I may have found an easy way to extend point mechanics, so as to include the scale effects.

Consider first replacing the traditional point xⁱ with a vector field xⁱ(l^j) = xⁱ(l,T). We call the xⁱ(l^j) the sizes of the spot. l^j measures the distance from the spot xⁱ(l^j) to an observer in the direction j. When the l^js tend to infinity, we are at the macroscopic scale and we expect the xⁱs to tend to zero: an observer that is far away from the object will see it as a “point”. On the opposite, when the l^js tend to zero, the observer is very close to the object, which appears very large to him: the xⁱs tend to infinity, meaning the object occupies all the vision field.

We can build a dynamics from these xⁱ(l). The first criterion is the velocity, we define as:

 

(1)               v_ij(l) = c(¶x_j/¶lⁱ - ¶x_i/¶l^j)

 

It’s a skew-symmetric 2-tensor field. Following that, the mass of a point-like body is replaced with a spectral mass density m^(l,T), still measured in kg/m³. It’s spectral because it’s a density function over spectral space-time. We’ll see a more conventional definition of the mass density later on. The integral of this spectral mass density over a finite spectral 3-volume V^ gives the spectral mass of the incident body:

 

(2)               m^(T) = ò_V^ m^(l,T)d³l

 

This quantity depends only on the spectral time T (the period in the signal description). It’s immediately global on original space-time. In particular, it no longer depends on the original time t. From this point of view, it’s always a conserved quantity with respect to evolution in original space-time.

Next step: we form the Maxwellian Lagrange density

 

(3)               £^ = ½ m^(l)v_ij(l)v^ij(l) + f_j(l)xⁱ(l)

 

As £^ is in J/m³, f_i(l) is in N/m³ (force density). In the field description, m^(l) is more suitable than m^(T). The momentum density is:

 

(4)               p_ij(l) = ¶£^/¶v^ij(l) = m^(l)v_ij(l)

 

and the equations of motion:

 

(5)               c¶p_ij(l)/¶l_i = f_j(l)

 

They give the evolution of the spot xⁱ(l) in spectral space-time, knowing the external constraints f_i(l). We have the usual gauge invariance on the velocity field:

 

(6)               x_i(l) -> x_i(l) + ¶g^(l)/¶lⁱ

 

with g^(l) an arbitrary spectral scalar field expressed in m². This freedom enables us to set up the condition:

 

(7)               ¶xⁱ(l)/¶lⁱ = 0

 

What’s interesting here is that, not only original space x but also original time t are now allowed to change with the 4 lⁱ, i.e. from scale to scale (or wavelength to wavelength in the signal description). We knew they were both relative with respect to the velocity of a moving frame, we now see they can also, in addition, be relative with respect to the characteristic scales in all 4 directions of spectral space-time: none of the equations of motions (5) nor the transversality condition (7) depend on x or t. The dynamics is global, not only on original space, but on the whole original space-time and everything actually happens in spectral space-time: for a “conscious” observer standing in original space-time, there’s no particular motion at all. Nothing happens. The only way he can deduce “something” is going on “somewhere” is to observe scale variations on the size of a physical object, in space and time directions. And still: most of the time, a scale variation is for our observer a mere homothety, that is, a forward or backward zoom on the size of the object he observes. Here, such a zoom would be a very particular case of a much wider class of behaviours xⁱ(l), according to the form of the external constraints f_i(l) acting upon a… a what? A spectral body, of course, since the matter distribution of the incident object we considered is spectral: m^(l,T).

It sounds a bit obvious, yet:

 

A SPECTRAL BODY MOVES IN SPECTRAL SPACE-TIME.

IN ORIGINAL SPACE-TIME, NOTHING HAPPENS (NO MOTION, NO MATTER).

 

Nothing to observe => no existence. Wrong deduction. It is not because we cannot observe something that this something cannot exist. Only the converse is true, as to know: no existence => nothing to observe.

More sophisticated models are of course possible, I only took the simplest one to illustrate.

Let’s try a “Newtonian” solution like:

 

(8)               x_i(l) = n_iV_pl/l^kl_k

 

where V_pl is the Planck volume and n_i a unit vector. We get:

 

(9)               v_ij(l) = cV_pl(l_in_j – l_jn_i)/(l^kl_k)²

(10)           p_ij(l) = m^(l)cV_pl(l_in_j – l_jn_i)/(l^kl_k)²

(11)           f_j(l) = c²V_pl{(l_in_j – l_jn_i)l^kl_k¶m^(l)/¶l_i + m^(l)(4lⁱl_jn_i - n_jl^kl_k)}/(l^kl_k)³

 

Gauge condition (7) would lead to additional, but subsidiary, n_ilⁱ = 0. Since we started from a solution, the equations of motion (5) must give us the related force density. Surprinsigly enough, this force density does not demand that m^(l) be point-like, i.e. of the form m^(l) = m^(T)d(l). Besides, as you can check, this last form doesn’t give anything interesting for f_j(l), except derivatives of the Dirac pulse, nobody really cares of… :)) This actually means the “point-like” distribution has nothing natural anymore.

