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.

 

 

 

Since quantum probabilities were assumed to be oscillating as all the rest, there are basic properties needing re-examination in order to understand the concept of statistical motion at the foundations of the microscopic “cement” of thermodynamics and heat transfers.

 

For two classical events A₁ and A₂ with probabilities of occurrence P₁ = P(A₁) and P₂ = P(A₂), the fundamental properties of probabilities were defined as such:

 

(1)               P₁ + P₂ = 1

(2)               P(A₁ AND A₂) = P₁P₂

(3)               P(A₁ XOR A₂) = P₁ + P₂

(4)               P(A₁ OR A₂) = P₁ + P₂ - P(A₁ AND A₂)

(5)               P(A₁|A₂)P(A₂) = P[(A₁|A₂) AND A₂] = P(A₂|A₁)P(A₁) = P[(A₂|A₁) AND A₁]

 

Some comments, now.

 

Property (1) is known as the “normalization condition”, it says that the sum of probabilities linked with each event must be equal to 1, that is, we can be sure at least one of them is to occur.

 

Property (2) says the probability for two independent events to conjointly occur is equal to the product of the probabilities for each event, separately.

 

Property (3) says the probability for two disjoint events to occur is simply the sum of the probability of each event to occur. There’s a natural limitation here, due to the fact that the result, P₁ + P₂, remaining a probability, must be found between 0 and 1. In the case where there are only two events, property (1) guarantees the result is exactly equal to 1.

 

Property (4) is already more complicated, it says that the probability for at least 1 over 2 independent events to occur is equal to the probability (3) minus the probability (2).

 

Finally, property (5) is known as the Bayes rule, it’s about conditional probabilities: A₁|A₂ stands for “the realization of event A₁ is submitted to that of event A₂”. P(A₁|A₂) then measures the chance that conditioned event A₁|A₂ is to occur. As you can see, there is a symmetry between the probability the conjoint event (A₁|A₂) AND A₂ and its “reciprocal” (A₂|A₁) AND A₁ in the sense they have equal chance to occur.

 

It must be emphasized here that these basic properties of classical probabilities are the same as “cardinal numbers” in set theory: in this mathematical theory, a “cardinal number” is a number that measures the total number of elements of a given set. In some way, probabilities of occurrence are a measure of the total number of elements of non-deterministic sets or “Borel sets”, reported back to the closed interval [0,1] (the deterministic situation corresponding to the Boolean pair {0,1}). As such, one expects they follow the same rules as cardinals, which reveals to be the case, as soon as events are then considered as algebraic sets. This to say that there’s not only a physical justification to properties 1 to 5, there’s also and overall a much more formal mathematical one.

 

Properties (1-4) easily generalize to N classical events A₁,… A_N with probes P₁,…,P_N:

 

(6)               S_i=1^N P_i = 1

(7)               P(AND_i=1^M A_i) = P₁…P_M             (1 =< M =< N) 

(8)               P(XOR_i=1^M A_i) = S_i=1^M P_i             (1 =< M =< N)

 

To generalize (4), a bit of explanation, as the process is iterative. For three events, one has:

 

P[(A₁ OR A₂) OR A₃] = P[(A₁ OR A₂)] + P(A₃) - P[(A₁ OR A₂)]P(A₃)

= (P₁ + P₂ + P₃) - (P₁P₂ + P₂P₃ + P₃P₁) + (P₁P₂P₃)

= P[A₁ OR (A₂ OR A₃)] = P(A₁ OR A₂ OR A₃)

 

For four events,

 

P(A₁ OR A₂ OR A₃ OR A₄) = (P₁ + P₂ + P₃ + P₄) - (P₁P₂ + P₂P₃ + P₃P₁ + P₁P₄ + P₂P₄ + P₃P₄)

                                                + (P₁P₂P₃ + P₁P₂P₄ + P₁P₃P₄ + P₂P₃P₄) - (P₁P₂P₃P₄)

 

One can see expressions are completely symmetric with respect to the events. The explanation lays in the fact that the terms I voluntary placed between brackets are the coefficients of the (algebraic) polynomial of degree M, (P’ - P₁)…(P’ - P_M). Indeed, for M = 2:

 

(P’ - P₁)(P’ - P₂) = P’² - (P₁ + P₂)P’ + P₁P₂ = P’² - c₁P’ + c₂

 

so that (4) rewrites P(A₁ OR A₂) = c₁ - c₂. For M = 3,

 

(P’ - P₁)(P’ - P₂)(P’ - P₃) = P’³ - (P₁ + P₂ + P₃)P’² + (P₁P₂ + P₂P₃ + P₃P₁)P’ - P₁P₂P₃

     = P’³ - c₁P’² + c₂P’ - c₃

 

and P(A₁ OR A₂ OR A₃) = c₁ - c₂ + c₃. You got it now: P(A₁ OR A₂ OR A₃ OR A₄) = c₁ - c₂ + c₃ - c₄ and so on, where the c_is all depend on the M probabilities P_j. As a result:

 

(9)               P(OR_i=1^M A_i) = S_k=1^M (-1)^k-1S_i(1)=1^M-k+1S_i(2)=i(1)+1^M-k+2…S_{i(k)=i(k-1)+1}^M P_i(1)…P_i(k)

(1 =< M =< N)

 

As for Bayes, it generalizes into:

 

(10)           P(A_i+1|A_i)P(A_i) = P(A_i|A_i+1)P(A_i+1)           (1 =< i =< N)

 

 

The now “cyclic” (…) question is: what does this all become in the quantum?

 

Let A_i(ALPHA_i), 1 =< i =< N, be N quantum events with probabilities of occurrence P_i(PI_i) = P(PI)[A_i(ALPHA_i)]. According to the rule on the sum of pairs:

 

(11)           S_i=1^N [P_i(0),PI_i] = [P(0),PI]

(12)           [P(0)]² = S_i=1^N [P_i(0)]² + 2S_i=1^N-1S_j=i+1^N P_i(0)P_j(0)cos(PI_i - PI_j)

     = [S_i=1^N P_i(0)]² - 4S_i=1^N-1S_j=i+1^N P_i(0)P_j(0)sin²[½(PI_i - PI_j)]

(13)           tan(PI) = [S_i=1^N P_i(0)sin(PI_i)]/[S_i=1^N P_i(0)cos(PI_i)]

 

P_i(0) = P(0)[A_i(0)] is the probability the classical event A_i(0) occurs. PI_i is the quantum state of the probability P_i(PI_i) = P(PI)[A_i(ALPHA_i)] the quantum event A_i(ALPHA_i) occurs (while ALPHA_i is the quantum state of this event itself). As:

 

(14)           (1,0)/[P(0),PI] = [1/P(0),-PI]

 

the quantum equivalent to the normalization condition (1) writes,

 

(15)           [1/P(0),-PI]S_i=1^N [P_i(0),PI_i] = S_i=1^N [P_i(0)/P(0),PI_i - PI] = (1,0)

 

We can check it includes negative probabilities. Indeed, for PI = pi, [P(0),pi] = -P(0), tan(pi) = 0 and according to (13), this corresponds [together with tan(0) = 0] to a “Fresnel-like” relation:

 

(16)           S_i=1^N P_i(0)sin(PI_i) = 0

 

Conversely, when all PI_i are equal and equal to pi, all P_i(PI_i) = -P_i(0), relation (16) is automatically fulfilled, P(0) = S_i=1^N P_i(0), PI = pi (it cannot be 0) and S_i=1^N [P_i(0)/P(0),0] = S_i=1^N P_i(0)/P(0) = (1,0) = 1 becomes a classical tautology. We therefore needs to precise that, if the total number of classical events likely to occur is N, then S_i=1^N P_i(0) = 1.

 

Extending (7) is easy. According to the quantum product, it’s simply:

 

(17)           P(PI)[AND_i=1^M A_i(ALPHA_i)] = P₁(PI₁)…P_M(PI_M)                       (1 =< M =< N)

(18)           P(0)[AND_i=1^M A_i(0)] = P₁(0)…P_M(0)

(19)           PI[AND_i=1^M ALPHA_i] = S_i=1^M PI_i

 

Amplitudes multiply, giving back classical property (7), while quantum states add.

