We now turn to a topic I want to study for quite a long time: thermodynamics. It may interest financers as well, as the principles of thermodynamics and statistical physics find applications in finance, through stochastic processes. As we’re going to see it, the extension of thermodynamics to the quantum leads to very different conclusions and requires a review of well-anchored pre-conceived ideas about probabilities, temperatures and entropies (or “degrees of disorder”).

 

We begin with the concept of probability.

 

Classically, a probability P(0) is the chance an event has to occur. It measures the lack of certainty we have before that event to occur or not. When we can be sure it’s going to occur, P(0) = 1, meaning the chance of occurrence is 100% and we call the event deterministic (i.e. such that it can be determined in advance with certainty). On the other end, when we’re sure it cannot occur, we call its realization impossible and P(0) = 0. So, classically, we can understand a negative probability would have no meaning, for it would mean that the event in question is “even more impossible to occur”, which is absurd in itself: once it’s proven impossible to occur, it’s simply impossible, period.

 

Geometrically now, any physical quantity can be attributed a space, which does not need to be physical. Non-physical spaces are called “abstract” and only serve to visualize things. So is the case for a 1D “probability space”, which is a purely mathematical one, identified with the closed interval [0,+1]. The classical theory of probabilities tells us that it’s symmetric, [-1,0], is to remain empty, geometrical translation of “there exists no negative probabilities”. So, we have kind of a “polarization” here, where the entire content is in [0,+1] and there’s nothing “on the other side of the zero”.

 

In the quantum, things become different. The classical probability P(0) only represents the amplitude of the quantum probability:

 

(1)               P(PI) = [P(0) , PI]

 

P(0) continues taking its values into [0,+1], but the presence of a quantum state PI now demands that there’s, not only a prolongation of this interval to a symmetric [-1,0], but that the so prolonged axis [-1,+1] be doubled. When PI = 0, we’re in the initial segment [0,+1]. According to what we saw in B140, when PI = pi/2, axis are permuted, so that this segment is found on the “wavy” axis, while the “corpuscular” probability falls down to 0. What this actually says is that the chance to see a quantum event displaying a substantial way is now null, so that this event is to be expected as purely “wavy”, with a probability of occurrence P(0). But, when PI = pi, axis are turned upside-down, so that the initial [0,+1] segment becomes the [-1,0] one. As a result, [0,+1] is now empty. It has been emptied of all its elements, transferred to its symmetric with respect to zero. So, when PI = pi, the classical situation is somehow “reversed” and P(pi) = [P(0),pi] probabilities are only defined between -1 and 0. Positive ones do not exist, as they would, in turn, be “even more unlikely than the impossible”, which would again sound absurd… :) Finally, when PI = 3pi/2, we have a combination of reversion and permutation of axis, the event is expected as purely wavy with a negative probability -P(0) of occurrence and no chance to appear under a substantial form.

 

In the quantum, the “positive” is simply the unsigned, the “negative” is a “positive” in phase opposition and the “positive”, the “negative” in phase opposition.

 

This is what is called in mathematics an involution:

 

-(-1) = +1 = 1

 

An operator (here “change of sign”) applied twice gives back the original result. In other words, it “neutralizes itself”. Here, a first application of “change the sign” turns 1 into -1; a second application annihilates the first action: --1 = +1 = 1. An involution can thus be seen as an alternance of creations and annihilations: ---1 = -1, etc.

 

If we project P(PI) onto its “corpuscular” and its “wavy” axis, we will find oscillating probabilities:

 

(2)               P¹(PI) = P(0)cos(PI)  ,  P²(PI) = P(0)sin(PI)

 

always smaller (in unsigned values) than the value classically calculated: in the quantum, the classical is always the “most optimistic” measure. So, there’s already a correction to be brought between probabilities of occurrence predicted on a classical approach and those predicted on a quantum one. That correction is obviously due to taking quantum states into account, which always review results to the decrease.

 

With the concept of probability comes that of entropy. Entropy is a measure of the lack of information about an event (Shannon’s ‘paradigm’, generalizing Boltzmann’s definition). Classically, it’s defined as:

 

(3)               s(0) = -k_BP(0)Ln[P(0)]

 

Ln(.) is the “natural” logarithm, the reciprocal function to exponentiation exp(.) or elevation of the irrational e = 2,718281828456… to a power. As P(0) always stands between 0 and 1 and Ln(1) = 0, the logarithm of P(0) is always negative, hence the minus sign in (5), for entropy needs to remain a non-negative quantity. k_B is Boltzmann’s constant, approximately 1,38 x 10^-23 J/K (Joule per Kelvin). Why should s(0) never turn negative? For always the same reason. If P(0) = 1, the event is certain and (5) gives s(0) = 0: no disorder, no lack of information. If P(0) = 0, the event is impossible (we’re sure of this too), zero elevated to the power zero gives 1 by convention (and graphic behavior), so that P(0)Ln[P(0)] = Ln[P(0)^P(0)] = Ln(1) = 0 and s(0) = 0 again: no more disorder than for P(0) = 1, the situation is just opposite in chances of realization [besides, considering the probability P’(0) = 1 - P(0) of non-realization and applying the entropy formula to it would bring you back to the P’(0) = 1 situation]. For s(0) to turn negative, we would need (as k_B is positive) Ln[P(0)^P(0)] > 0 and P(0)^P(0) > 1, which is impossible in the limits [0,1].

 

It’s a general feature of quantum theory that the question of the physical constants occurs. In the case of k_B, it becomes the amplitude k_B(0) = k_B of a quantum coefficient k_B = [k_B(0),kappa_B]. Is the quantum state kappa_B to be a constant as well or, on the contrary, should we let it vary? I don’t know. I know of no experiment where k_B could vary. But we can leave the phase.

 

The second point I noticed when working on the quantization process was that elementary mathematical functions defined first in the classical should be extended to the quantum and not be used as still-classical functions. Hence B139 and this is logical after all, as all complex-valued functions of a complex-valued variable develop as:

 

f(phi)[x(ksi)] = f(0)[x(0),ksi]exp{iphi[x(0),ksi]}

 

The amplitude f(0)[x(0),ksi] is not even the initial classically-defined function: that one is re-obtained fixing the phase ksi of the variable to zero and the phase phi[x(0),ksi], which then takes the particular value phi[x(0),0] to the UV limit 0 or pi (so as to allow both signs). These are quite severe conditions, showing the set of quantum functions of a quantum variable is much wider than the set of classical functions of a classical variable, making the use of classical functions of quantum variables a dubious extension (I didn’t mention it again in my discussion about Riemann surfaces in B140, but classical functions are still used in that geometrical description as well…).

 

It appears that what seems to be the most proper way to define quantum entropy is to propose the following formula (in polar representation, see B139 example 2 for technical details):

 

(4)               s(sigma) = [s(0),sigma]

(5)               s(0)[P(0)] = k_B(0)P(0)|Ln[P(0)]|

(6)               sigma[P(0),PI] = kappa_B + PI + Lambdaeta[P(0),PI] + pi

 

In the definition (5) of the amplitude of quantum entropy, I used the classical logarithm of the classical probability P(0). This is not a problem at all, since s(0) and sigma are classical components anyway and it’s always possible to use the amplitude of the quantum logarithm reversing formula (B139-32):

 

(7)               {Ln[P(0)]}² = {Ln(0)[P(0),PI]}² - PI² >= 0

 

Why not using Ln(0)[P(0),PI]? Because of (B139-36a), which predicts an amplitude s(0)(1,PI) = k_BPI for classically sure events. In itself, there’s nothing absurd as, for PI = 0, s(0)(1,0) = 0 as in the classical. But, for PI = pi, P(pi) = -1 would be given a “lack of information” s(0)(1,pi) = k_Bpi, whereas this event is still considered as certain to occur. So, this would actually break the symmetry we’ve just obtained between positively-counted probabilities and negatively-counted ones, while nothing physical would justify it (on the contrary). That’s why, with the concern of keeping that symmetry, I preferred to slightly modify the classical expression, first, to take the quantum state PI of the probability into account in the expression of quantum entropy and, second, to retrieve the same results as in the classical for P(0) = 0 and P(0) = 1, if to obtain an s(0) independent of PI. As for the former minus sign in the classical expression (5), it’s now included in the quantum state sigma of quantum entropy, since -1 = (1,pi).

 

Consider a quantum system. Then, s(sigma) is a (quantum) measure of its disorder,

 

(8)               s₁(sigma)[P(0),PI] = s(0)[P(0)]cos{sigma[P(0),PI]}

 

is a measure of its substantial disorder and

 

(9)               s₂(sigma)[P(0),PI] = s(0)[P(0)]sin{sigma[P(0),PI]}

 

a measure of its wavy disorder. Both “projective” measures can now turn negative. As for probabilities, we can understand where it comes from: a phase opposition on the quantum state of entropy. Indeed:

 

(10)           s₁(sigma + pi)[P(0),PI] = s(0)[P(0)]cos{sigma[P(0),PI] + pi}

        = -s₁(sigma)[P(0),PI]

 

and

 

(11)           s₂(sigma + pi)[P(0),PI] = s(0)[P(0)]sin{sigma[P(0),PI] + pi}

        = -s₂(sigma)[P(0),PI]

 

Below is a summary of the most specific situations.

