 # The Quantum Relations Principle

The Quantum Relations Principle.

Managing our Futur in the Age of Intelligent Machines.

Hardy F. Schloer, Mihai I.Spariosu

Chapter 2. Why is Quantum Relations Quantum? The term “quantum” as used in physics is derived from the Latin word quantus, meaning “how many?” It was first used in quantum physics to describe the behavior of systems in which only certain levels were permitted. In classical physics, a particle of light can have any amount of energy, but in quantum physics, only certain levels are allowed. In turn, these levels are highly constrained by the laws governing the atom that emitted the photon. The basic unit of which all other values must be a multiple is called a “quantum.” The term “quantized” means that a system can take on only certain values, and those are chosen according to particular rules. As an example, U.S. currency is quantized, and the quantum is the penny. In everyday life, nothing costs a fraction of a penny. What justifies the use of the name Quantum Relations for the principles and techniques outlined in this book? This is a reasonable question since ‘quantum’ has a particular meaning in scientific discourse, usually signifying some discrete particle for which further reduction is outside the theory. Therefore, the present chapter explains why this particular name was chosen and describes some of the theoretical underpinnings of the Quantum Relations Principle. Scientific puzzles discovered in the 19th century led to two revolutionary theories during the early years of the 20th century: quantum mechanics and the theory of relativity. These were more than empirical discoveries; they were both conceptual frameworks within which to remap systems. Relativity greatly sharpened the concept of frames of reference, and quantum mechanics contributed to the development of a well-defined theory of states and state vectors. Furthermore, quantum mechanics was framed in terms of operators (a
special method of describing functions) and of probability distributions over function spaces. All of these conceptualizations apply to Quantum Relations. A Five-Cent Tour of Quantum Mechanics For the non-mathematically inclined reader, be assured that nothing in this section involves mathematical manipulations or formulae. Instead, our descriptions are conceptual, to give the flavor of the subject. Quantum mechanics (QM) is a theory of probabilities. It is a model, a conceptual framework used in physics to calculate probabilities. A probability is simply the chance that some event will be observed. For example, the chance that two dice rolled randomly will show 11 is about 1/18. A probability relates to future observations (or is used to explain past observations). Probabilities are either calculated or measured. To get to the number 2/9, we could enumerate the different outcomes ({1, 1}, {1, 2} … {6, 6}) and count the number of each that gave particular sums, or we could roll the two dice a few billion times and mark the number of times that each total appeared. Either method should give pretty much the same answer. Classical (i.e., non-quantum) mechanics uses ordinary arithmetic, applied to ordinary “real” numbers like 3, 2.7, pi, or seventeen billion. Classical probabilities of events are computable with these numbers, using addition, subtraction, multiplication, and division. If you have enough information about a system, you can calculate any result exactly, and there is nothing in principle, except time and lack of brain cells, to prevent you from gaining a perfect level of information. You can observe, and enumerate or put into some order, all the different states of any system. Single definite states evolve over some time span to other single, definite outcomes. Quantum mechanics, on the other hand, uses a different kind of number, and a different kind of arithmetic. Additionally, states are not necessarily
observable but can still affect outcomes. In QM, all that can be observed are outcomes. Because QM uses a different kind of number, and a different kind of arithmetic, the ordinary way of calculating probabilities will not work. QM has a different method. QM numbers have two different and independent “ordinary” numbers for every “state” of a system.1 In many cases, the predictions of the system may not predict a single outcome—there is still only one prediction, as in classical mechanics, but that single prediction may have multiple outcomes, only one of which will be actually observed. For each prediction, every possible outcome will have some probability assigned to it. So, QM, like classical mechanics, makes exact predictions, but these predictions are not predictions of the outcome of some process. In classical mechanics, there can be one, and only one, correctly predicted observable. In quantum mechanics, there can be one, and only one, correct set of outcomes, each with one, and only one, probability associated with each outcome. Quantum Mechanics is exact in its predictions of sets, but ‘fuzzy’ in predictions of any particular outcome. And, worse, one cannot, even in theory, get enough information about the system to make the calculations of a particular outcome exact. Fundamental to 20th century Quantum Mechanics are two concepts: observables, which are simply quantities capable of being measured; and some function (typically the ‘wave function’), which returns the probability (amplitude) of measuring each particular value of the observable. In classical mechanics, one calculates the outcome. In QM, there may be one, two, or even an infinite number of different outcomes, but for each, a probability of

1 State appears in quotations in this sentence because QM states are not necessarily observable, and therefore are internal theoretical entities that may or may not correspond to anything in the ‘real’ world.