Solution (8) has two poles: l^k = 0 and l^kl_k = 0. We discussed l^k = 0. When the scale 4-vector is light-like, everything happens as in the microscopic. Yet, c²T² - l² = 0 only gives v_ph = c! So, it only means the (spectral) light cone is singular towards the Newtonian solution, making the spot size diverge.

 

We have a “reverse” model, permutating the xⁱs and the lⁱs. We then find familiar field theory using the space-time coordinates as field parameters. Instead of the spectral mass density, we find an original mass density m(x,t) and an original mass m(t) for an original volume V of original 3-space (these repetitions, despite necessary, have nothing original anymore… lol).

Mechanical features of this kind of motion (in original space-time – okay, tomorrow, I stop) are the field velocity V_ij(x), the momentum density P_ij(x) and the force density F_j(x) (all originals – STOP IT!). A Newtonian solution is now l_i(x) = n_iV_pl/x^kx_k, with singularity on the (…) light cone. At the origin x^k = 0 (got ya, mate!), l_i(0) = ¥: we’re macro; same on the (…) light cone, whereas x^k -> ¥ gives lⁱ -> 0: micro.

I find this reverse model less interesting, as l_i(x) describes local changes of scales. Moreover, matter distributions being in m(x,t), they describe only scale-invariant (…) matter, which does not include biological bodies as we saw it in the former bidouille. I mentioned it anyway, just to be complete.

 

Non deformable solids are of course mere idealizations: they don’t exist in Nature. All natural bodies sustain elastic deformations and this is precisely what this extension of the “point” expresses: that natural bodies can never be reduced to their centre of gravity or, if we prefer to keep this approach, then this cog can no longer be considered as point-like, but must be viewed as a spot, with sizes varying according to how far it is from an external observer.

We then see a spectral mechanics naturally emerges from such an extension.

 

We can mix both in a (careful!) 8D fictitious space-time with 4 space dimensions, 4 time dimensions and Minkowskian metric:

 

(12)           ds₈² = dxⁱdx_i – dlⁱdl_i = c²dt² - dx² - c²dT² + dl² = c²dt²(1 – v²/c²) – c²dT²(1 – v_gr²/c²)

 

When v < c and v_gr < c, the original space-time is causal (observable), but the spectral space-time is not (dlⁱdl_i > 0 => -dlⁱdl_i < 0).

When v < c and v_gr > c, the whole 8D space-time is causal, but an observer in the original space-time will receive no information from the spectral.

When v > c and v_gr < c, the whole 8D space-time is no longer observable (ds₈² < 0).

Finally, when v > c and v_gr > c, the spectral space-time is causal (observable), but the original space-time is not.

Furthermore, we have:

 

(13)           ds₈² = 0  <=>  dxⁱdx_i – dlⁱdl_i  <=>  (dT/dt)² = (1 – v²/c²)/(1 – v_gr²/c²)

 

The physical reality is much more a 4D space-time with “original state” on “one side of the light cone” and “spectral state” on “the other side”. A 2-state space-time. The metric (12) then expresses the fact that, when one state is observable, the other is not, for they are separated by light. On one side, coordinates are expressed by xⁱ; on the other side, by lⁱ. But these are all distances, all in meters. We could right as well x^ia = (xⁱ,lⁱ), where a = 1,2 is the state label.

 

 

The attentive reader will have noticed (for long) that, throughout these papers, i kind of repeat myself many times, always turning around the very same items. Only the approaches change. Thus, in bidouille 10 to 12, you’ll find the same argumentation (quantum, discrete space and time, stochasticity, wavepackets) as in the last bidouilles.

 

Could we actually be victims of appearances?...

 

Prigogine himself explained in great details the similitude between “classical” and “quantum” complexity: both are founded on a distribution law in the suitable phase space. I think I remember a bidouille where I showed that even classical mechanics could be rewritten in terms of operators acting on dynamical variables, that is, a non-commutative way…

 

I worked a little bit on different forms of distribution laws this week and I finally asked myself:

 

IS PARAPSYCHOLOGY A QUANTUM SCIENCE OR RATHER A SCIENCE OF COMPLEXITY?

 

For it appeared to me that:

 

QUANTUM PHYSICS CAN BE VIEWED AS A PARTICULAR CASE OF COMPLEXITY THEORY, WHEN THE SCALES lⁱ REPRESENT WAVELENGTHS.

 

I’ve been looking for explanations in the quantum theory. Is it the right frame or isn’t it rather a question of complexity? Probability laws are constructed the same…

Consider a distribution of matter m(x,t) in original 3-space. Such a distribution depends on x, but not on l nor T. It describes a matter that is inhomogeneous (x-dependence) but scale-invariante (l-independence), that is, homogeneous from one scale to the other: should you change scale, the distribution of this matter would remain exactly the same, at each point, each time.

Consider now the spectral image m^(l,T) in spectral 3-space. Such a distribution depends on l, but not on x nor t. It describes this time a matter that is homogeneous (x-independence) but inhomogeneous from one scale to the other: should you change position in space-time, the distribution of this matter would remain exactly the same, at each scale, each period.