 

a)      order corresponds in the classical to s = 0 which, from (3) is reached only for P = 0 or P = 1. In the quantum, it corresponds to s(0)[P(0)] = 0 which, from (5) is reached for the same values of P(0);

b)      Systems with s = cte non zero are classically isentropic and (3) leads to P = cte. Systems with s(sigma) = cte non zero, i.e. s(0) = cte and sigma = cte are quantum isentropic. However, setting both (5) and (6) to constant values induces a functional relation between P(0) and PI. Now, these two components are independent. So, the final result is again P(0) = cte, together with PI = cte;

c)      All other entropic situations feature disordered systems;

d)      An anti-disorder is a disorder in phase opposition. As a disorder is measured with a positive entropy, an anti-disorder is to be measured with a negative entropy. But, it’s still a disorder!

 

Following these fundamentals:

 

a)      sigma[P(0),PI] = 0, substantial disorder, non-substantial order;

b)      sigma[P(0),PI] = pi/2, substantial order, non-substantial disorder;

c)      sigma[P(0),PI] = pi, substantial anti-disorder, non-substantial order;

d)      sigma[P(0),PI] = 3pi/2, substantial order, non-substantial anti-disorder.

 

 

Non-reversibility in the quantum

 

We now come to a central point of the general theory of disordered systems: non-reversibility.

 

Classical non-reversibility is expressed by Boltzmann’s “H” theorem which concludes by saying that the entropy of a classical system should always increase with time:

 

(12)           ds(t)/dt > 0

 

The quantum formulation of this “instantaneous variation” is D_t(tau)s(sigma)[t(tau)] and it’s obviously a quantum grandeur. As such, it carries “no sign or all of them at the same time”, so that inequalities like D_t(tau)s(sigma)[t(tau)] > 0 or < 0 would completely be meaningless: one can only compare classical numbers. The only relation that holds in the quantum is:

 

(13)           D_t(tau)s(sigma)[t(tau)] = 0

 

and it deals with isentropic quantum systems. When we look at quantum functions of a quantum variable, f(phi)[x(ksi)] = {f(0)[x(0),ksi],phi[x(0),ksi]}, we find no less than four variations in place of a single one for classical functions f(x) of a classical variable:

 

-         D_x(0)f(0) = ratio between the differential of the function amplitude and that of the variable amplitude;

-         D_ksif(0) = ratio between the differential of the function amplitude and that of the variable quantum state;

-         D_x(0)phi = ratio between the differential of the function quantum state and that of the variable amplitude;

-         and D_ksiphi = ratio between the differential of the function quantum state and that of the variable quantum state.

 

In mechanics, for instance, a quantum motion x(ksi)[t(tau)] through quantum space induces four velocities. It speaks better in planar:

 

-         D_t1(tau)x₁(ksi) = v₁₁(t₁,t₂), “corpuscular (instantaneous) velocity”;

-         D_t1(tau)x₂(ksi) = v₁₂(t₁,t₂) = instantaneous variation of the “wavy” motion or wavelength reported to that of the “corpuscular time”;

-         D_t2(tau)x₁(ksi) = v₂₁(t₁,t₂) = instantaneous variation of the “corpuscular” motion reported to that of the “wavy time” or period of a signal;

-         and D_t2(tau)x₂(ksi) = v₂₂(t₁,t₂) = instantaneous variation of the wavelength reported to that of period = group velocity;

 

So, we find these two notions of the velocity of a “solid”, v₁₁, and the velocity of a “wavepacket” in a signal, v₂₂, familiar to physicists, plus two “mixed” or “crossed” velocities, v₁₂ and v₂₁.

 

In the case of quantum entropy, the quantity that interests us is the total variation:

 

(14)           ds(0) = D_t(0)s(0)dt(0) + D_taus(0)dtau

 

of the amplitude s(0)[t(0),tau] when t(0) varies a infinitesimal quantity dt(0) and dtau, an infinitesimal angle dtau. It’s legitimate again, here, to use the classical differential d, as we’re to deal with classical grandeurs only: t(0), tau and s(0)[t(0),tau]. There’ll be an increase in entropy (and therefore, in disorder) when ds(0) > 0 and this will happen for:

 

(15)           D_t(0)s(0)dt(0) > -D_taus(0)dtau

 

This condition has nothing restrictive anymore, because both dt(0) and dtau can be either positive, null or negative. We have no other choice here but to be a bit repetitive, because we have no less than eight situations to examine. We will also assume that the derivatives are everywhere regular, so that both D_t(0)s(0) and D_taus(0) are finite ratios. The ninth situation, dt(0) = 0 and dtau = 0 is of no interest here.

 

i) dt(0) > 0, dtau = 0 (i.e. tau = cte):

 

(16)           D_t(0)s(0) > 0

 

says the amplitude of entropy must increase with that of time. Same result as in the classical.

 

ii) dt(0) < 0, dtau = 0:

 

(17)           D_t(0)s(0) < 0

 

As the amplitude of time decreases, that of entropy must increase: increase of disorder as we go back in the past.

 

iii) dt(0) > 0, dtau > 0:

 

(18)           D_t(0)s(0) > -D_taus(0)dtau/dt(0)

 

is the condition. As the ratio dtau/dt(0) > 0, if D_taus(0) < 0, s(0) decreases (resp. increases) with decreasing (resp. increasing) tau and (18) then says that s(0) must meanwhile increase as t(0) increases or decrease as t(0) decreases, respecting the lower bound -D_taus(0)dtau/dt(0). If D_taus(0) > 0, D_t(0)s(0) is only required to remain higher than a negative value, allowing the case D_t(0)s(0) < 0.

 

iv) dt(0) < 0, dtau > 0:

 

(19)           D_t(0)s(0) < -D_taus(0)dtau/dt(0)

 

same as above, but reversed.

 

v) dt(0) = 0, dtau > 0:

 

(20)           D_taus(0) > 0

 

wow… at constant time amplitude, as we now have a quantum state, the condition only holds on the variation of s(0) with respect to tau: if tau increases (resp. decreases), so has to do s(0), no matter what the variation of t(0) is. So, if this holds, it will in the future as in the past.

 

vi) dt(0) > 0, dtau < 0:

 

(21)           D_t(0)s(0) > -D_taus(0)dtau/dt(0)

 

hm… same as iii), except that, the sign of D_taus(0) is reversed.

 

vii) dt(0) < 0, dtau < 0:

 

(22)           D_t(0)s(0) < -D_taus(0)dtau/dt(0)

 

same as iv).

 

viii) dt(0) = 0, dtau < 0:

 

(23)           D_taus(0) < 0

 

or v) reversed.

 

These are the eight conditions to fulfill for an increase of disorder in the quantum. For an increase of order, ds(0) < 0, would lead to eight symmetric conditions.

 

If you see anything really restricting in these conditions, whether on the setting of disorder or of order, thanks in advance for showing me, I would have missed it…

 

(can leave comments at the end of articles)

 

Well, folks, what we do not see being not forbidden, it leaves… space… (and a large one) to self-regenerative systems… :)

 

Indeed, for a system to regenerate by itself, it suffices that, after increasing its internal disorder, it’s able to reorder things. Apparently this does not contradict quantum thermo-dynamics at all. You notice in passing that an increase (resp. decrease) of disorder in the future doesn’t mean for as much an increase (resp. decrease) of order in the past: you can check, for instance, that between iii) and iv), there’s no time reversal in the entropic behavior, precisely because of the presence of D_taus(0) that can take both signs in each case. As a result, reversing time while maintaining the same variation for its quantum state does not reverse the process for as much.

 

This may be the best indicator that self-regeneration can be taken for serious in the quantum. You can now find systems the degree of disorder of which will increase [ds(0) > 0] for a certain amount of time t(0), then decrease [ds(0) < 0] for another amount of time, while still pointing towards future!

 

Nobody talks here about identically rebuilding a body: self-regeneration has never been about this, such an idea is an “extrapolation” of it… :) In mechanics, a self-regenerative system is basically a system able to renew the amount of energy it lost: call H the total energy of a mechanical system; if the system dissipates part of its energy, dH/dt < 0 (the amount of energy decreases as time increases) and a regeneration would bring energy in (dH/dt > 0) so that the system retrieves its initial amount. This is the case for all the “living”, by the way, from the biological cell and the unicellular body up to evolved mammals: they spend energy and renew it feeding… They don’t need to travel back in time to get the energy they lost back… :))

 

Here, it’s the same, except that it’s now allowed even to inert substance…

 

There’s an extremely important and general feature of the quantum environment I’d like to discuss about in this bidouille and I’m really sorry I can’t make any drawing that would or wouldn’t be readable to everybody, because visualizing things would make it much easier and straighter to understand. It’s about progressions and regressions in the quantum world. There’s a general picture we all need to understand (me included) in order to be able to make meaningful reasonings.