occurrence can be calculated. In other words, we do not know whether the answer will be 11, and no amount of knowledge about the system will help us know that, but we can predict that if we keep rolling the dice, eventually 1/18th of the total number of rolls will show an 11. This idea that only probability (not outcome) is calculable is not unique to quantum mechanics. The way of determining it, however, is. Years of detailed experiments have convinced physicists that simple numbers (3, 2.7, pi, a billion) are not sufficient to explain how things are in the atomic world. However, all the physics known at that time could be explained using pairs of numbers (such pairs are unfortunately often called complex numbers—in truth there is nothing complex about them). Since QM numbers are pairs, a new way of doing arithmetic with them had to be devised.2 Numbers now looked like {3, 2}, rather than simply 3 or simply 2. How do you add {3, 3} to {4, -1}? Multiplication got its own rules too: {3, 0}* {3, 0} gives {9, 0}, but {3, 1}*{3, -1} also gives {9, 0}. And, how does one calculate the square root of {6, -3}? QM arithmetic provides an answer, although for our purposes, it is not important what that answer might be. The QM numbers like {3, 2} are generally called vectors, but vectors can also have more than two elements. A four-element vector would look like {3, 2, 1, 0}. Ordinary (classical) numbers can be thought of as vectors of length 1, i.e., {3} but there is nothing to gain by not merely calling that quantity 3. Classical mechanics also uses vectors, but the elements of these vectors are usually ordinary numbers. But in QM, to repeat, the basic elements are not simple
2 “New” is likely the incorrect word here, because mathematicians already knew how to do such manipulations of complex numbers and matrices. In the early part of the 20th century, Heisenberg and Schrödinger independently discovered the value of these artifices to explain and predict experiments in atomic physics.
numbers. Fortunately for physicists, two element numbers appear sufficient to explain physical problems, using somewhat odd definitions of addition, subtraction, multiplication and division. There is no obvious reason for this to be true, but it seems that it is so. A Five-Cent Tour of Quantum Probabilities A probability is the chance that something will occur. There is a 20% chance of probability of rain tomorrow. Earlier we said that QM predicts sets of probabilities. This is not quite true—QM predicts sets of probability amplitudes. Thus, QM is even a little farther away from classical mechanics. There is nothing difficult or particularly strange about probability amplitudes—a probability amplitude is just the square root of a probability, using the arithmetic of QM. QM also uses probability density. Probability density can be thought of as a map of ‘how likely’ some measurement is to be found with some particular value (some particular probability amplitude). For example, the probability of finding me on Christmas Eve is more dense around my home, in London, England, than around Melbourne, Australia.
Amplitudes of probability of distance from home on Christmas Eve
home sister‘s house
Melbourne
Figure 2 – (not to scale) Probability Amplitudes Distribution Quantum systems, as well as classical ones, have states. Everyone knows what a state is: My car is either in the state of running or the state of not running. A state is also a location in some abstract space—in the car example, the distinct space is comprised of two points, running and not running. Using the example of Christmas Eve, the space could be the surface of the earth, and both Melbourne and my home in London would be regions within space. So, my states could be the group consisting of {near home, near Melbourne}. The probability density would be higher in the ‘near home’ region than the ‘near Melbourne’ region. Technically, the probability density is a relationship between quantum states and the (amplitude of a) probability of finding the observable (me) in that state.