Consider finally both, m(x,t,l,T), and assume m explicitly depends on the ratios xⁱ/lⁱ (i = 1,2,3,4, x⁴ = ct, l⁴ = cT), for reasons of physical units. Such a distribution of matter is inhomogeneous in both original and spectral 3-spaces: not only does it change from one point to the other, but also from one scale to the other.

The distribution m(x,t) is not complex: it’s the same as at the microscopic level (lⁱ = 0). We usually use it to describe corpuscular matter.

The distribution m^(l,T) is complex and global.

As for m(x,t,l,T), we use it to describe “quantum” matter when the lⁱ’s are wavelengths. Then, the sum of the ratios xⁱ/lⁱ represents the phase of a monochromatic wave and å_i=1⁴ò dxⁱ/lⁱ(x) the phase of a polychromatic wave, in a parametric representation lⁱ(x) (and up to a 2p factor).

 

When the lⁱ’s are not especially wavelengths, we find complexity theory.

 

If quantum theory was “merely” a particular case of complexity theory, what kind of physical link could we build between them?

m(x,t) models the corpuscle. It’s clearly elementary (what’s not complex is elementary – or simple…).

m^(l,T) models its spectra. It carries all the complexity (and nothing else!).

So, if m(x,t,l,T) is supposed to model quantum matter than, in such matter, all the complexity is carried by the wavepacket.

That would mean that, at super-high energies (l⁴ -> 0), we indeed retrieve the “corpuscular approximate” (almost-Dirac); at super-low energies (l⁴ -> ¥), we retrieve the “wave approximate” (almost-flat). In between, we are “quantum”, i.e. a “mixing of corpuscular and wavy behaviours”.

So-called “intrication” proceeds through lⁱ-dependence. When lⁱ -> 0, we’re at the “geometrical limit” of classical optics (“rays”), no intrication.

What do we actually call the “microscopic”? It relates to the average size of physical objects, with respect to “our” sizes. Particles and atoms are therefore “microscopic” to us. We can see here the neat distinction with l⁴ -> 0, which relates to the energy level of the object. If quantum physics was confined into “the microscopic”, we would find no “macroscopic” effects, i.e. effects on physical objects with average sizes much greater than ours.

Once again, confusion is easy between lⁱ as wavelengths, quantum-associated with energy and momentum and lⁱ as complexity scales or sizes: in the quantum context, the “size” of an object is the width of its wavepacket…

There’s a possible way to find “quantum disintrication”, at least as an interpretation of things, if we assume, as we saw above that the property “in a quantum object, the corpuscle have zero complexity, while the wavepacket has full complexity” is typical of objects of microscopic sizes, while the “reverse” property “in a classical object, organized matter have full complexity, while the wavepacket has zero complexity” is typical of objects of macroscopic sizes. There’s then a “complexity transfert” from the wavepacket to substantial matter as the system organizes and expands into space. At small distances, we would then observe quantum effects, whereas at large distances, we would observe substantial effects and the wavy ones would have disappeared from our observation, unless we observe a collective phenomenon, i.e. as long as we’re not critical.

Anyway, it’s really all a question of scale, since we observe quantum effects again as soon as we zoom forward back to the microscopic… J

This, again, can be seen as the property of the quantum to be scale-dependent, as any other complex systems. Notice a point-like body is not scale-dependent: it remains a point-like body at all scales. A corpuscle remains a corpuscle. Wave mechanics add a wave to it, but don’t expand it!...

 

Let’s now make a little calculation. We will limit ourselves to orders of magnitude, it’s far enough. The mean mass of a living cell is around 10^-8 kg (that’s about ten micrograms). Taking it circular (for simplicity, but it doesn’t change a lot the result), its ray is about 1 micron (10^-6m) and its thickness, about 1 nanometer (10^-9m). This gives a volume pr²h around 3 x 10^-21 m³ and a mean density m_cell around 3 x 10^-12 kg/m³. The mean mass of a human body is around 70 kgs. Taking it axial, its ray is about 0.5 m and its height, 1.7m. This gives a volume of order 1.335 m³ and a mean density m_human around 52.4 kgs/m³. The ratio between the two densities is 52.4/3 x 10^-12 = 1.75 x 10¹³ = (4.2 x 10⁶)². The ratio between the scale of a cell and that of a human being of order 10⁶, we come to the conclusion that the density of biological matter strongly depends on the scale, roughly as 1/l².

Is it due to some quantum expansion?

Obviously not. It’s rather due to complexification.

Hence my first question above: is parapsychology a question of quantum behaviour or rather a complexity problem, if complexity theory includes quantum theory?

With quantum theory, we face problems, difficulties and legitimate oppositions.

With complexity theory, we no longer face such problems, but we have to look for other evolution stages of organized matter. A new class of problems…

I can keep all the work I’ve done so far on the quantum context, but send it back to the microscopic domain where it belongs in non-critical situations.

And I enlarge the context replacing it with complexity theory.

 

Matter evolves. When reaching a certain complexity level, it’s evolved enough to become autonomous: matter becomes “alive”. It manages itself, it reproduces… but it still strongly depends on thermodynamics. It “gets old”.

