₊(x) = ½ (x² + 2bx + c²) = ½ [(x + b)² - (b² - c²)]

 

As the quantity b² - c² is no longer of definite sign, the usual procedure is to set the condition b² > c² if we want to find two distinct real-valued roots.

 

Now… this isn’t the only possibility. b² - c² being a hyperbolic square, it can always be written as a product:

 

b² - c² = (b + c)(b - c) = D₁D₂

 

Let’s set y = x + b and develop:

 

½ (y + D₁)(y - D₂) + ½ (y - D₁)(y + D₂) = y² - D₁D₂

 

So, with:

 

(5)               D₁ = b + c  ,  D₂ = b - c

 

type II reduces into,

 

(6)               P₂₊(x) = ¼ [(x + c)(x + 2b + c) + (x - c)(x + 2b - c)]

 

This is not a completely factored expression as in type I, but a sum of two completely factored expressions, illustrating the “splitting” from a single D in type I to two Ds in type II.

 

The zeros of (6) correspond to:

 

(7)               (x + b)² = D₁D₂

 

When 0 =< |c| < |b|, D₁D₂ > 0 and (7) has two distinct real-valued roots:

 

(8)               x₁ = -[b - (D₁D₂)^1/2]  ,  x₂ = -[b + (D₁D₂)^1/2]

 

When |c| > |b|, D₁D₂ < 0 and (7) has no root in R. This is of course because the curve P₂₊(x) is entirely contained above the x axis.

 

The novelty here is in the presence of two discriminants D₁ and D₂ in type II, in place of the traditional single determinant D for both types. That last determinant, D = (b² +/- c²)^1/2, was non-linear in the coefficients b and c, whereas D₁ and D₂ are both linear. If it does not change the nature and existence of the solutions, it does change the structure of the polynomial, first distinguishing two types and, second, introducing two discriminants.

 

This is a quick remark about “spinors and space-time”.
 
It is usually assumed (or did I get it wrong?) that the special status of the physical space-time to be four-dimensional allows one-to-one correspondences between it and non-commutative 2-dimensional complex structures known as “spinors”.
 
I strongly disagree with that argument. The correspondence in question:
 

1. y^a = theta*^Asigma^a_ABtheta^B

 
where small Latin indices run from 1 to 4 (or 0 to 3) and capital ones, from 1 to 2, is a contracted invariant product over the last ones. So, it can be used in any dimension and shows nothing “specific” to the dimension 4. Instead of C² as the “spin space” and SU(2) as its invariant group, we can equivalently consider Cⁿ and SU(n), for any n in N, it won’t change anything to the above formula, which can also be applied to any commutative and real-valued manifold of dimension d:
 

1. y^a = theta*^Asigma^a_ABtheta^B (a = 1,…,d; A,B = 1,…,n)

 
and this is consistent with the well-known fact that “any particle with spin s is represented in its reference frame at rest as a symmetric spinor of rank 2s with 2s+1 components, whatever the value of s”. For s = ½, one finds 2-component vectors; for s = 1, M₂(C) symmetric matrices with 3 independent components; etc.
 
It follows that the above correspondence, not only have no specificity with the “external” dimension 4 (in terms of symmetries), it also makes no difference between spinors and tensors, that is, between fermions and bosons…
 
Geometrically, it means it does not define any “anti-commutative sub-structure to the (pseudo-)Euclidian structure of Minkowski space-time or E⁴ after performing a Wick rotation”.
 
In practice, it means it brings me nothing more able to be used to “extend” or “refine” the properties of “classical space-time”… K
 
Hence this remark.
 
If I use Pauli’s original spin-space, it will be endowed with a skew-symmetric metric J_AB = -J_BA, associated with a spin ½. If I use a spin 1, I’ll simply double Pauli’s indices, obtaining matrix coordinates theta^AB in M₂(C) in place of the former theta^A (symmetric, 3-component, analogue to a vector of E_C³, the 3D complexified Euclidian space) and metric J_ABCD = -J_CDAB = J_BACD = J_ABDC (3 components as well). The Grassmann property will write V^AW_A = -V_AW^A for spin ½ and V^ABW_AB = -V_ABW^AB for spin 1… The first one will imply V^AV_A = 0, while the second one will give V^ABV_AB = 0, which is not equivalent to V² = 0 since, in Euclidian 3-space, the metric is symmetric.
 
Actually, V^ABC…V_ABC… = 0 under a symplectic structure is perfectly normal for any completely symmetric V of rank 2s, whereas it leads to V^ABC… = 0 under a Riemannian structure [and a null cone under a pseudo-Riemannian one with signature (1,n)].
 

I’ve been turning around the pot since the very beginning of this blog, several years ago (except, of course, for articles about finance). My central concern is to find the proper universal frame that will complement space-time. This shows the hardest task. I tried many approaches, quantum physics, space-time relativity,… yet couldn’t find anything satisfying me enough. Indeed, as I repeated it many times, our best “witness” for parapsychological events is the Near Death Experiment (NDE). And the process seems formal on one point: in order to understand what can happen then while staying consistent with neurobiological datas, we need two bodies. Mind cannot be the candidate. Mind is a purely neurochemical process, it’s fully part of the biological one.