Quantum Mechanics, Quantum Relations, and Measurement When making some physical measurement, the probability of “finding” the observable in some particular state (i.e. having some particular value) is proportional to the square root (in QM arithmetic) of the probability amplitude.3 That is, QM does not predict the value of any measurement.
3 This squaring of the probability amplitude is related to the fact that, in physics, quantum states are located in a complex vector space (a Hilbert space), and complex numbers have two measures (a + bi where I is the square root of -1). Since probabilities are (by definition) real numbers, positive and with only one measure, squaring the probability amplitude and Instead, it computes the probability amplitude of measuring some result. An example would be, as shown above (Figure 2), the set of probabilities of finding me at some distance from my home on Christmas Eve. One fundamental feature of quantum mechanics is that the sum of all the probabilities measured over all the possible values of the observable must add up to one. For example, in Figure 2, there could be, say, 59% probability that I will be near home, and 39% probability that I’ll be near my sister’s house, and a 2% probability that I’ll be somewhere else. These total probabilities add up to 1.0.
As mentioned at the beginning of this chapter, Quantum Mechanics is ‘quantum’ because it posits ‘quanta’, which are tiny indivisible entities that have states. A photon, for example, is a light quantum, and it has a state that represents its position in physical space. A photon can also be right-handed or left-handed, and or it can be partly in one state and partly in another, although the two states are entirely distinct.4 Once all the reachable states of a system have been enumerated, plus the probability amplitudes of transitioning from one state to another, QM knows everything possible about that system. So, the taking its absolute value converts probability density within the Hilbert space into a real number. 4 This ability to be in two distinct states at the same time distinguishes quantum logic from Aristotelian logic, violating the principle of the excluded middle. It is part of the apparent weirdness of quantum arithmetic.
idea of state is tightly connected to the idea of quanta. Quanta are elements that cannot be further reduced, within the theory.
In turn, Quantum Relations (QR) also has quanta. A quantum in QR is a “Data Fusion Object” (DFO). Data Fusion Objects can be constructed from other DFOs, and many DFOs can be deconstructed into their components. However, in any QR system, there is a point at which DFO’s cannot be further deconstructed. These baseline DFOs are the quanta of the particular QR system.
QR also has its own arithmetic. A system comprised of two DFOs will behave in ways that depend on how the properties (i.e. observables) combine.
DFOs can be added, subtracted, multiplied, and combined, and, just as in QM, more complicated combinations are possible, such as grouping, encapsulating, and splitting. Still, there comes a point in QR where a DFO cannot be further reduced, and therefore QR is entitled to be called a non-physical quantum theory.
Indeterminacy, Random Variables and Probability in QM and QR Popular recitations of Quantum Mechanics emphasize indeterminacy, including the famous Heisenberg uncertainty principle that we have already mentioned. In QM physics, the quanta are exceedingly small, many times smaller than atoms. When working out problems in QM arithmetic with real world particles, the indeterminacy results from the basic definition of QM
multiplication.
The uncertainty principle of quantum physics is worth further  explanation, because it reflects a more general principle of certain algebras (in QR terms, arithmetic). As mentioned earlier, QM uses the concept of a  probability density. It can be thought of as being denser where things are likely,
and less dense where things are unlikely. Generally, the probability density is  given by some function, called the probability density function, or simply the  pdf.
Scientists who study QM or the mathematics of probability have devised all sorts of ways of calculating the pdf for starting in one particular state of a particular system, then after some time, reaching another particular state.
None of these ways is important here, but once the pdf is known, you can calculate lots of other interesting things that describe states of the system and their evolution over time.
We shall now break our promise of avoiding mathematical formulae. But, the non-mathematical reader can simply skip this section and move to the next, keeping in mind only that Quantum Relations, like QM, is a probabilistic theory with some basic, irreducible quanta. The consequence of this, from a purely mathematical viewpoint, is that both systems have built-in and unavoidable uncertainty. For QM, the uncertainties are tiny compared to real-world events, and they are unlikely to be observed by anyone other than high-energy physicists. QM uncertainties do not explain psychic phenomena, they do not explain magic, and they do not explain why people forget their house keys.