 

My question now is: is there another complexity level above which “dead” matter do not depend on thermodynamics anymore?

Surely the physical state of such matter would be fundamentally different from that of “alive” matter.

We yet have a guideline, as the properties reported by NDEs seem to show that “dead” matter has many properties similar to those of quantum media.

Now, we can use quantum equations to help us move forward in complexity theory. So we’re (in principle) allowed to draw some similitudes with quantum matter, as long as they remain similitudes.

 

 

Call me stupid… I’ve been talking about « quantum » kernels for many articles and i didn’t even remember that l was a scale… L

The spectral approach seems to be the right one… if i perform it right.

First, I made a mistake that I want to correct right now. In the 1D Laplace kernel r(x,l) = exp(-x/l)/l, l is obviously independent from the position variable x, since the integral equality:

 

(1)               l = ò₀^+¥ xr(x,l)dx = L(x) = Laplace transform of x

 

is a continuous summation over all distances x from zero to infinity. So, the result does not depend on any particular value x in this interval.

Second, x is “local”, while l is “global”. This, again, is obvious in (1), as integration is a global operation.

Third, x and l definitely have different roles. There can be no kind of “inverse kernel” that could be obtained permutating x and l. The inverse Laplace transform L^-1(l) giving back x reverses the kernel r(x,l) and integrates over l (Cauchy’s formula).

All this holds for more sophisticated kernels. Besides, (1) is equivalent to an integration over the differential of the kernel:

 

(2)               r(x,l)dx = -ldr(x,l)  =>  l = -l ò₀^+¥ dr(x,l) = -l[r(x,l)]_x=0^x=+¥

 

This property of the kernel also implies that:

 

(3)               dr(x,l)/dx = instantaneous variation of r in space at scale l =

= -r(x,l)/l = -(mean scale variation of r at the same point x)

 

which means that, going from differential calculus and thus geometry to algebraic calculus, integral geometry and thus topology, is only due to the property of the kernel. It’s only because this kernel verifies (3), which is a bridge between differential geometry and discrete geometry, that the instantaneous variations of any continuous function f of the distance variable x is Laplace-transformed into a discrete derivative of the Laplace spectrum of f with respect to the Laplace spectrum of x, that is, l.

We have here full confirmation, if necessary, that all properties of the Laplace transform are contained in its kernel. More complicated algebraic expressions are obtained with other transforms, but the result is the same: all the properties obtained are contained in kernels.

 

So: 1) x and l are independent quantities; so good; 2) x and l have same physical units (here, meters).

As I said previously, the wrong interpretation would be to deduce from that an extended 6D space or 8D space-time: l is a length, as is x and they both belong to the same space or space-time. The pair (x,l) does not double the number of physical dimensions.

I asked myself why. Why couldn’t l be used as an additional space dimension?

I simply lacked a geometrical representation.

Fix a coordinate system and an origin O. x is the distance of any point in space from O. The distance between the origin and itself is x = 0. x varies from point to point.

l is different. It’s a scale of distances. Say l = 1m, for instance. This choice will mean we include all distances x from x = 0 to x = 1m. So, l is actually not a space variable. It’s a length. The length of a segment [0,l]. It’s the boundary value of a distance interval. When l varies, the length of this interval varies.

 

Geometrically speaking, the introduction of a scale means the ordinary point in space is replaced with a segment of given length l.

 

The substitution is similar to the one in string theory, where a point-like body is “extended” into a small string. However, the geometrical interpretation in spectral analysis is different:

On the one hand, we have point-like bodies located at points x in space; these bodies remain point-like. And, on the other hand, we have distances scales l, representing segments of space. The spectral image of a point x in “original space” is a segment of length l in “spectral space”. Conversely, the original image of a segment of length l in spectral space is a point x in original space.

There’s no “spatial extensions” of point-like bodies. There’s a correspondence between an “original space (or space-time)” with “original (point-like) bodies” and a “spectral space (or space-time)” with “spectral bodies”. These “spectral bodies” are point-like only when l = 0. Otherwise, they have “characteristic size l”.

 

What does that mean?

On the geometrical point of view, this first mean points have to be replaced with segments [0,l] of variable sizes. Consequently, curves have to be replaced with surfaces, surfaces with volumes, volumes with 4D hypervolumes and so on.

On the material viewpoint, only non-deformable solid bodies can be represented as “point-like” (the “hard sphere” model), since all their points move with their centre of gravity at the same time, so that their motions in space(-time) can be brought back to their cog alone.

If a material body shows a spanning in space, it cannot be so. Whether it’s deformable or it cannot be considered as solid.

This last point is very important, since it has important consequences on the internal structure of bodies: if an “original” body can always be considered “point-like” as soon as it’s in a solid and non deformable state, a “spectral” body can be considered as such only for l = 0, that is, when its size is zero. As no substantial body with zero size exists in Nature, we deduce that:

 

NO SPECTRAL BODY CAN BE VIEWED AS A SOLID, NON DEFORMABLE SUBSTANTIAL BODY: WHETHER IT’S SUBJECT TO DEFORMATIONS OR IT CANNOT BE CONSIDERED AS BEING IN A SOLID STATE AS WE CONCEIVE IT.