 

To quantize (9), we need to extend (alternated) sums of products. S_i1=1^M P_i1(PI_i1) is done. Let’s call it P’₁(PI’₁) = [P’(0),PI’₁], not to confuse it with P₁(PI₁). Then:

 

S_i1^M-1S_i2=i1+1^M P_i(1)(PI_i(1))P_i(2)(PI_i(2)) = [P’₂(PI’₂)]² = {[P’₂(0)]²,2PI’₂},

S_i1^M-2S_i2=i1+1^M-1S_i3=i2+1^M P_i(1)(PI_i(1))P_i(2)(PI_i(2))P_i(3)(PI_i(3)) = [P’₃(PI’₃)]³ = {[P’₃(0)]³,3PI’₃},

…

S_i(1)=1^M-k+1S_i(2)=i(1)+1^M-k+2…S_{i(k)=i(k-1)+1}^M P_i(1)…P_i(k) = [P’_k(PI’_k)]^k = {[P’_k(0)]^k,kPI’_k},

…

 

and the highest contribution, k = M, was done too, in (17). Finally:

 

(20)           P(PI)[OR_i=1^M A_i(ALPHA_i)] = S_k=1^M (-1)^k-1[P’_k(PI’_k)]^k                 (1 =< M =< N)

    = -S_k=1^M [P’_k(PI’_k + pi)]^k

 

is the quantum formulation for the probability of M quantum events to occur or not. Decomposing it into classical amplitudes and quantum states won’t lead to simple formulas at all, being given that the classical formulation (9) is already rather complicated.

 

Bayes becomes:

 

(21)           P(PI)[A_i+1(ALPHA_i+1)|A_i(ALPHA_i)]P(PI)[A_i(ALPHA_i)] = P(PI)[A_i(ALPHA_i)|A_i+1(ALPHA_i+1)]P(PI)[A_i+1(ALPHA_i+1)]        (1 =< i =< N)

 

Amplitudes gives classical Bayes back, while quantum states verify:

 

(22)           PI(ALPHA_i+1|ALPHA_i) + PI(ALPHA_i) = PI(ALPHA_i|ALPHA_i+1) + PI(ALPHA_i+1)

 

which is automatically satisfied for:

 

(23)           PI(ALPHA_i|ALPHA_i+1) = -PI(ALPHA_i+1|ALPHA_i)

 

giving,

 

(24)           PI(ALPHA_i+1|ALPHA_i) = ½ [PI(ALPHA_i+1) - PI(ALPHA_i)]

 

with the straightforward consequence that,

 

(25)           PI(ALPHA_i|ALPHA_i) = 0

 

A normal result after all, since an event cannot be conditioned to itself prior to occur…

 

(classical Bayes says nothing about this, as the relation reduces to a mere identity)

 

 

Means values, variances and higher momenta

 

Let now x stand for a classical statistical variable able to take N discrete values x_i with the probability P_i = P(x_i). The (statistical) mean value of x is the number:

 

(26)           <x> = S_i=1^N x_iP_i

 

and the momentum of order m of x (or “mth-momentum) is the statistical mean value of the mth-power of x:

 

(27)           <x^m> = S_i=1^N (x_i)^mP_i

 

These very general definitions can be readily extended to the quantum under the form:

 

(28)           <[x(ksi)]^m> = S_i=1^N [x_i(ksi_i)]^mP_i(PI_i)  ,  P_i(PI_i) = P[x_i(ksi_i)]

 

assuming we now consider a quantum statistical variable x(ksi) = [x(0),ksi] likely to take N discrete values x_i(ksi_i) = [x_i(0),ksi_i] with probabilities P_i(PI_i), I = 1,…,N. However, we need be careful of something, as x is no longer deterministic, but statistical, that is, random: opposite to a deterministic variable with a series of N values to take, we’re no longer sure in advance the value x_i (in the classical) or x_i(ksi_i) (in the quantum) is going to occur. We can only predict it will, with a chance of realization P_i [or P_i(PI_i)]. If, in the classical, this has no other consequence than “being unknown in advance”, it does have in the quantum, when we’re going to evaluate mean values, because of the summation over the N states. As we know, this summation is going to induce interferences between terms. Now, intuitively, how can we conceive an interference between a value that is indeed going to occur (in a near future) and another, that will not? Or, worse, between two values that both won’t?... :|