Furthermore, QR uncertainties are not caused by QM uncertainty and, except for some mathematical parallelism, they are probably not related to it. But in QR, the calculations of uncertainties have a more dramatic effect, on a much larger scale than in QM, and this results primarily from the inability to precisely specify DFOs and DFO arithmetic that are computable using today’s computing machinery. The best we can do is to use powerful supercomputers to create approximations. Having said that, the non-mathematically inclined reader can safely skip to the next section.
For the more adventuresome, let us get right down to the relevant math.5 Much of this will seem like a review of elementary QM, but certain particular features are worth highlighting. We define some quantity 𝑿 as a “random variable”. This means that an exact value for 𝑿 is unknown, and perhaps even unknowable. Every time you measure 𝑿 you can get a different 5 We are indebted to Professor Paul J. Nathin for the form of the argument set out below, which can be found in considerably more detail in Chapter 5 of his excellent book, Dr. Euler’s Fabulous Formula, Princeton University Press, 2006, updated 2011.
and unpredictable value. 𝑿 is like your eccentric uncle at dinner—you can never predict what his next unlikely political statement will be. The random variable 𝑿 does have some well-defined properties. One is its probability density function 𝑓𝑿(𝑥), which is defined such that: (i) 𝑓𝑿(𝑥) is never negative regardless of 𝑥, and (ii) 𝑝’,)= 𝑓𝑿(𝑥)𝑑𝑥 ’ ) is the probability that 𝑿 has a value somewhere in t he interval [𝑎,𝑏], that is, 𝑎≤𝑥≤𝑏. The integral sign simply means that every possible value is tested in the region [𝑎,𝑏]. Since we want to determine 𝑿 empirically, that is, by making measurements, we also require that with probability 1 (i.e., certainty), 𝑿 has some value (perhaps including zero) somewhere. This requirement is stated mathematically as: (iii) 𝑓𝑿(𝑥)𝑑𝑥 12 32 =1 As already stated, there are many ways of deriving a probability density function (pdf) for a particular system, whether it be probability theory or QM or QR. For now, we will assume that the pdf has already been calculated, that it has some well-defined form, and meets the requirements above. There are some interesting things we can learn from every such pdf.
First, we can calculate the mean of the pdf, which is essentially its average value. We write the mean as 𝑿 and calculate it in the obvious way,
adding all the possible values of 𝑿 and dividing by the number of such values. Even if the number of possible values is infinite, we can perform this calculation by using an infinite sum, such as:
𝑥𝑓𝑿(𝑥)𝑑𝑥1 2 32 It is not obvious, but it is true that we can always make a system in which 𝑿 is equal to zero, and our new system will be equivalent to the original. To do this, we simply define a new 𝑿 that is equal to the old one, except we subtract it from 𝑿. That is, we work with 𝑿− 𝑿 , and simply call it 𝑿. A little thought will show that since 𝑿 is a random variable, but 𝑿 is simply a (real) number, nothing has changed with this substitution. Mathematically, 𝑿− 𝑿 = (𝑥− 𝑿)𝑓𝑿𝑥𝑑𝑥 12 32 = 𝑥𝑓𝑿𝑥𝑑𝑥 12 32 − ( 𝑿)𝑓𝑿𝑥𝑑𝑥 12 32 = 𝑥𝑓𝑿𝑥𝑑𝑥 12 32 −( 𝑿) 𝑓𝑿𝑥𝑑𝑥 12 32 =(𝑿 -𝑿)=0. R emembering that 𝑿 is the average value, and 𝑿 is the particular value given by the system at random, it should be obvious that if 𝑿 does not vary from 𝑿 by very much very often, then 𝑿 is a pretty good indicator of the typical value of 𝑿. Therefore, it is useful to know when this is true, and we can define a
quantity called the variance of 𝑿 as a measure of it. This variance is written as 𝜎𝑿 9=(𝑿 −𝑿)9, and it measures the square of the variation of the random variable around its average value. The primary purpose of squaring the variation is to eliminate negative values, which would cancel out positive ones. Combining this result with the definition that makes 𝑿=𝟎, and taking the square root, we get 𝜎𝑿 = 𝑥9𝑓𝑿(𝑥)𝑑𝑥 12 32 This has a great deal to do with quantum mechanics, but also with Quantum Relations. Nothing in the above discussion concerned any physical quantities, just some formal manipulation of a random variable. But, if you think about 𝜎𝑿 for a few minutes, you will see that it represents, in a very real way, the uncertainty that we have in the value of 𝑿. A “small” 𝜎𝑿 means that 𝑿 is pretty certain, not very uncertain, while a lar ge 𝜎𝑿 means that 𝑿 is all over the map.