 

For an “original” (biological) observer, it looks like something “ethered”.

Anyway, as we saw it in B94, on “our” side of light, spectral bodies are unobservable (since l² = lⁱl_i < 0). On “the other side” of light, they are and biological bodies aren’t anymore (k² = kⁱk_i < 0). When we have to deal with 4 scales lⁱ, a point xⁱ in original space-time has to be replaced with a 4D hypervolume in spectral space-time. Hence the easy confusion with a 8D “superspace”, since to any geometrical structure, we add 4 dimensions… J

Actually, it’s all fictuous. The physical frame remains 4D, but the structure of bodies is directly impacted, as the consequence of the change in the nature of space (and time).

Hence too, the confusion with wavepackets: we’re not dealing with wavepackets, but with spectra. Sure, associating x and l can help build wavepackets. But oscillating motions as well!

 

A wave can never be substantial. A spectral body can. Technical subtleties making huge differences in the end…

For substances, we find “originals” and “spectra”.

For links between the “original” state of space-time and the “spectral one”, we can use quantum objects. Because they play on both sets of variables.

 

To sum up, I would say that, in parapsychology, everything that relates to physical substances is not what we call “quantum”. Possible communications between the two sides of light, i.e. between “livings” and “deads” are “quantum”. Rather “wavy”.

 

For all these reasons, I’ll no longer be too severe with technicians seeing in “superstrings”, “extra-dimensions” or other sophisticated quantum devices possible “doors” or “gates” to the “paranormal”. I may be repeating, but confusions are very easy and subtleties much harder to distinguish! And it’s always much easier to add dimensions and “see what it’s gonna give” than to stay in good old 4D space-time.

 

I’d like to bring a final correction to B94:

 

WHEN GOING FROM ONE SIDE OF LIGHT TO THE OTHER, IT’S NOT A QUESTION OF GETTING SUBSTANTIAL, THE SPECTRAL BODY IS SUBSTANTIAL ON BOTH SIDES, IT’S A QUESTION OF BECOMING OBSERVABLE.

 

It all turns around questions of observation and scales…

If you’re spectral, you can be fully substantial and consistent, while seeming unconsistent and therefore unsubstantial to an « original » observer’s conception, because your matter looks like « dispatched into space » (and even time) to him.

You no longer appear as a geometrical object, but as a topological one.

 

Everything but simple, all this stuff…

 

Yep! There’s unfortunately still something that doesn’t fit…

We find exactly the same relations in the corpuscular as for waves. Nothing original? Not sure. For I found no clear explanation about this in my references.

Look.

Take a point-like body B with instantaneous velocity vector v(t) = v(t)n. His energy and momentum in motion are respectively:

 

(1)               E(t) = E₀/[1 – v²(t)/c²]^1/2  ,  p(t) = E₀v(t)/c²[1 – v²(t)/c²]^1/2 = E(t)v(t)/c²

 

From what we deduce that:

 

(2)               E(t)/p(t) = c²/v(t)

 

whereas:

 

(3)               dE(t)/dp(t) = v(t)

 

If the second relation is well understood, my question is:

 

WHAT REPRESENTS THIS SECOND VELOCITY IN (2)?

 

Stupid as usual, isn’t it? We cannot argue c²/v(t) is a phase velocity, since we’re not dealing with waves, but with corpuscles. Hence my (stupid) question. E(t)/p(t) looks like a mean value. Just like Canada Dry, (2) “looks like” a mean velocity, but is not. For letting c²/v(t) = v_moy(t) = x(t)/t would lead to d[x²(t)] = d(c²t²), that is, to c²t² - x²(t) = c²t₀² in the causal region or c²t² - x²(t) = -c²t₀² in the tachyonic one. A quadratic relation analogue to E²(t)/c² - p²(t) = E₀²/c². I turned to the Laplace transform, obviously with no result, since the algebraic quotient of two Laplace spectra sends back to a convolution formula for the originals. Nothing I tried satisfied me. The only relations I found were:

 

(4)               v(t) = original of v_ph(T) = l(T)/T by e^-t/Tdt/T with x(0) = 0;

(5)               c²/v(t) = original of v_gr(T) = dl(T)/dT by e^-x/ldx/l with t(0) = 0;

 

since v(t) = dx(t)/dt <=> 1/v(t) = dt(x)/dx, and:

 

(6)               v_gr(T) = ò₀^+¥ v(t)exp[-v(t)/v_gr(T)]dv(t)/v_gr(T)

(7)               v_ph(T) = ò₀^+¥ v_moy(t)exp[-v_moy(t)/v_ph(T)]dv_moy(t)/v_ph(T)

 

in galilean relativity, while:

 

(8)               V_gr(T) = ò₀^+¥ V(t)exp[-V(t)/V_gr(T)]dV(t)/V_gr(T)