 

Let’s take N = 2 and m = 1 as an illustration. We then have that mean value:

 

(29)           <x(ksi)> = x₁(ksi₁)P₁(PI₁) + x₂(ksi₂)P₂(PI₂)

         = [x₁(0)P₁(0) , ksi₁ + PI₁] + [x₂(0)P₂(0) , ksi₂ + PI₂]

 

According to (12b) for N = 2, the amplitude of that mean value is therefore:

 

(30)           <x(0)>² = [x₁(0)P₁(0) + x₂(0)P₂(0)]² -

- 4x₁(0)x₂(0)P₁(0)P₂(0)sin²{½[ksi₁ - ksi₂ + PI₁ - PI₂]}

 

and its quantum state:

 

(31)           tan(<ksi>) = [x₁(0)P₁(0)sin(ksi₁ + PI₁) + x₂(0)P₂(0)sin(ksi₂ + PI₂)] /

[x₁(0)P₁(0)cos(ksi₁ + PI₁) + x₂(0)P₂(0)cos(ksi₂ + PI₂)]

 

These are the values we expect. Mean values are tendencies. After observation, what if x₁(0) occurs, but not x₂(0)? Then, the result we’ll observe will have become P₁(0) = 1, P₂(0) = 0 and <x(0)> = x₁(0), tan(<ksi>) = tan(ksi₁ + PI₁) = tan(ksi₁) since PI₁ will be zero and finally, <ksi> = ksi₁ modulo pi, so that <x(ksi)> will either be x₁(ksi₁) or -x₁(ksi₁). So, here we are, with a value x₂(ksi₂) we predicted (because we couldn’t do otherwise) but did not concretize.

 

Where does the interference term in (30) actually come from, then?

 

From our own prediction process and nowhere else. Should we have made no prediction, should we simply have awaited for the results to come out, we would have found no interference anywhere, because results are submitted to a chance of realization. So:

 

Only in the deterministic are interferences unavoidable.

In the statistical, because results are pondered with chances of concrete realization, interferences are only due to the prediction the observer makes.

 

When none of these two events occur, our theoretical predictions (30-31) fall completely aside, since the observed result is then <x(0)> = 0 and tan(<ksi>) = 0 implying <x(ksi)> = 0…

 

So, extremely careful with probabilities, because we only try to guess results. And, if we take them too much for granted, in the quantum, it will even induce false models containing artificial interferences… :| Instead, always keep in mind that:

 

THE PREDICTION IS NOT THE RESULT.

 

And the worst of the worst would be to “transfer probabilities from the initial values they were affected to to the trigonometric function characterizing the interference term”: this would be total nonsense… :(

 

Well, maybe here and to the disappointment of technicians, the technical impossibility for us to measure correlated quantities with maximum accuracy led the 20^th-century physicists onto an inappropriate road, when they assimilated that quantum measurement was to be accepted as fundamentally statistical, just because “one couldn’t know in advance”, despite they, meanwhile, accepted the fact that statistics based on that spectroscopic limitation “had nothing quantum in itself”. Statistics is found everywhere in the Universe… The motion of meteors in the solar system is entirely statistical… Nature doesn’t “all of a sudden” turn statistical because we cannot measure both the signal and its frequency spectrum on correlated quantities… I even go up to think that, if we had a more powerful mathematical tool than the linear integral, maybe (I say: maybe) we could find exact solutions to the many-body problem over 2 bodies without needing to introduce statistics… Poincaré the first recognized that motions in systems with more than 2 bodies couldn’t be determined because the problem was “not integrable in quadratures”… so that we had no deterministic tool to show us the shape of the general solution… This doesn’t mean that because we lack such tools, Nature should be statistical…

 

“Quantum” means “naturally, spontaneously, oscillating”. It doesn’t mean “statistical”, “non-commutative” or else… Again, in the solar system around us, some motions are non-commutative, because they’re bounded!... :)

 