Quantum Relations, Fourier Transformations and Uncertainty In the world of Quantum Relations, it is often difficult to express quantities in some “natural” way, as time series of observable quantities. It may be much easier to express them as frequencies of occurrence, with amplitudes associated with frequencies, or as frequencies and phases. Without getting into the methods of such representations, it will be obvious to the mathematically inclined that this last way, representing a series or a time-amplitude function as frequency and phase, is equivalent to using the Fourier transform of the original series or function.
A function and its Fourier transformation are the same thing, just represented in different ways. A time series function (whether continuous or discrete) is a map of value amplitudes at different times, whereas its Fourier transform can be used as a map of where the energy is at different frequencies, that is, at different periodic time intervals. To say it in a different way, time series show time value pairs and the Fourier transform of a time series shows energy interval pairs. Consider a particular function 𝑓(𝑡) and its Fourier pair ℱ(𝜔). We could have 𝑓(𝑡) represent some location in time and ℱ(𝜔) represent energy at some frequency. Location in time means that “most” of 𝑓(𝑡) occurs within some interval of time and “some frequency” means that “most” of ℱ(𝜔) occurs within some interval of frequencies.
The term energy needs some definition. We can use the classical mechanical definition quite nicely: The energy of some function 𝑔(𝑥) is defined as 𝑊= 𝑔9𝑥𝑑𝑥 2 32
As mathematician Hermann Weyl pointed out in 1928, referring to the structure of QM, we can show by dividing both sides of the above equation by W that: 𝑔9𝑥 𝑊𝑑𝑥23 2 =1 This is reminiscent of condition (iii) for a random variable. That is, substituting time (𝑡) for x, we can justifiably treat @AB C as a random variable in time. Therefore, we can compute its mean and variance, and applying the formula given above, 𝑔(𝑥) has an uncertainty of 𝜎𝒙 = 𝑔9(𝑥) 𝑊𝑑𝑥1 2 32 For our purposes, and following Weyl, we substitute 𝑓𝑡for 𝑔(𝑥) and get 𝜎𝒕 = 𝑓9(𝑡) 𝑊𝑑𝑡1 2 32 This is quite a remarkable approach. We are treating time itself as a random variable taking values in the interval −∞,∞, and having an average value of zero we derive a probability distribution function of @A(B) C . Next, we use Rayleigh’s energy formula relating to Fourier transforms, giving
𝑊=1 2𝜋 𝜔9ℱ(𝜔)9 2 32
or
ℱ(𝜔)9 2𝜋𝑊23 2 =1 where for all 𝜔, ℱ(𝜔)9 2𝜋𝑊≥0 What we have shown is that ℱ(J)A 9KC looks just like the probability distribution function of a random variable that takes on values in the interval [−∞<𝜔<∞], and which (consequently) has an average value of zero. Therefore, just as we did with time, we can write 𝜎J = 𝜔9ℱ(𝜔)9 2𝜋𝑊 𝑑𝜔1 2 32 which gives us the uncertainty in frequency.