(9)               V_ph(T) = ò₀^+¥ V_moy(t)exp[-V_moy(t)/V_ph(T)]dV_moy(t)/V_ph(T)

(10)           V(t) = v(t)/[1 – v²(t)/c²]^1/2  ,  V_gr(T) = v_gr(T)/[1 – v_gr²(T)/c²]^1/2

(11)           V_moy(t) = v_moy(t)/[1 – v_moy²(t)/c²]^1/2  ,  V_ph(T) = v_ph(T)/[1 – v_ph²(T)/c²]^1/2

 

in einsteinian relativity. That’s all. Besides, for variable velocities, when performing the wave-corpuscle duality, we let aside one more “tiny detail”:

 

(12)           v = v_gr

 

Great. But, formally, this only holds for constant velocities. For variable ones, v depends on t (the “original time”), while v_gr depends on the period T (the “spectral time”)! We can make no mistake between them, since T is the Laplace spectral image of t. So, we’re definitely not dealing with the same time variables. The only thing we can say is, that, in numerical values, v and v_gr are of the same magnitude.

In one sense, this reinforces what I support in favour of the existence of two bodies, a double nature of things: on one “side” of light, what is described by a dynamics based on x(t), v(t), E(t), p(t) and so forth and on “the other side”, what is described by a dynamics based on l(T), v_gr(T), E^(T) and p^(T), spectral transforms of E(t) and p(t).

So, why shouldn’t I be fully satisfied?

Because if we really were in presence of spectral transforms, the original motion x(t) and the spectral motion l(T) would never be independent… As an example, l(T) can simply be realized two ways:

 

(13)           l(T) = ò₀^+¥ x(t)e^-t/Tdt/T = L_t[x(t)]

 

Laplace transform of x(t) with respect to time(s), or:

 

(14)           l(T) = ò₀^+¥ x(t)exp[-x(t)/l(T)]dx(t)/l(T) = L_x(t)[x(t)]

 

the same transform, but with respect to x(t), i.e. in a functional space of corpuscular trajectories spanning from 0 to the infinity.

In any case, x(t) = x(0) for all t, that is, a fixed position in original space(-time) implies l(T) = cte = l(0) for all T, that is, a corresponding fixed position in spectral space(-time), due to the normalization of the Laplace kernel: ò₀^+¥ e^-t/Tdt/T = 1 and ò₀^+¥ exp[-x(t)/l(T)]dx(t)/l(T) = 1.

In civilized language, this means that, when the biological body is fixed in space, the spectral body should remain fixed in spectral space… and conversely.

I suggested, that’s right, to go beyond spectral transforms but, this time, it demands two completely independent set of dynamical variables and parameters: the set {x,t,x(t),f(x,t),…} and the set {l,T,l(T),F(l,T)…} on each side of light. Setting up a duality between them would lead to waves, wavepackets and finally the quantum.

But what about oscillating motions?...

We can have a substantial pendulum moving according to x(t) = x(0)cos(t/T) for instance… such a motion makes use of both set of variables and still represents a “corpuscular” motion (the motion of a corpuscle)… When oscillating, the pendulum behaves like a wave. More precisely, its motion in space resembles a wave.

 

I’m a bit lost for we haven’t been rigorous enough in the mathematical description of our contemporary physics. We neglected too many things. Let me just recall that the quantization procedure is still not completely proven. I’d be very annoyed to be forced to leave the quantum frame, not only because it would demand to develop a brand new physics “based on nobody knows what”, but also because many aspects of parapsychology are in sound adequation with quantum behaviours. For instance, this feeling of “heat” in NDEs: the superfluid motion is very sensitive to heat. There are numerous details such as this one going in favour of quantum theory.

Yet, there remains a lot of technical points that do not satisfy me: not all oscillating motions are quantum; spectral dualities involve dependent motions; corpuscular dynamics having the very same relations as wave dynamics;…

As I said above, we haven’t built a mathematical frame solid enough to be able to move on “quietly” in parapsychology. Too many aspects of “well-established” physics are actually more than shallow…

“it works” for it’s “in agreement with experiments” so far.

Now, in biophysics, it doesn’t work. Or it does, but not properly.

I don’t want “hypothesis” nor “postulates”. I want proves. I want things, reasonings, to be proven.

Now, the only kind of solid proves we have in quantum physics is… experiments.

And as we still don’t have solid protocols in parapsychology, we can only base ourselves on theoretical work.

The snake bites its toe…

I had some successes, found some explanations, but there remains too many open questions that should have found an answer if our theoretical basis in physics were tough.

This is the most frustrating in all and I cannot move on mainly because of that.

I cannot be sure of what I propose for, each time, there’s some “tiny correction” or “completion” to bring…

Instead of moving on, I spend my time trying to correct existing physics!!! 8((((

 

What’s really unbelievable… is this.

 

As promised, let’s now turn to technical details and see the consequence for NDEs.

We’ve already reviewed spectral analysis several times on this blog, so I won’t write all the calculations down again. For our present purpose, anyway, the Laplace transform is sufficient. More sophisticated transforms would only complicate things, but wouldn’t change anything to the principles.