 

Consider three classical values x_-(0), x(0) and x₊(0) such that 0 =< x_-(0) < x(0) < x₊(0). In a linear progression, they’re graphically represented as succeeding points along a straight line: x_-(0) being the smallest value is the first point, then comes x(0) and finally x₊(0). In a non-linear progression, these three points still succeed to one another (as it’s a progression), but stand on a curve instead of a straight line. If we now take x(0) as a reference point, then we usually consider that x_-(0) comes “before” or “precede” x(0) and that x₊(0) comes “after” or “succeed to” x(0). This is how we build “arrows” in physics: from a numerical ordering of values. When these points represent positions in space and x(0) is used as an observer’s location, then this observer considers x_-(0) stands “behind” him/her and x₊(0) “ahead” of him/her: our observer orientates distances around him/her, ordering space. When these points represent instants in time and x(0) is used as “present”, then x_-(0) is assumed to be “in the past of x(0)” and x₊(0) in its “future”: same ordering, same notion of an “orientation” that gives birth to an “arrow of time” or “succession of instants” or else, “time flow”. This is in the classical, where comparisons hold: “greater than or equal to, >=”, “smaller than or equal to, =<”, “strictly greater than, >” and “strictly smaller than, <”.

 

How about in the quantum?

 

In the quantum, a classical point x(0) becomes a quantum point x(ksi) = [x(0),ksi] in polar representation or [x₁(ksi),x₂(ksi)] in planar rep. This means that both x(0) and ksi or x₁(ksi) and x₂(ksi) are fixed in the classical plane. If we let ksi free of varying now, the continuous family of quantum points x(ksi) draws a circle in classical plane with radius x(0) neither x₁(ksi) nor x₂(ksi) can exceed anyway. Physically, again, it means the classical value x(0) is also the highest value both projections can reach.

 

If we now take two distinct quantum points x(ksi) = [x(0),ksi] and x₊(ksi) = [x₊(0),ksi] with same inclination ksi in the classical plane, these points will stand on a line inclined an angle ksi with respect to the “corpuscular” axis and, if we set x₊(0) > x(0) >= 0, then x₊(ksi) will come after x(ksi), as it will be further away from the origin (0,ksi). When this happens, we can say x₊(ksi) succeeds to x(ksi). If we take a third quantum point x_-(ksi) = [x_-(0),ksi] such that 0 =< x_-(0) < x(0), than x_-(ksi) will stand on the same line and precede x(ksi).

 

What about if we take two quantum points x(ksi) = [x(0),ksi] and x₊(ksi₊) = [x₊(0),ksi₊] with still x₊(0) > x(0) >= 0, but different inclinations ksi and ksi₊? What can we say about x₊(ksi₊) regarding x(ksi)?

 

Distances to a given origin in the classical plane are determined by amplitudes: x(0) is the distance of x(ksi) to that origin and x₊(0) is that of x₊(ksi₊), no matter their quantum states. Because, if you take ksi and ksi₊ into account, you can find x₁(ksi) smaller or larger than x₁₊(ksi₊) and same for x₂(ksi) compared to x₂₊(ksi₊) as long as the equalities:

 

(1)               [x₁(ksi)]² + [x₂(ksi)]² = [x(0)]²

(2)               [x₁₊(ksi)]² + [x₂₊(ksi)]² = [x₊(0)]²

 

and the inequality,

 

(3)               0 =< x(0) < x₊(0)

 

are respected. So, it all comes back, in the end, to amplitudes and this raises an important question:

 

What is the more significant in the quantum, the notion of points or that of circles?

 

With the notion of points, we work on fixed amplitudes and fixed quantum states and we go from one fixed pair to another. Then, x(ksi) represents the fixed pair [x(0),ksi].

 

With the notion of circles, we work on fixed amplitudes, but free quantum states and we go from one circle with a fixed radius to another. Then, x(ksi) represents the circle (1) above.

 

If x(0) remains the only meaningful distance to an origin and ksi “only” serves as orienting a point with respect to a pair of axis intersecting through that origin, then it appears much better to use circles instead of points, because on a circle centered on the origin of a 2D frame, all points equally stand at a distance x(0) from that origin, so that all quantum states are included instead of a single one.

 

But this also means we have to move from a topology of points to that of circles, which are no longer sizeless geometrical objects, since their size is given by x(0)… Their size, not their length: their length is given by 2pix(0) and their area, by pi[x(0)]² (or, more correctly, the area of the disk delimited by that circle).

 

The size of a circle is given by the amplitude of the quantum grandeur related to it.

 

Using that new topology, we can extend our previous three classical values x_-(0), x(0) and x₊(0) into three circles x_-(ksi_-), x(ksi) and x₊(ksi₊) with totally independent quantum states. The former inequalities 0 =< x_-(0) < x(0) < x₊(0) will then enable us to assert that the quantum circle x_-(ksi_-) will be the smallest of all three, then will come x(ksi) and finally, x₊(ksi₊) will be the largest one. All three circles, once centered on the origin of the frame, will be circumvent: x_-(ksi_-) -> x(ksi) -> x₊(ksi₊), thanks to the classical ordering of their radii. The smallest circle possible is the circle with zero radius, which identifies with a point.

 

So, already, as amplitudes can never turn negative, you retrieve the fact that nothing can be smaller than a point, that no size can be negatively counted, so that you’ll have to modify your conception of “successions / preceedings”, as you can no longer take zero as a reference point, only as a universal reference:

 

As we can no longer translate the origin anywhere we want, as in the classical, the quantum zero is no longer a relative value, but becomes an “absolute” one (anew?).

 

There’s only one zero in the quantum and no negative sizes, nothing smaller than it.

 

Despite this apparently drastic reduction, we can still rebuild a notion of “successions” and “preceedings” in the following way.

 

Classical 1D space is a straight line, it goes from -oo (infinity) to +oo and its therefore unlimited at both ends.

 

Quantum space as a whole is also 1D and unlimited, but in size: its radius is infinite. Now, we need to understand this last dimension is also a quantum one. And, as we defined quantum grandeurs as pairs of classical ones, this quantum dimension corresponds to two classical ones. So, geometrically, the picture is not that of a classical plane, as generally assumed, but that of an unlimited circle: the whole quantum space is naturally cyclic and the circle is defined as a close line and thus, as a 1D object only. But an object still inscribed in a 2D classical plane, hence the correspondence between classical dimension 2 and quantum dimension 1: each quantum dimension “is equivalent to”, in idea, to two classical dimensions. So, when you go from the quantum down to the classical, you multiply dimensions by two and, when you extend the classical to the quantum, you divide dimensions by two.

 

However, the fact that the geometry of quantum space is cyclic completely changes the properties of classical space. In classical space, once you start from a point taken as the origin, as long as you progress, you can always go further and further away from your departure point, in space as in time. Whether you move deeper in space or in the future or deeper in space (opposite direction) or in the past. Mechanically, you can always go backwards, but you need make a U-turn. In quantum space, you go round. It means that, at least in principle, you will always be back to your point of departure after “going round the quantum universe”. If this has no particular consequence on space motion, but for avoiding you the U-turn, it does have on time motion, because it means that, still in principle, you can move into the “future”, than “go back to the present through the past”… which, set in those terms, doesn’t mean a lot… :|

 

This is because we still reasoned based on motions from points to points. At the best, it leads to the disappearance of the notion of “time arrow” and “space orientation”; at the worst, it leads to absurdities and paradoxes: how could you reach the present back always going deeper in the future, round the quantum universe?... Where’s the past in that picture?... :|

 

Instead, if we accept to reason with circles in place of points, we retrieve consistent reasonings, to the cost of more technical difficulties.