We have derived an uncertainty in location in time, and we have derived an uncertainty in location in energy. The question that occurs to every student of Heisenberg is: what can we say about the product of these, i.e., 𝜎J 𝜎𝒕 ? Mathematically, and again following Weyl, we use the Cauchy-Schwartz inequality: If ℎ(𝑡) and 𝑠(𝑡) are two real-valued functions, then assuming the integrals actually exist: ℎ𝑡𝑠𝑡𝑑𝑡23 2 9≤ ℎ𝑡𝑑𝑡 2 32 𝑠𝑡𝑑𝑡 2 32
For our purposes, we substitute 𝑡𝑓(𝑡) for 𝑠(𝑡) and for ℎ(𝑡) we substitute O@ OB. With these substitutions, the above inequality becomes: 𝑡𝑓(𝑡)𝑑𝑔 𝑑𝑡𝑑𝑡23 2 9≤ 𝑡9𝑓9(𝑡)𝑑𝑡 2 32 𝑑𝑔 𝑑𝑡9𝑑𝑡23 2 Where to go from here is not entirely obvious, but since 𝑔𝑡O@ OB=O@A(B)/9 OB , then 𝑡𝑓(𝑡)𝑑𝑔 𝑑𝑡𝑑𝑡23 2 = 𝑡𝑑(𝑔9𝑡 2)𝑑𝑔 𝑑𝑡 𝑑𝑡23 2 and the integral on the right of the equality can be integrated by parts: 𝑡𝑑(𝑔9𝑡 2)𝑑𝑔 𝑑𝑡 𝑑𝑡23 2 =(𝑡𝑔9𝑡 2 |∞ −∞− 𝑔9(𝑡) 2𝑑𝑡23 2 And because g(t) goes to zero much faster than 1𝑡, then lim| B|→2𝑡(𝑔9(𝑡) =0 which means that 𝑡𝑔(𝑡) 𝑑𝑡𝑑𝑡23 2 =− 𝑔9(𝑡) 2𝑑𝑡23 2 This is almost our classical definition of energy (negated and divided by 2), so
that
𝑡𝑔(𝑡) 𝑑𝑡𝑑𝑡23 2 =−1 2𝑊
and this is the right-hand side of the Cauchy-Schwartz inequality. The left-hand side is simple by comparison. By definition, 𝑡9𝑔923 2 𝑡𝑑𝑡=𝑊𝜎9𝒕 and we use Raleigh’s energy formula in this way: 𝑑𝑔 𝑑𝑡9𝑑𝑡=1 2𝜋 𝜔9ℱ(𝜔)9𝑑𝑤 2 32 which is equal to 𝑊𝜎J 9 where 𝜎J is the uncertainty in frequency, that is: 𝜎J = 𝜔9ℱ(𝜔)9 2𝜋𝑊 𝑑𝜔1 2 32 And now we can state the Cauchy-Schwartz inequality as an uncertainty principle, relating measurements of time and frequency: −1 2𝑊9≤(𝑊𝜎B9)(𝑊𝜎J 9 which simplifies to 𝜎B𝜎J≥1 2 This result is important —it means that any attempt to determine future values by modeling past measurements with random variables is subject to a fundamental limitation in its accuracy. And yet, the real world (particularly in
QM) appears at some very fundamental level to be best modeled by such methods. We accept it as true that Quantum Relations is subject to the same limitation, and that the use of time series, whether discrete or not, and models such as the Data Fusion Object (DFO) have definite limitations that cannot be overcome by merely adding computing power to them. The result says that the more one focuses on narrowing the precise time slice of a system, the less one knows about what will happen during that time slice.
What Do QM, Relativity, and QR Calculate?
Assume that some set of probability amplitudes has been predicted by a Quantum Mechanics calculation. Nothing is certain about the actual outcome (an observation) except the assigned probabilities. However, once a particular outcome has been observed, no matter how small the probability assigned to that outcome, any subsequent observation will show the same value. This is extremely mysterious—the complicated set of probability amplitudes has suddenly vanished, ‘collapsed’ into one single and simple, ordinary real number.
QM arithmetic assures us that there are no ‘hidden variables’, that is, it is not as simple as the system having one and only one internal state that we have not observed. Instead, there is a set of values, not just one, and the simple act of observation, by anyone, collapses this set into one, and only one, value.6
6 This is one of many ways of explaining this mysterious behavior, and roughly follows the so-called Copenhagen interpretation. Others have argued for different interpretations (including the many-worlds approach), but these are not relevant to the present discussion.