In bidouille 37, eq. (7), is given the equation for rather general kernels r(x/l). According to (1), bidouille 93, the equation for the 3-volume of a body B’ is the equation for r(x/l) in 3 dimensions with only a change of sign before the identity, that is: -3Id in place of +3Id (nId in n dimensions). Moreover, for any smooth kernel r(x/l) = Kexp[-a(x/l)]/l¹l²l³, where x/l is short writing for (x¹/l¹,x²/l²,x³/l³) and a(x/l) = 0 at x = 0, one finds V(0) = K^-1l¹l²l³ to be the volume of B’ around B, when B is located at the origin of a chosen reference frame. This alone shows that V(0) is spectral.

To deduce from this result that the physical space in which parapsychological phenomena have to be studied would be 6D, 3 xⁱ for “ordinary” space + 3 lⁱ for “spectral” space would be a mistake. Indeed, I’ve already pointed out the fact that, in more than 3 space dimensions, physical bodies should keep a 3D basis. This is not what NDEs tell us and to say that would be a wrong interpretation of them.

Instead, NDEs show us we’re in front of two complementary states of the same 4D Universe: an “ordinary” state with coordinates xⁱ and a “spectral” state with coordinates lⁱ (i = 0,1,2,3). When taking these two states into account, we get a mechanical description of a quantum Universe: giving us coordinates (xⁱ,lⁱ) is actually equivalent to giving us coordinates xⁱ alone and wavepackets y(xⁱ,lⁱ) with squared amplitude r(xⁱ,lⁱ), as these wavepackets connect physical objects depending on the xⁱs (and called “originals” in the language of spectral analysis) to physical objects depending on the lⁱs (and called “spectral images” or simply “spectra”), through a global, integral, transform. As a practical example, we have the Laplace transform of a space trajectory x(t):

 

(1)               l(T) = L[x(t)] = ò₀^+¥ x(t)e^-t/Tdt/T

 

which gives a spectral trajectory l(T) in spectral space. Simple calculation then shows that the instantaneous velocity v(t) = dx(t)/dt of a substantial body B moving into “original” 3-space is transformed into v_ph(T) = l(T)/T, the phase velocity of a signal “associated with B”, according to the initial formulation of De Broglie’s wave-corpuscle duality. We now see that v_ph(T) is actually the mean velocity of a “spectral” body B’ complementing B in the quantum picture. It’s merely the equivalent of the mean velocity v_moy(t) = x(t)/t in “spectral” space, i.e. in the spectral state of (quantum) 3-space.

Let’s now go back one more time to the spectral relation k² = kⁱk_i, with kⁱ = 2p/lⁱ the components of the 4-wave vector. We know that:

 

(2)               k² > 0 => kⁱ time-like => v_ph > c <=> v_gr < c

(3)               k² = 0 => kⁱ light-like => v_ph = c <=> v_gr = c

(4)               k² < 0 => kⁱ space-like => v_ph < c <=> v_gr > c

 

since

 

(5)               v_ph(T).v_gr(T) = c²

 

for a group velocity v_gr(T) = dl(T)/dT, whatever the original motion in original 3-space, uniform or not. Now, v_ph(T) = l(T)/T implies:

 

(6)               k² > 0 => v_ph > c => l² = lⁱl_i < 0

(7)               k² = 0 => v_ph = c => l² = 0

(8)               k² < 0 => v_ph < c => l² > 0

 

Truly, there’s absolutely nothing new in what we’re talking at the moment. These are all elementary and well-known results of spectral analysis in space-time.

We’re used to say a signal is “causal” when k² > 0. It therefore means l² < 0, i.e. lⁱ is space-like or “tachyonic”, meaning it stands “on the other side of the light cone”.

We consider a signal to be “tachyonic” when k² < 0, meaning l² > 0, i.e. standing “on ‘our’ side of the light cone”.

And we consider a signal to be light-like when k² = 0, meaning l² = 0, i.e. standing exactly on the light cone.

Are you beginning to see what I’m undermeaning in all that?

In our reclassified interpretation of the wave-corpuscle duality, we find a body B’ in place of the signal.

 

LET B BE A “SUBSTANTIAL” BODY IN ORIGINAL 4-SPACE-TIME AND B’ ITS “SPECTRAL EXTENSION” IN QUANTUM 4-SPACE-TIME.

WHEN k² > 0, l² < 0 AND B’ IS “NON SUBSTANTIAL” IN ORIGINAL 4-SPACE-TIME.

WHEN k² = 0, l² = 0 AND B’, AS B, IS “IN BETWEEN”.

WHEN k² < 0, l² > 0 AND B’ IS “SUBSTANTIAL” IN SPECTRAL 4-SPACE-TIME.

 

What do we have at the precise moment when the patient’s blood pressure p falls down to zero? An “OBE”. Well, that’s not really what spectral physics tells us: nothing needs to “get out” of the biological body. Instead, we have a double “quantum” body, from the beginning, with a “substantial” part B and a “spectral extension” B’. Both are “alive”. But, when B is thermodynamically active, we have no direct perception of our B’, for a simple reason: B’ alone is thermodynamically reversible. And just like in superfluids, there exist exchanges between B and B’, but implying no mass transfert, i.e. no energy-momentum transfert. If there’s no perceptible changes in such exchanges, we perceive nothing. Or only a “single” body.