 

If our three circles x_-(ksi_-), x(ksi) and x₊(ksi₊) represent “quantum distances” and x(ksi) is taken as the “reference position”, then x_-(ksi_-) being smaller than x(ksi) will stand “behind” it, while x₊(ksi₊) being larger than x(ksi) will stand “ahead” of it, no matter the positions occupied on these circles. Instead, we now have this two-way correspondence:

 

0 =< x_-(0) < x(0) < x₊(0)  <=>  “x_-(ksi_-) smaller than x(ksi), itself smaller than x₊(ksi₊)”

 

We could apply this to a “quantum time” as well. However, B141-theorem shows that this is not even necessary, as it suffices to pair a space variable with a time one to replace any space-time with a continuous family of spaces-only. See the example. It easily generalizes to any space-time with p space variables and q time ones, with p > q: we make q pairings to eliminate the q time variables, this gives q angles serving as as many continuous parameters in a family of (p+q)-dimensional spaces-only…

 

If, now, for purely conceptual reasons, we’d prefer to keep a notion of time in the quantum, same two-way correspondence as above, now specifically writing:

 

0 =< t_-(0) < t(0) < t₊(0)  <=>  “t_-(tau_-) in the quantum past of the quantum present t(tau), itself

in the quantum past of the quantum future t₊(tau₊)”

 

From what was previously seen, you can now see that:

 

The “furthest quantum past” is zero, the universal time origin, and all other quantum moment is located in its quantum future.

 

We now retrieve the notion of a “universal space origin” and that of a “universal time origin”. None of them are subject to relativization. So good, actually, because it replaces relativism within the observer’s choice and not as an inherent physical property of the world…

 

Take all observers away from the frame, what remains is a comparison of sizes, not orientations, not “arrows”…

 

 

An interesting remark now.

 

It is a very general feature common to both the classical and the quantum that the dynamical notion of a motion can be equivalently represented whether as bodies moving into a fixed frame or as bodies occupying fixed positions in a moving frame.

 

And this holds for space as for time. I can assume I’m moving inside fixed 3D space and even fixed time, both fixed “once and for all”, i.e. from the birth of the universe, or I can equivalently assume that I make no motion on my own but, instead, both space and time move around me, carrying bodies with them. The result will be the same, but the geometrical picture will be completely different.

 

In fixed frames, you move from one location to another whether a continuous or a discontinuous way (“jumps”) and from one instant to another: x_-(0) -> x(0) -> x₊(0) or t_-(0) -> t(0) -> t₊(0). It’s a succession of locations and moments and you explore them the one after the other. In this picture, the frame is passive and you are the actor. This is generally what we assume as “motions”.

 

In moving frames, you stand still, you’re steady, and the frame inflates [x_-(0) -> x(0) -> x₊(0)] or deflates [x₊(0) -> x(0) -> x_-(0)]. You can also occupy a steady position [x(0) -> x(0)], whether permanently or for a certain duration only. This second picture is now about scalings and rescalings of both space and time. Starting from (idealistically) 0, a first inflation brings space to a size x_-(0) > 0. As you stand on a circle with radius x_-(0), you’ll be brought from the initial position zero to the position x_-(0). The angle ksi (the quantum state space is in - notice: space, not you) can then “affine” that positioning of you (this time), despite you made no action to displace from one location to another: you’re now the one to remain passive. If it deflates of the same amount, you’ll be back to zero. If it inflates again, you’ll be brought to a new location x(0) > x_-(0). And so on.

 

The same holds with time. It’s even a much better representation of “time flow” than trying to represent it as a “passive time” the traveler would “visit” one instant after the other. In a moving time, the traveler visits nothing, he/she just let him/her being “carried”, “transported” by time inflations and deflations, changes of scales. Time flows… :)

 

Finally, in a moving quantum frame, in addition to that common (re)scaling of amplitudes (radii), we find a rotation of the frame in the quantum states. Indeed, if you’re found in a “quantum position” [x(0),ksi] in numerical values and, later on, in a position [x(0),ksi’], with ksi’ different from ksi, space around you didn’t move but it rotated an angle ksi’ - ksi. As a result, you still moved, even if you didn’t actually go further away.

 

Regarding, now, a classical system of axis, one “corpuscular” (“horizontal”) and one “wavy” (“vertical”), a rotation of space of pi/2 permutes these axis: the corpuscular becomes wavy and the wavy, corpuscular. A rotation of pi will turn these axis upside-down: the “positive” becomes the “negative” and the “negative”, the “positive”, but the corpuscular remains corpuscular and the way remains wavy.

 

So, as you can see, this is all purely conventional, in the end… :) In the quantum, there simply can’t be anything like “the substantial”, the “non-substantial”, the “positive” or the “negative”. It simply has no sense… (if I may use that metaphor…)

 

If you combine both, inflation/deflation with rotation, you obtain a motion known in mathematics as a similitude. You’ll then give an observer reasoning in a fixed frame the perception that you went from point [x(0),ksi] to another point [x’(0),ksi’]. In reality, you didn’t move, the frame did it for you. But this is ultimately irrelevant, because the result is the same… :)

 

Or is it really? Because, now, the geometrical picture is radically different.

 

In a fixed frame, a quantum motion x(ksi)[t(tau)] = {x(0)[t(0),tau],ksi[t(0),tau]} is graphically represented as a Riemann surface. In other words, the quantum curve x(ksi)[t(tau)] corresponds to a pair of classical surfaces, making that “Riemann surface”. To each classical instant t(0) and each quantum state tau of quantum time corresponds a location x(0)[t(0),tau] with quantum state ksi[t(0),tau] on that surface. The picture in itself is interesting, giving a time surface developing through some 2n-dimensional space, where n is the number of classical dimensions, but is it the truly quantum picture? No: the truly quantum one is represented as x(ksi)[t(tau)], it’s a quantum time curve developing in n-dimensional quantum space, where n is now the number of quantum dimensions…

 

In a moving frame, x(ksi)[t(tau)] represents instead a “space circle function of a time one”. Both families of circles are centered at the same “universal” origin (0,0), the only point in quantum space. What happens now is: as the time radius t(0) of t(tau) increases, there’s a corresponding space radius x(0). The set of all radii t(0) between t(0) = 0 and some final instant t(0) = T(0) that can be pushed to infinity as well, this set makes a dense family of concentric time circles all centered at (0,0). So does the set of all corresponding radii x(0) between the initial and final instants. Consequently, if we draw a system of two perpendicular planes on a sheet of paper, one horizontal and one vertical, on the horizontal plane, you’ll see growing concentric time circles and, on the vertical plane, corresponding concentric space circles. There’s no “motion” whatsoever in the space surrounded by these two planes. The “motion”, i.e. the succession of expansions and contractions, entirely occurs in the vertical space plane.

 

Let’s take an example to illustrate this geometrical difference and close this article.

 

Consider the quantum parabolic motion x(ksi)[t(tau)] = ½ [t(tau)]².

 

In a fixed frame, it corresponds to the Riemann surface {x(0) = ½ [t(0)]² , ksi = 2tau}. Quantum objects submitted to this law will move on that surface, starting at (conventional!) t(0) = 0, at {x(0) = 0, ksi = 2tau}. So, first, even at the departure point, you still need to precise the initial quantum state tau in order to be able to deduce ksi. Then, for each value of tau, your body will trace a parabola in (classical!) space: x(0) = ½ [t(0)]². So, even for a portion only of angles, say 0 =< tau =< TAU =< 2pi, you’ll get a dense sheaf of parabolas, one for each value of tau. And, as tau is periodic, when it reaches pi radians, ksi reaches 2pi radians and we’re back to the parabola at ksi = 0. Here’s the geometrical picture you get of this particular motion when assuming that bodies move in a fixed frame.

 

In a moving frame, you find concentric time circles, all centered at t(0) = 0, and concentric space circles, all centered at x(0) = 0. As time radii grow, so do space ones, following a square law. You don’t need to worry anymore trying to know at each step what the quantum states of time and space are. If tau is given to you, then you can add to your knowledge of the quantum motion that the position on the corresponding space circle will be twice that value. But it’s no longer necessary to understand the motion. All we need to know is that, in a time inflation, we have a quadratic (or parabolic) space inflation and, in a time deflation, a parabolic space deflation, so that space inflates or deflates much quicker than time.

 

N.B.1: in example given, physical units have to be restored.

N.B.1: in the case of a much more general motion like

 

x(ksi)[t(tau)] - x₀(ksi₀) = ½ a(alpha)[t(tau)]² + b(beta)t(tau) + c(khi)

 

space and time origins can always be translated back to (0,0) using shifts

 

t’(tau’) = t(tau) - t₀(tau₀)

x’(ksi’) = x(ksi)[t’(tau’)] - x₀(ksi₀)

 

where t₀(tau₀) is given by the timeless coefficients a(alpha), b(beta) and c(khi). So it changes nothing to our purpose.

 

This bidouille, very technical, about quantum extensions of classical variables and functions.