From a computational point of view, the collapse of the set of probability amplitudes into one simple number makes predictions of complex events much easier. Whenever an observation is made, we substitute the result of the observation for the complicated set of probability amplitudes and continue with the computations. Thus, in Quantum Relations models, probability trees with potentially trillions of leaves equally collapse into a few hundred leafless branches, well within the computing power of today’s machinery. We can similarly trim the trees by eliminating the less possible branches (those with lower probability) which, while decreasing the accuracy of our ultimate predictions, makes computation of likely outcomes possible in reasonable lengths of time.
Relativity, on the other hand, is another method used by physics to calculate expected observables. Relativity, as we have seen, is not concerned with probability, but rather with the relationship of physical laws from different frames of reference. A physical principle, according to Galileo Galilei, does not change depending on the observer’s frame of reference. In Galileo’s book Dialogue Concerning the Two Chief World Systems (1632), the physical principle was the relativity of motion—specifically the location of a sack of grain on a moving ship. From the shore, the sack moves. For an observer on the boat, the sack is motionless. However, Galileo argued, there is only one physical principle that must apply in both frames of reference. Similarly, when the ship is moving, a rock dropped from the mast will appear, to the ship’s captain, to fall straight down, but, from the shore, it will follow a curved path. And yet, Galileo argued, there is only one physical principle that governs both. One cannot understand any observation, then, without taking into account the frame of reference of the observer.
Quantum Relations uses this principle as well. But It does not use (nor does it claim to use) the specific mathematics that governs physical laws such as electromagnetism or gravitation. Fundamentally, QR is the modeling of systems by putting their components into frames of reference and discerning the sets of invariants and symmetries between the observations in different frames and in different combinations of frames. QR is not concerned with Lorenz transformations, or the value of the speed of light, or whether c is a universal speed limit. Nor is it concerned with mass and energy in the physical sense, although it uses similar terms where they help explain a set of relationships or transformations. Rather, it utilizes the methods of relativity in the conception of how different relationships (“laws”) can be framed in ways that are independent of particular frames of reference. These relationships are then used to calculate particular results of particular interactions. The theory is successful to the extent that these results (predictions) are successful, i.e., correspond to observed outcomes.
To sum up the discussion in this chapter, Quantum Mechanics is about computing probabilities and using a particular set of machinery to perform
these computations. Similarly, Quantum Relations is about probabilities, and not about determined outcomes. QR and QM do not use exactly the same set of mathematical operators, but the mathematical methods behind QR were inspired by the approach of QM. QR is called Quantum Relations because, like Quantum Mechanics, it rejects a privileged frame of reference. Finally, QR does not claim to synthesize either Quantum Mechanics or Relativity Theory. Instead, it borrows mechanical tools (including mathematical ideas) and valuable concepts from both and uses them to calculate real-world results, generally human, sociological, political and economic predications.
Models in QR are valuable to the extent that they are falsifiable, at least in principle. As a relatively simple, common example from the world of finance, consider a set of actors who are modeled in an economic simulation as interacting “wants” and “needs,” with limits on some set of internal variables or observables (e.g., wealth). The interactions within some global frame of reference can be calculated, following certain patterns and behaving in certain ways. These ways are often surprising, even to the model designers. When applied to real-world actors, measured values are supplied for the “wants” and “needs,” and the simulations can be run on a computer. The results then predict the outcome of the economic transaction. If the result is proved incorrect, then either the supplied values do not represent real values, or the model is incorrect or incomplete.
The benefit of using QR for this type of simulation is that methods of interaction can be described algorithmically (or functionally). These should and
can be independent of particular frames of reference. That makes these methods of interaction useful in the same way that other algorithmic or functional processes are useful, as reusable components and building blocks for larger systems. In the next chapter, we shall describe the principles of computation and several specific computational tools derived from the QR principle.

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