Things begin to change when p = 0, for B becomes inert. Surprisingly enough, the patient then “becomes conscious he/she gets a ‘second’ body, ‘of a different nature’”.

He/she just becomes conscious his/her “spectral extension” B’ is alive… because p = 0 does not concern it… J Or, much better, autonomy is transferred from B to B’. Whatever really happens, B’ is now autonomous and conscious.

 

So, what’s the need for a “Tunnel” with a “Great White Light” at the end?

 

This is where things become really extraordinary. Look at (6) to (8): what does spectral physics tells us?

 

THE SPECTRAL BODY NEEDS TO GO THROUGH LIGHT TO GET SUBSTANCE.

TO BECOME FULLY CONSISTENT.

 

Incredible, isn’t it?

When this “fake OBE” happens, B’ is “wavy”, “unsubstantial”, for it stands in the same “causal” region of the Universe, on “our” side of the light cone, where it appears to us as a signal. But this is the region of pretended “substantial matter”. He’s not in his region! His region is “on the other side of the light cone”. This is where it appears “substantial”, where he gets “consistence”, whereas previous “substantial matter” there becomes “unsubstantial”! J

It’s all about the choice of observers: to an observer in the “causal” region k² > 0 of space-time, l² being < 0, everything with l² < 0 looks non substantial, as matter “cannot move faster than light” and waves “move at the speed of light”… J. To an observer in the “tachyonic” region k² < 0, l² > 0 and everything inside the previous “causal” region looses all substance, for the very same reasons, as this “tachyonic” region is actually “spectral causal”.

There’s a complementarity between the two dynamical regions of space-time.

Statically, now, we find three states of the same Universe: the “original” state (k² > 0 , l² < 0), the “spectral state” (k² < 0 , l² > 0) and the “quantum” state, gluing together these two states.

 

So, B’ has to move from “this” side of the light cone to “the other” to get consistence. And this can only be done with the help of a wormhole (Bidouille 89) and the velocity (!) condition G_iGⁱ = c² on the gravitational potentials.

It all goes around velocities…

 

ONCE B’ IS “ON THE OTHER SIDE OF THE LIGHT CONE”,

HE CANNOT COME BACK.

 

There can be no “return from the Deads”. This is not what happens. Again, it’s a feeling.

B’ cannot come back, for he’s now substantial and no substance can go through the light cone. Now, to “come back” in the “biological region” he left, he would need to go back through the light cone. Impossible for substantial matter.

 

BESIDES, NO “TUNNEL BACK” IS REPORTED IN NDEs.

 

The Tunnel was possible for B’ was not substantial yet. Now he is, it’s no longer possible…

 

BUT THIS APPARENT CONSTRAINT ACTUALLY EXPLAINS THE RADICAL CHANGE IN THE PATIENT’S BEHAVIOUR WHEN HE WAKES UP:

BECAUSE WHAT HE WAS BEFORE p = 0, I.E. A BIOLOGICAL BODY WITH A SPIRITUAL INCONSISTENT EXTENSION, HAS BECOME A DOUBLE CONSISTENT BODY, WITH A CONSISTENT BIOLOGICAL PART “ON ONE SIDE OF THE LIGHT CONE” AND A CONSISTENT SPIRITUAL PART “ON THE OTHER SIDE OF THE LIGHT CONE”

HIS “SOUL” TOOK CONSISTENCE.

 

And that makes the system completely different from what he was before…

Because the two bodies can communicate through the light cone.

How?

 

THERE’S A VERY INTERESTING SPECIFICITY OF NEURON CELLS, THAT IS TO PRODUCE AN ELECTROMAGNETIC ACTIVITY. AND THIS ACTIVITY PRECISELY PROPAGATES AT THE SPEED OF LIGHT.

AS A CONSEQUENCE, THE BIOLOGICAL BODY AND THE SPECTRAL ONE (HIS “SOUL”) STAY IN CONNEXION AND CAN COMMUNICATE THROUGH THESE ELECTROMAGNETIC FIELDS.

AND THIS CAN EXPLAIN WHY A NDE CAN BE INTERPRETED AS A “CONSCIOUSNESS EXPERIMENT”: BECAUSE THE “LINK” BETWEEN THE BIO AND THE PSI, IN ANIMALS, CAN ONLY BE THEIR NERVOUS SYSTEM AND THEREFORE THEIR CONSCIOUSNESS.

 

I’m particularly satisfied with these results, as they first reconciliate all points of view, the biologists’, the spiritualists’ and the physicists’ and because they gratify a hard work done throughout these last two years, showing all I developed so far was kind of founded and that the blame was actually on the incompleteness of quantum physics. It was these incompleteness that led me into dead ends or to unadapted scenario like phase transitions.

 

I’m glad I managed to understand a tiny, but central, part of what’s susceptible to happen “when we die” and to bring a physically consistent scenario to NDEs.