Another pre-conceived idea in quantum theory was that the “non-relativistic vacuum state was Gaussian”. “Non-relativistic” in the sense it followed Galileo’s relativity of space only. The “vacuum state” was the state of lowest energy, with no field particle present, only (guess what?) “statistical fluctuations”. And Gaussian? I’m sorry, but you take any introductory book to probabilities and statistics, you’ll find written in it that the Gaussian distribution (bell shape) is an approximation (only an approximation) of the much more general binomial distribution, when the total number of samples is very high and values, extremely closed to their mean value… That’s too significant (not to say “severe”) restrictions. To put it differently, the Gaussian distribution has nothing universal at all… and we made it a rather universal feature of quantum vacuums… In the theory of the “quantum oscillator”, for instance, it clearly appears the “wavefunction” of the vacuum is a Gaussian and “excited modes” (where particles are produced), derivatives of that Gaussian… wow… and people were surprised, by the end of the 1990s, that the whole building collapsed when confronted to astronomical datas… :|

 

I’m sure a lot think from the beginning I’m “wasting my time” re-examining, one by one, the fundamentals, the basis, of quantum theory. But it appears that we went very, very far away from all these fundamentals. And what this global re-examination is showing me up to now is that I don’t get the same lecture of equations as the one I can find in all my literature about quantum theory, “relativistic” or not… :|

 

For “intellectual recreation”, you can still calculate <[x(ksi)]²> and the variance, for N events. You’ll come up with the same conclusion: artificial interferences, but for all probas to 0 or 1… Why don’t I do it? For always the same argument: because taking probabilities as a physical reality and not as a mere mathematical tool as it should be induces that fake image of mean values being “bits of mixed values”. Look again at (26): if we consider P_i as having some physical content, then we’re led to believe that x_iP_i is “a bit of x_i”; as a consequence, we’ll get that picture, maybe unconsciously, that <x> is a “sum of bits”, giving another “bit”… which is absolutely not reality. It only gives us a global tendency of a reality we expect.

 

Where it becomes really concerning and even dreadful is when, in finance, social or economics, it’s turned as master rules… K As if it was the way flows behave… As if forecasting the weather over five days, despite now, a physical limitation due to chaos, was “natural” because our statistical models say it…

 

Wait… we’re making here predictions a reality… :| we’re making expectations certainties… “it’s got to be this way, because all tendencies converge to this, our models say…”

 

We feel it easier, here, to see the world around us as we’d like it to be rather than as it is, whether we like it or not…

 

“oh, look at these formulas: quantum theory extremely complicated, especially when you introduce time relativity, lengthy expressions you find nowhere else, not even at the IRS… (did I mention it? No, I pay my taxes like anybody else, no worry…), it requires the latest super-computers… and still, you make them smoke…”

 

:|… The fundamental can only be poor… :) and when you take away from quantum theory all that is not quantum… what remains look basic, childish… as any basic environment… J

 

The world wasn’t made complicated, it was made simple.

 

And it rather seems that the most complicated for the human mind is to make simple… J

 

We built a society where, if you make too simple, you’re asked: “what did you study for, all these years?...”

 

And an “Eastern-style answer” like: “I’ve studied to understand, work this knowledge, to realize, in the end, that I had to go back to the most fundamentals and do it all again” is just not tolerated… J

 

Instead, the “popular” reaction is: “it’s impossible, it just can’t be as simple as that…”

 

Except that Nature 1) was born well before us and 2) doesn’t care at all what we think… J

 

We’re mere observers. We learn from it, not the converse.

 

Nature has nothing to learn from us so far, but about our genuine appetite for (self)-destruction, absurdities, useless complications… and self-satisfactions.

 

I’m satisfied with doing as simple as I can. And if I could do even simpler, I’d do it.

 

As R.P. Feynman used to say in his lectures: “the equation of Nature is U = 0; the problem is, we don’t have a clue what U is…” J

 

This could sound as a criticism, when I was younger, I’d have agreed, now I grew older, it’s not, it’s a mere constatation. And, somewhere, maybe… it’s worse. J

It’s worse, because we’re destroying a civilization that also made great things. Only because “we wanna be God before God”… L but that’s the way it is, I fear we went too far to go back and that’ll be my final word.