 

We begin with variables. A classical variable is a variable x(0) element of R. It’s defined up to a conventional sign. As we need a start to build anything, the quantum extension of x(0) is built, whether as the element of C:

 

(1)               x(ksi) = x(0)e^iksi

(2)               i² = -1 = e^ipi  ,  i = e^ipi/2

 

using the classical irrational e and elevating it to the power a purely imaginary number iksi, or has an element of R²:

 

(3)               x(ksi) = [x(0),ksi]                          (polar representation)

(4)               x(ksi) = [x¹(ksi),x²(ksi)]                 (planar representation)

 

endowed with the addition and multiplication on pairs of reals,

 

(5)               x(ksi) + y(psi) = [x¹(ksi) + y¹(psi) , x²(ksi) + y²(psi)]          (planar rep)

(6)               x(ksi)y(psi) = [x(0)y(0) , ksi + psi]                         (polar rep)

 

The one-to-one correspondence is guaranteed through:

 

(7)               x¹(ksi) = x(0)cos(ksi)  ,  x²(ksi) = x(0)sin(ksi)

(8)               x(0) = {[x¹(ksi)]² + [x²(ksi)]²}^1/2  ,  ksi = Arctan[x¹(ksi)/x²(ksi)]    (mod pi)

 

By using the second approach, we only call for classical trigonometric functions on building classical projections of the quantum x(ksi), which is then defined as the pair (4) of such classicals. We don’t need to combine a classical irrational e with a purely quantum iksi, as when using the de Moivre formula: (7) and (8) are fully classical. What becomes quantum are basic arithmetic operations on pairs of classicals: the left side of (5) is a quantum addition; the left side of (6), a quantum multiplication. What their right sides give are equivalences with the originally classical operations.

 

Let’s now define the quantum function f(phi) of a quantum variable x(ksi) following that procedure. We find 4 possible pairings, corresponding to as many representations:

 

(9)               f(phi)[x(ksi)] = {f(0)[x(0),ksi] , phi[x(0),ksi]}                                            (polar-polar)

(10)           f(phi)[x(ksi)] = {f(0)[x¹(ksi),x²(ksi)] , phi[x¹(ksi),x²(ksi)]}              (polar-planar)

(11)           f(phi)[x(ksi)] = {f¹(phi)[x(0),ksi] , f²(phi)[x(0),ksi]}                                   (planar-polar)

(12)           f(phi)[x(ksi)] = {f¹(phi)[x¹(ksi),x²(ksi)] , f²(phi)[x¹(ksi),x²(ksi)]}       (planar-planar)

 

As above, all four representations are fully classical in components and quantum appears in pairings. phi(.,.) is a 2pi-periodic function, geometrically describing a 2D torus in R³, as ksi is 2pi-cyclic as well. f(0)(.,.) is a non-negative function and geometrically describes a 2D open tube in R³. Now, f(phi)[x(ksi)] = y(psi), which only has two representations. So, the rule is actually the following one:

 

No matter the representation of the variable, that of the result will copy the representation of the function.

 

As a result of this rule, (9) and (10) will lead to the same polar representation [y(0),psi], while (11) and (12) will lead to the same planar representation [y¹(psi),y²(psi)]. It’s kind of a “Markov process”: the procedure “forgets” about all former representations and only keeps the last one.

 

A quantum operator T(TAU) will be defined the same, as an application transforming a quantum function into another. It’s therefore a “function of a function” or a “functional”:

 

(13)           T(TAU).f(phi) = T(TAU)[f(phi)] = g(gamma)

 

Applying the “representation rule” leaves g(gamma) the representation of T(TAU), no matter that of f(phi). So, “sequentially”:

 

(14)           T(TAU){f(phi)[x(ksi)]} = T(TAU)[y(psi)] = g(gamma)[x(ksi)]

 

y(psi) keeps the representation of f(phi), no matter that of x(ksi); g(gamma), that of T(TAU), no matter y(psi) and x(ksi), as a consequence of the intermediary step.

 

A reciprocal quantum function is a quantum function f^-1(-phi) such that:

 

(15)           f^-1(-phi).f¹(phi) = f¹(phi).f^-1(-phi) = Id(O)

 

gives the quantum identity function Id(O), where “O” stands for the “zero state” or a “universal quantum vacuum state”. Applied to a quantum variable x(ksi) element of the definition domains of both f¹(phi) and f^-1(-phi), it gives that variable back:

 

(16)           f^-1(-phi).f¹(phi)[x(ksi)] = f¹(phi).f^-1(-phi)[x(ksi)] = x(ksi)

 

It appears that Id(O) has no particular representation (hence it’s universality, by the way), since, as a function, it should follow (9) to (12) but, as it gives the same result as the initial variables, the representations of Id(O) in fact faithfully follow that of the variable. It suffices to look at (9) and (10) to get convinced of this:

 

Id(O)[x(ksi)] = {Id(0)[x(0),ksi] , O[x(0),ksi]} = [x(0),ksi]

 

Implies both

 

Id(0)[x(0),ksi] = x(0)  whatever the value of ksi

 

and

 

O[x(0),ksi] = ksi  whatever the value of x(0)

 

while

 

Id(O)[x(ksi)] = {Id(0)[x¹(ksi),x²(ksi)] , O[x¹(ksi),x²(ksi)]} = [x¹(ksi),x²(ksi)]

 

applying the identity operation, or

 

Id(O)[x(ksi)] = {Id(0)[x¹(ksi),x²(ksi)] , O[x¹(ksi),x²(ksi)]} = [x(0),ksi]

 

applying the representation rule (the variable follows the polar representation of the function). It follows from this “non-representativity” of Id(O) that:

 

The representations of f(phi) and of its reciprocal f^-1(-phi) “neutralize”.

 

Here are now two practical examples on how to construct quantum extensions of classical functions. These examples will serve us in the next bidouille. The general idea is:

 

We keep the properties of classical functions, we simply replace them with better-suited quantum ones, when applying to quantum variables.

 

As we’re going to see it, this apparent “insignificant change” actually modifies everything and brings additional informations anyway, since quantum states appear in both variables and functions…

 

 

Example 1: the quantum exponential

 

The classical exponential was defined as the elevation of the irrational e to a power a real-valued number x: e^x. By power properties, this function verified:

 

e^x.e^y = e^x+y , (e^x)^y = (e^y)^x = e^xy

 

while x⁰ was set to +1 whatever the value of x (including x = 0), by convention (and graphical confirmation). Similarly, we will define the quantum exponential e(epsilon) as the function who satisfies:

 

(17)           e(epsilon)[x(ksi)].e(epsilon)[y(psi)] = e(epsilon)[x(ksi) + y(psi)]

(18)           {e(epsilon[x(ksi)]}^y(psi) = {e(epsilon)[y(psi)]}^x(ksi) = e(epsilon)[x(ksi)y(psi)]

 

for any two quantum variables x(ksi) and y(psi). In particular, the previous complex-valued representation e^iksi, which combined e with the imaginary unit i, is to be replaced with the better-suited e(epsilon)^iksi. The polar representations are the same: (1,ksi). In that representation, former exp[x(0)e^iksi] = e^x(0)cos(ksi)e^{ix(0)sin(ksi)} is replaced with:

 

(19)           e(epsilon)[x(ksi)] = e(epsilon)[x(0)e(epsilon)(iksi)]

= {e(0)[x(0),ksi] , epsilon[x(0),ksi]}

(20)           e(0)[x(0),ksi] = e^x(0)cos(ksi) >= 0

(21)           epsilon[x(0),ksi] = x(0)sin(ksi)

 

Particular values are:

 

(22)           e(0)[x(0),0] = e^x(0) , e(0)[x(0),pi] = e^-x(0)

(23)           e(0)[x(0),pi/2] = e(0)[x(0),3pi/2] = 1                    for all x(0)

(24)           epsilon[x(0),0] = epsilon[x(0),pi] = 0                     for all x(0)

(25)           epsilon[x(0),pi/2] = epsilon[x(0),3pi/2] = -x(0)

(26)           e(0)(0,ksi) = 1 , epsilon(0,ksi) = 0             for all ksi

 

As a result, the quantum function (19) will have the particular polar representations:

 

(27)           e(epsilon)[x(0)e(epsilon)(i0)] = {e^x(0) , 0}

(28)           e(epsilon)[x(0)e(epsilon)(ipi)] = {e^-x(0) , 0}

 

corresponding to the classical exponential and its inverse. Additionally:

 

(29)           e(epsilon)[x(0)e(epsilon)(ipi/2)] = {1 , x(0)}

(30)           e(epsilon)[x(0)e(epsilon)(3ipi/2)] = {1 , -x(0)}

 

corresponding, this time, to the same amplitude unity, but opposite quantum states. Also notice, for instance:

 

e(epsilon)[x(0)e(epsilon)(ipi/4)] = {e^x(0)/sqr(2) , x(0)/sqr(2)} , sqr(.) = “square root”

 

 

Example 2: the quantum (natural) logarithm

 

This is classically the reciprocal to the exponential. Ln(.) verifies:

 

Ln(xy) = Ln(x) + Ln(y) , Ln(x^-1) = -Ln(x) , Ln(x^y) = yLn(x) , Ln(0⁺) = -oo , Ln(1) = 0

 