 

But we also need two physical frames or we wouldn’t be able to explain why the biological body would not be involved in the NDE process, while the experiencer would discover a “second body, of a different nature”. And that second body is apparently not perceived by the medical team around. Now, directly observable or not, if that second body was in the same space-time as the biological one, its presence alone in the same room as the medics would be enough to induce “disturbances” in the room they would perceive, even if absorbed in their task. Make that simple experience again: look straight to the neck of someone walking ahead of you and, more than 4 times over 5, that person will suddenly turn back. Worth trying if you never did. It works and pretty well. So, if this can work despite there’s no direct “influence” (no “field effect” we say in physics) between the observer and the observed, you can easily convince yourselves (and these are laws of physics) that, exerting a direct influence around you through a “field of forces” would be perceived “almost for sure”, especially in a “confined” room…

 

Now, this is not what is reported, neither by patients, nor by the medical staff. Instead, patients feel themselves “floating above their (biological) body”. However, they can see and hear everything going on inside the room, they can even see under the table; some left the room, went in corridors and still saw and heard everything,… but, in none of these circumstances did anybody report he/she “felt an unobserved presence” back.

 

There’s an apparent “contradiction” somewhere, right? On one hand, we should have an “aetheric body leaving the biological one”, which would suggest they originally were inside the same space-time and, on another hand, we have the same aetheric body who would be like behind a “semi-transparent mirror”, able to see and hear everything, yet nothing passing through.

 

The only physically consistent way out would be to consider two space-times, one where the biological body is and one where the “aetheric” body is.

 

But this is not as simple, as it would still not explain why the aetheric body could perceive while “biological livings” wouldn’t (or even couldn’t!). Hence that intensive search for this second space-time and for a larger universe too, that would include both space-times and both bodies.

 

The difficulty now is to find a second frame that would be as universal as space-time. Physics says a lot about specific frames, but almost nothing about another universal one. Here’s the general context, common to ”classical” as to “quantum” physics: there now exists a legion of physical field inside 4D space-time, these are all parametrizations of the form f(x), where x is a space-time coordinate. Such parametrizations can send back to generalizations of the initial Galilean motion x(t) in 3-space. We can find fields like f with many components, not necessarily linked with space-time. Comparing f(x) to x(t) may incite to think of the object “f” as a coordinate in another frame, different from space-time, since fields are usually not measured as lengths. The point is: each “additional frame” built this way is specific. For instance, the “electromagnetic space-time” using the four Maxwell potentials A_i is specific; Einstein’s “gravitational space-time” using the ten “potentials” g_ij is specific; Pauli’s “spinor space” using the two complex-valued psi^A is specific… I thought once that the one-to-one correspondence between spinor coordinates theta^A and space-time ones yⁱ, yⁱ = theta*^Asigmaⁱ_ABtheta^B, could serve as a “second space-time”, but this construction actually refers to the original space-time itself: it says that, “under” the 4D commutative macroscopic structure of space-time described by “classical” physics”, there’s a more fundamental, 2D, anti-commutative and wavy microscopic sub-structure that is spinor and which is actually able to generate that “continuous” 4D “space-time tissue” at large scales… In other words: the correspondence between spinors and 4-vectors can be used in the same space-time, it does not require nor generate a “second one”… K

 

Physics thus offers a plethora of “non-space-time” possible frames, but nearly all of them have nothing “universal”, they all refer to producing sources… This doesn’t make a frame. Space-time is something that can stand by itself, even in the classical approach: it’s an environment that can be completely empty and still be, proof that it’s not related to any source. You’ll tell me: “but fields in the vacuum are waves and they therefore depend on no characteristic like mass, charge,… of sources; they could become a candidate…”

I’ll reply: “no, because your ‘waves’ actually aren’t… I thought there was, there isn’t anything like a ‘source-free field’. This is again a classical idealization. If you look only at semi-classical interacting models, you’ll immediately see that, taking vacuum states into account eliminates all ‘waves’, because vacuum states interact with fields and act as a source term…”

The concept of “waves” only comes from the fact that the vacuum is neglected in the classical approach and associated with “nothingness”…

In fact, they are purely mathematical solutions, due to determinism. As soon as you take a statistical approach, you find fluctuations and those fluctuations, that do not vanish, act as a source.

It’s even so blatant that vacuum fluctuations can change the configuration of a system!

They can make it flip from one state to another…

 

No. I went back and forth, round and round, again and again and the only frame I’ve heard of that meets the requirements is the spectral one… That one is universal. There are former bidouilles about it, but I’d like to make another synthesis, because I feel I didn’t go deep enough in the physical content or I didn’t interpret it in the suitable way. We can actually make a geometrical synthesis between at least three approaches: oscillations, complex-number theory and spectral analysis.