Originally, this function wasn’t defined for negative values of the variable. It was then extended as such to the complex domain through the formula:

 

Ln[x(0)e^iksi] = Ln[x(0)] + iksi = {Ln[x(0)],ksi}            (planar representation)

 

Again, we suggest Ln(.) to be replaced with a Ln(Lambdaeta), while conserving the same properties. This gives:

 

(31)           Ln(Lambdaeta)[x(ksi)] = {Ln(0)[x(0),ksi] , Lambdaeta[x(0),ksi]} = {Ln[x(0)],ksi}

(polar rep)                                           (planar rep)

 

since we need retrieve the same result as previously. Hence, immediately:

 

(32)           {Ln(0)[x(0),ksi]}² = {Ln[x(0)]}² + ksi²

(33)           Lambdaeta[x(0),ksi] = Arctan{ksi/Ln[x(0)]}         (mod pi)

 

with the particular values,

 

(34)           Ln(0)[x(0),0] = |Ln[x(0)]| , Lambdaeta[x(0),0] = 0           (mod pi), for all x(0)

(35)           Ln(0)(0⁺,ksi) = +oo  for all ksi , Lambdaeta(0⁺,ksi) = 0 (mod pi)

(36)           Ln(0)(1,ksi) = |ksi| , Lambdaeta(1⁺,ksi) = pi/2 , Lambdaeta(1^-,ksi) = 3pi/2 (mod 2pi)

(37)           Ln(0)(e,ksi) = (1 + ksi²)^1/2 , Lambdaeta(e,ksi) = Arctan(ksi) (mod pi)

 

As a result, the quantum logarithm will have the following polar representations:

 

(38)           Ln(0)[x(0)] = {|Ln[x(0)]| , 0} = |Ln[x(0)]|

(39)           Ln(pi)[x(0)] = {|Ln[x(0)]| , pi} = -|Ln[x(0)]|

(40)           Ln(0)(0⁺) = {+oo , 0} = +oo , Ln(pi)(0⁺) = {+oo , pi} = -oo

(41)           Ln(psi/2)[{1,ksi}] = |ksi| , Ln(3psi/2)[{1,ksi}] = -|ksi|

(42)           Ln(Lambdaeta)[{e,ksi}] = {(1 + ksi²)^1/2 , Arctan(ksi) (mod pi)}

 

This is a pretty unexpected result (one more?) that should greatly help us in our research. The

non-technician can skip the proof and go directly to the practical application of it, where the theorem is explained.

 

 

THEOREM:

 

Let X be a real manifold with dimension 2n and signature (n,n). Then, X can be made conformally Euclidian near each of its point.

 

 

In “civilized language”, what this theorem says is that any 2n-dimensional space with an open geometry can be closed, at least locally, and therefore, compactified.

 

Before giving the general proof, let us show it on a practical example. This is a transformation I never thought about and never saw anywhere else either.

 

 

Example:

 

Let’s consider a 2-dimensional plane space-time with then a single dimension of space and a metric:

 

(1)               ds² = c²dt² - dx²

 

Usually (i.e. in all publications I read so far about special and general relativity), this expression is factorized into:

 

(2)               ds² = (1 - v²/c²)c²dt²

 

where v = dx/dt is the instantaneous velocity around point x. If we introduce a polar parametrization:

 

(3)               cdt = cos(dksi)dr  ,  dx = sin(dksi)dr

 

where dksi is an angle between 0 and 2pi, not only will we be guaranteed the absolute values of both cdt and dr will stay between 0 and dr > 0, but ds² and v will write:

 

(4)               v = tan(dksi)

(5)               ds² = cos(2dksi)dr² = [(1 - v²/c²)/(1 + v²c²)]dr²

 

As for dr², it will take the Euclidian form:

 

(6)               dr² = c²dt² + dr²

 

in planar representation. Now, the crucial point is that the metrical tensor (field) is defined as belonging to the space tangent to a given base space, which is here, our 2-dimensional Minkowski space-time. So, it’s very natural, from the geometrical viewpoint, to link the metrical coefficient g(dksi), which explicitly depend on dksi, to the velocity (4) in (5):

 

(7)               g(dksi) = cos(2dksi)  or, equivalently,  g(v) = (1 - v²/c²)/(1 + v²c²)

 

But, then, our original 2-dimensional Minkowski space-time is made equivalent to a dense family of conformal 2-dimensional Euclidian spaces. As long as v will remain between 0 and c (in pure value), g(v) will remain positive and the resulting Euclidian manifold as a whole will be causal. It will be made of that set of all points (ct,x) of our original 2D Minkowski space-time such that, in the immediate neighborhood of each of these points, 0 =< v =< c and g(v) >= 0. The rest will be made of those points where v > c, g(v) < 0, which will give another locally conformal Euclidian space, a priori non observable.

 

Last but not least, the conformal factor g(v) remains bounded: g(0) = +1, g(c) = 0 and, for infinite velocities, g(v) -> -1.

 

Let’s now give the general proof.

 

 

Proof of the theorem:

 

Let X(KSI) be a complex Riemannian n-dimensional manifold of Cⁿ and TX(KSI) its tangent bundle. If x(ksi) = [x¹(ksi¹),…,xⁿ(ksiⁿ)] is a point of X(KSI), the quantum differential at point x(ksi) = x(0)exp(iksi) is that infinitesimal complex-valued quantity of order 1 of TX(KSI) defined as follows:

 

(8)               d(delta)x^a(ksi^a) = exp{ideltaksi^a[x(0),ksi]}d(0)x^a(0)                       (a = 1,…,n)

 

where d(0)x^a(0) is always a non-negative infinitesimal quantity of order 1 we will call the classical differential of x^a(0) at classical point x(0) = [x¹(0),…,xⁿ(0)]. and ideltaksi^a[x(0),ksi] is a quantity between 0 and 2pi. From that definition of d(delta), it’s possible to build the second quadratic form of X(KSI) as that infinitesimal complex-valued element of the symmetric tensor product of TX_x(ksi)(KSI) with itself:

 

(9)               [d(delta)s(sigma)]² = g_ab(2gamma_ab)[x(ksi)]d(delta)x^a(ksi^a)d(delta)x^b(ksi^b)

 

where, as usual in tensor calculus, we use the Einstein’s summation convention as long as we can. In polar representation,

 

(10)           g_ab(2gamma_ab)[x(ksi)] = g_ab(0)[x(0),ksi]exp{2igamma_ab[x(0),ksi]}

(11)           g_ab(0)[x(0),ksi] = g_ba(0)[x(0),ksi]

(12)           gamma_ab[x(0),ksi] = gamma_ba[x(0),ksi]

 

As a result, (9) takes the form:

 

(13)           [d(delta)s(sigma)]² = [d(delta)s(sigma)_D]² + [d(delta)s(sigma)_ND]²

 = SS_a=<b=1ⁿ exp{2isigma_ab[x(0),ksi]}[d(0)s_ab(0)]²

(14)           2sigma_ab = 2gamma_ab + deltaksi^a + deltaksi^b

(15)           [d(0)s_ab(0)]² = g_ab(0)[x(0),ksi]d(0)x^a(0)d(0)x^b(0)   (a,b = 1,…,n)

 

We’re only interested in the diagonal contribution [d(delta)s(sigma)_D]², because it groups all the n main directions of the quadric (9) and, therefore, contains its signature:

 

(16)           [d(delta)s(sigma)_D]² = S_a=1ⁿ exp{2isigma_aa[x(0),ksi]}[d(0)s_aa(0)]²

(17)           sigma_aa = gamma_aa + deltaksi^a

(18)           [d(0)s_aa(0)]² = g_aa(0)[x(0),ksi][d(0)x^a(0)]²  (a = 1,…,n)

 

The other contribution concerns planes:

 

(19)           [d(delta)s(sigma)_ND]² = SS_a<b=1ⁿ exp{2isigma_ab[x(0),ksi]}[d(0)s_ab(0)]²

 

Developed into its real and imaginary parts, (16) gives two real-valued second quadratic forms:

 

(20)           [d(delta)s(sigma)_D1]² = S_a=1ⁿ cos{2sigma_aa[x(0),ksi]}[d(0)s_aa(0)]²

(21)           [d(delta)s(sigma)_D2]² = S_a=1ⁿ sin{2sigma_aa[x(0),ksi]}[d(0)s_aa(0)]²

 

Both seems to be of hyperbolic type, which would lead to two 2n-dimensional manifolds X¹(KSI¹) and X²(KSI²) of R²ⁿ with the topology of open spaces. These spaces are known as not being compacts. However, a closer look at (20-21) shows that, since [d(0)s_aa(0)]² is the square of a real-valued quantity and, therefore, never negative, signs of the metrical components are only due to the trigonometric functions. The argument of these functions lays on the tangent bundle of X(KSI), not on X(KSI) itself. This is a first important aspect to remind: the information about the signature of a manifold does not belong to it, but to its tangent bundle. The second and, this time, crucial aspect to keep in mind is that, in the quantum, i.e. in mathematically complex spaces, the negative sign is formally equivalent (and even originates from) a phase opposition. Otherwise, there exists no such thing as a “negative sign”. This last aspect means that, at points x(ksi) of X(KSI) where a given metrical component appears with a negative sign, we’re in front of a phase opposition, i.e. a shift of pi. As a consequence, there is no such thing as a “causal” and a “non-causal” domain as this is the case in the classical (i.e. real geometry) any longer, everything becomes “causal”, “observable”. Some things only display “in phase” and others, “in phase opposition”. It follows that, in order to find a definite signature on X¹(KSI¹), we need all terms to have same sign and this is submitted to n non-restrictive conditions we’re going to explicit.

 

In sector I on TX(KSI):

 

(22)           0 =< sigma_aa[x(0),ksi] =< pi/4                               (a = 1,…,n)

 

Both (20) and (21) are positive-definite, so that all points [x(0),ksi] satisfying this set of conditions make two “causal” manifolds X¹₊(KSI¹) and X²₊(KSI²) of R²ⁿ. “Causal”, because (20), as (21), have real-valued square roots.

 

In sector II on TX(KSI):

 

(23)           pi/4 =< sigma_aa[x(0),ksi] =< pi/2                            (a = 1,…,n)

 

(20) is negative-definite and (21), positive-definite: the generated manifolds are an “anti-causal” X¹_-(KSI¹) and a “causal” X²₊(KSI²). “Anti-causal”, because “causal, but in phase opposition”.

 

In sector III on TX(KSI):

 

(24)           pi/2 =< sigma_aa[x(0),ksi] =< 3pi/4              (a = 1,…,n)

 

Both (20) and (21) are negative-definite, leading to two “anti-causal” X¹_-(KSI¹) and X²_-(KSI²).

 

Finally, in sector IV of TX(KSI):

 

(25)           3pi/4 =< sigma_aa[x(0),ksi] =< pi                             (a = 1,…,n)

 

we get a “causal” X¹₊(KSI¹) and an “anti-causal” X²_-(KSI²).

 

In each of these four possible situations, the real manifolds so obtained have the topology of a closed space and can therefore be compactified, at least locally, i.e. from point to point.

 

And this ends our proof of the theorem. Indeed, local coordinate transforms do change the values of both the amplitude g_aa(0)[x(0),ksi] and the phase sigma_aa[x(0),ksi] covariantly, however, they concern X(KSI), not its tangent bundle (only the partial derivatives of these transforms do). So, as long as the n conditions (22), (23), (24) or (25) are fulfilled on TX(KSI), changing representation on X(KSI) does not change anything to the topology of the induced real manifolds, which remain Euclidian.

 

 

Practical application:

Observing quantum shapes

 

A shape is a volume surrounded by a closed surface. It is not necessary to know that volume, it only suffices to observe its bounding surface to have a clue of the shape contained in that volume. By extension, a quantum shape is a quantum volume surrounded by a quantum surface, all naturally oscillating as a consequence. What the theorem above says is that any quantum shape gives birth to two classical projections, as any other quantum object or process, a “corpuscular” or “substantially-perceived” projection and a “wavy” or “non-substantially-perceived” projection. Apparently, these two projections seem “open”, i.e. non-compact. If it was the case, we would have troubles, not only in defining quantum shapes, but also with remaining in agreement with observations. Indeed, observations show that, even if oscillating, any physical body with quantum properties keep a delimited shape, but maybe for gases, but classical gases have no definite shape either, except if they are trapped inside a delimited solid area, in which case, they spouse the shape of the area filling it. Fortunately, the theorem says otherwise and shows that, under non-restrictive conditions, both projections can actually have a closed geometry and, therefore, be compact, should this process was to be performed point to point.

 

This being said, dimension n = 2 corresponds to surfaces. As here, we’re not dealing with classical, but with quantum surfaces, the starting point is a quantum surface (surrounding a quantum shape) in quantum dimension 2. Now, as anything is grouped into pairs to make a quantum, a quantum dimension n is mathematically (but not physically!) equivalent to a classical dimension 2n (n classical pairings make n quantum components). So, first of all, with classical projections, we’ll be to deal with even-dimensional surfaces (always, a result of pairing). This poses an immediate problem to the classical observer. A quantum surface gives two classical projections of classical dimension 4, each. Now, our observer is in a 3D space only. At least, this is the way he/she perceives things around him/her. So, any quantum surface is already “too large”, “too dimensional” to be classically observed. Even if our observer was conscious of this, the only thing he/she could do would be to use perspective to try and reproduce a quantum shape re-projecting its two 4D classical projections into his/her 3D world, which wouldn’t give him/her the true aspect of the shape (just think of that exercise at school consisting in drawing a 3D cube onto a 2D sheet of paper: if perspective maintains the feeling of a volume, it can never faithfully reproduce that volume as it is in the 3D world).

 

To go round the difficult, the technician has a solution: cuts. The topographer does the same on maps.

 

For n = 2, we have two quantum coordinates x¹(ksi¹) and x²(ksi²), making four classical ones, namely x¹(0), ksi¹, x²(0) and ksi². We have at least one “in excess”. However, what the experimenter does most of the time is that he/she observes a quantum object in a definite quantum state. That’s actually fixing the values of the two angles ksi¹ and ksi². The dimensional obstruction occurs when one tries to reproduce the quantum shape in all its possible quantum states. Opposite to this, if we admit that “we’re having a lack of physical dimensions so that were going to use the cut technic and observe that shape quantum state by quantum state”, then, for each given pair of values (ksi¹,ksi²) of the angles, our two projected surfaces will only depend on the amplitudes x¹(0) and x²(0), which will give back a pair of classical surfaces in classical 3-space. What we’ll then deduce from this method is that the quantum shape is made of a doubly-continuous infinity (as ksi¹ and ksi² go independently from 0 to 2pi) of “substantially-perceived” classical shapes X¹_ksi1,ksi2(KSI¹) and “non-substantially-perceived classical shapes X²_ksi1,ksi2(KSI²), “causal” or “anti-causal”.

 

This is probably the best way to reproduce a quantum shape.

 

We start from (ksi¹,ksi²) = (0,0), it gives a first “substantial” surface X¹_0,0(KSI¹) corresponding to the behavior of the quantum shape at the UV limit. We then make ksi¹ and ksi² independently vary along the unit-radius circle and, each time, it gives a still “substantial” surface X¹_ksi1,ksi2(KSI¹), but not necessarily of the same shape. And all these shapes glued together give the “substantially-perceived” shape X¹(KSI¹), which is 4D and cannot be faithfully reproduced, even in 3-space. But, at least, we can get “cuts”, and continuous ones, of it. We do exactly the same to build X²(KSI²).

 

 

What does it imply to biology?

 

This is rather simple, isn’t it? The biological organism is a (living) classical shape. The biologists perceives it at the ultra-violet limit, meaning for cuts (ksi¹,ksi²) = (0,0) and (ksi¹,ksi²) = (pi,pi) only. And still, it may even reduce to (ksi¹,ksi²) = (0,0). That’s only one shape over a continuous infinity of others and this, anyway, only gives the substantial behavior of a quantum being. You still have to add to this another continuous infinity of non-substantial shapes, giving the wavy behavior of the quantum being in question.

 

And, still, after all this… it only remains projections. It doesn’t reproduce the quantum being an inch. His nature is quantum, his environment is quantum, there’s nothing reducible to the classical in that. Reducing it to classical projections, than to 2D “sections” is only aimed at trying to make an approximate picture of him. Get “a vague idea of how he may look like” when projected into a lower-dimensional world.

 

So, of course, saying there would be “a continuous infinity of biological bodies of different shapes” is actually meaningless, it’s only a matter of trying to interpret things. It does not correspond to the physical reality at all (just like the “multiverse” in quantum astrophysics has to be taken as one of the many interpretations of the so-named “wavefunction of the universe”). What a continuous parameter shows us is not that we have a continuous infinity of “parallel worlds”, but simply that we have to take a new physical dimension into account… :)

 

So, what the picture shows isn’t a “countless multitude of parallel biological organisms, each one belonging to a classical 3D world”, this is fantasy… :) (it is!), it only says “what we observe as a ‘biological organism’ is not even the tip of the iceberg, it’s only the UV limit of a much wider 4D but still classical shape, surrounding a 5D still classical body and this only makes the substantial aspect we’d be able to perceive of a quantum organism if we could have direct access, as conscious observers, to the 3D quantum world”.

 

That quantum world is entirely cyclic, it has nothing to do anymore with our classical perception of the world. It’s a geometrical environment where loops play the role of “straight lines”; tubes, the role of open curves and tori, that of closed curves. “Negativity” becomes “phase opposition”. There is nothing common to our daily life anymore. The physical laws are different, time is cyclic, everything is totally stranger to what we’re used to.

 

Despite all these fundamental differences, physics tells us that this is actually the world we live in, since our conception…

 

And we can’t even perceive a fraction of it…

 

So, where else would be the limitation, if not in our brain?... :)

 

We’re back to B135 and we’re now going to talk about quantum areas and volumes. We start in dimension 1. We first need to precise what kind of physical dimension we are to work in: if it’s classical dimension 1 then, indeed, there’s a single one; if it’s quantum dimension 1, there are 2 classical ones (remember we need double everything). The “wavy” dimension, also referred to as the “P²-projection”, is assumed to be located “above” the “corpuscular” one.

 

What does it means, “above”?

 

We’ll get a much better picture if we now embed ourselves in our much more familiar 3 dimensions of space. Classically, we feel we’re able to move anywhere in the three classical dimensions of space: length, width, height. Still classically, we can understand we extend this to wavelengths, considering anisotropic waves: as those waves do not propagate the same in all 3 directions, we can give them 3 independent wavelengths, one along each direction. The conceptual difficulty arises when we attempt to “glue” these 3 “wavy” dimensions to our 3 “corpuscular ones”, as we generally do not perceive these “extra-3” in current daylife: we can’t move along them, can we? So, a 6-dimensional world doesn’t really speak to us, does it? Even classical. And claiming it’s to be made equivalent to a quantum 3-dimensional one does not clear the situation at all… :) So, by pure convention, we usually assume that these 3 wavy dimensions are located “above” our 3 familiar ones, which is not true: in reality, all 6 stand on an equal footing and we are only limited in our perceptions of space around us (as usual). We cannot visualize non-solid waves, but we still can feel their effects, so that we remain conscious that waves exist, but we still cannot link them to anything “dimensional”. The picture we have of them is that they “undulate through classical 3-space”. But this is only good for classical waves. It does not represent the quantum reality. The quantum reality is 6-dimensional (space) or 8-dimensional (space-time).

 

This is one thing: additional dimensions. Then, we have the question of areas and surfaces.

 

A classically-perceived plane is a 2-dimensional space. If we consider a square inside that plane, with side x(0), than its area s(0) = [x(0)]² will always be a non-negative quantity. Negative areas cannot exist in classical space geometry, where they would be interpreted as areas “smaller than a point”, which is an object of null size, and this would lead to an absurdity.

 

Things are different in a geometry like that of classical space-time or, now, in the quantum. A quantum plane is schematized as a 2-dimensional plane delimited by that “horizontal corpuscular axis” and that “vertical wavy axis”: they’re similar, but not of the same physical nature at all (as the time dimension in special relativity was similar to any of the 3 space dimensions, but not of the same nature at all). If x(ksi) is now the size of a quantum square inside our quantum plane, then its quantum area is to be calculated as:

 

(1)               s(sigma) = [s(0) , sigma] = [x(ksi)]² = {[x(0)]² , 2ksi}

 

so that,

 

(2)               s(0) = [x(0)]²

 

remains a non-negative quantity, as the “pure area” of our quantum square, while

 

(3)               sigma = 2ksi

 

gives the quantum state our quantum area is found in when its side is found in the state ksi.

 

If a experimenter wants to measure the “corpuscular amount” of s(sigma), he/she’ll measure its P¹-projection:

 

(4)               s¹(sigma) = [x(0)]²cos(2ksi) = [x¹(ksi)]² – [x²(ksi)]²

 

If he/she wants to measure the “wavy amount”, he/she’ll measure the P²-projection:

 

(5)               s²(sigma) = [x(0)]²sin(2ksi) = 2x¹(ksi)x²(ksi)

 

According to the sector the quantum state sigma (that of the object we’re studying) is in, both projections will be either positively-counted, zero or negatively-counted.

 

If 0 < sigma < pi/2 (sector I), 0 < ksi < pi/4 (45°), then both s¹(sigma) and s²(sigma) will be measured positive.

If pi/2 < sigma < pi (sector II), pi/4 < ksi < pi/2, then s¹(sigma) will be measured negative while s²(sigma) will remain positive.

If pi < sigma < 3pi/2 (sector III), pi/2 < ksi < 3pi/4, then both s¹(sigma) and s²(sigma) will be measured negative.

And, if 3pi/2 < sigma < 2pi (sector IV), 3pi/4 < ksi < pi, then s¹(sigma) will be measured positive while s²(sigma) will be negative.

 

You’ll have noticed that, opposite to “classical” multiplication, “quantum” multiplication entangles the corpuscular and the wavy projections of the quantum side x(ksi). Despite this, s¹(sigma) remains the “corpuscular” projection of the quantum square and s²(sigma), the “wavy” one. The planar representation could therefore lead to easy confusion. The polar representation is much clearer, as it precises no projection, it instead gives the amplitude and the quantum state.

 

Let’s start from ksi = 0, that’s sigma = 0. Then, s¹(0) = [x(0)]² is obviously maximal and corresponds to the value given by classical geometry, while s²(0) = 0, confirming that, from a strictly classical viewpoint, the square is entirely “corpuscular”, since its “wavy side” is reduced to a point. As ksi increases, we go deeper inside the quantum plane, s¹(sigma) decreases while s²(sigma) increases: our quantum area acquires more and more “wavy content” and leaves more and more “substantial content”. When ksi reaches 45°, sigma = 90°, we stand on the P²-axis, s¹(sigma) = 0 and s²(sigma) = [x(0)]² is now maximal: a classical P²-observer would come to the same conclusion as our previous P¹-observer.

 

Let’s keep on increasing ksi. Then, sigma becomes greater than 90°, we change sector on the quantum plane, the “wavy content” of x(ksi) becomes greater than its “corpuscular” one, forcing s¹(sigma) to decrease under the value zero and turn negative. However, the physical context is very different from the one found in space-time relativity. In space-time relativity, the “absolute area” s² = c²t² - x² = c²t²(1 – v_moy²/c²), where v_moy = x/t stood for the pure value of the mean velocity of a moving body, couldn’t turn negative without going out of the observation scope. This was because the body would then move faster than the signal it produces, arriving always before it. Now, physical bodies are observed through the signal they emit. If they arrive before it, they’re non-observable… Here, nothing of this happens. What happens instead is we’re in a space with an open geometry (technically, “of hyperbolic type” – archetype: the horse saddle), like space-time, but without any specific restriction. In comparison, classical space had a closed geometry (“of elliptic type” – archetype: the rugby ball). Such a geometry allows only one sign to areas, the positive one.

 

The same holds for s²(sigma). We can always transform it noticing that:

 

2x¹(ksi)x²(ksi) = ½ {[x¹(ksi) + x²(ksi)]² – [x¹(ksi) – x²(ksi)]²}

 

which exhibits the same structure than s¹(sigma). And it goes on changing sign as quantum states go round the unit-radius circle. There’s no conceptual objection to finding negative areas in the quantum context, even projections, because there’s no definite sign in either s¹(sigma) or s²(sigma), it’s now only a question of which behavior predominates on the other in the side x(ksi) of the quantum square.

 

You can straightforwardly generalize this to quantum rectangles. Taking two quantum sides x(ksi) and y(psi), the area of the quantum rectangle will be:

 

(6)               s(sigma) = x(ksi)y(psi) = [x(0)y(0) , ksi + psi]

 

Then, you examine sigma sector by sector: results will be the same. It’s just a bit more complicated because you now deal with two quantum states ksi and psi instead of a single one. You find more combinations for a given sigma, namely:

 

(7)               sigma = ksi + psi

 

instead of (3). So, instead of finding a single value ksi = sigma/2 as for the square, you find a continuous infinity of possibilities psi = sigma – ksi for each value of sigma.

 

Quantum volumes proceed the same. In place of (1), you find the quantum volume of the quantum cube:

 

(8)               v(stigma) = [x(ksi)]³ = [v(0) , stigma]

(9)               v(0) = [x(0)]³

(10)           stigma = 3ksi

 

As x(0) is never negative, nor is v(0) but, according to the sector stigma will be found, projections v¹(stigma) and v²(stigma) will be positively or negatively counted or even be zero.

 

For a quantum parallelepiped:

 

(11)           v(stigma) = x(ksi)y(psi)z(zeta) = [v(0) , stigma]

(12)           v(0) = x(0)y(0)z(0)

(13)           stigma = ksi + psi + zeta

 

which, for each given value of stigma, draws a straight line, not in a “2D quantum state” anymore, but in a “3D” one. That’s a double continuous infinity of possibilities for zeta = stigma – ksi – psi.

 

In comparison, (9) or (12) show you again that classical volumes can only be found positive or zero.