A Notable Quasi-Relativistic Wave Equation and Its Relation to the Schrödinger, Klein-Gordon, and Dirac Equations

An intriguing quasi-relativistic wave equation, which is useful between the range of applications of the Schrödinger and the Klein-Gordon equations, is discussed. This equation allows for a quantum description of a constant number of spin-0 particles moving at quasi-relativistic energies. It is shown how to obtain a Pauli-like version of this equation from the Dirac equation. This Pauli-like quasi-relativistic wave equation allows for a quantum description of a constant number of spin-1/2 particles moving at quasi-relativistic energies and interacting with an external electromagnetic field. In addition, it was found an excellent agreement between the energies of the electron in heavy Hydrogen-like atoms obtained using the Dirac equation, and the energies calculated using a perturbation approach based on the quasi-relativistic wave equation. Finally, it is argued that the notable quasi-relativistic wave equation discussed in this work provides interesting pedagogical opportunities for a fresh approach to the introduction to relativistic effects in introductory quantum mechanics courses.


Introduction
As brash as this may sound, claims that probabilistic specifications of quantum mechanics are inconsistent with local realism and defy Bell's inequality are just plain wrong. This may be difficult to accept, depending on how wedded one is to the outlook that gives rise to defiance of the inequality as it is widely understood.
In the eyes of the professional physics community, the matter is now closed. The eminent journal Nature [1] flamboyantly announced the "Death by experiment for local realism" as an introduction to its publication of ambitious experimental results achieved at the Technical University of Delft. These were proclaimed to have closed simultaneously all seven loopholes that had been suggested as possible explanations for the purported violations of the inequality. My claim is that the touted violation of the inequality derives from a mathematical mistake, an error of neglect. Moreover, consequences of the error run through both the analytical development of the defiance structure and the statistical assessment of relevant matters. Its recognition relies only on a basic understanding of functions of many variables and on standard features of applied linear algebra. This presentation is prepared so to be read not only by physicists but by any sophisticated reader who has followed this issue at least at the level of popular description of scientific activity and who is not put off by equations per se. The defiance of Bell's inequality is one of the seminal results underlying the understanding of quantum theory as it has developed over the past half century. More than seven thousand references to pertinent discussion can be found on Google Scholar, and I do not intend to review the corpus. Most physicists regard the inequality as a condition on freely determined expectation values and their estimates from physical experiments, calculated according to the QM formalism. I consider this viewpoint to be mistaken, for reasons I shall detail herein. However, while surely in a minority camp of objectors, my analysis can be situated within a literature of creditable researchers whose perspective I share. The recent review article of Kupczynski [2] surveys and references more than eighty technical publications in this vein, both longstanding and recent, notably recognized contributions of Fine Experimental choices of the two polarizing directions yield a specific relative angle between them at stations A and B in any given experiment. Using Aspect's notation that parentheses around a pair of directions denotes the relative angle between them, the experimental detection angle settings ( ) In order to view the relative angles we are talking about, mentally we would need to swing the ( ) , x y plane around by 180˚ as it is viewed by the photon directed to station A, and superimpose it on the ( ) , x y plane as it is viewed by the photon directed to station B. In this manner we can understand the size and meaning of the relative angles between the various values of polarization orientations * a and * b as seen here in Figure 2, which follows.

The Physical Setup of Four Experiments Providing Context for Bell's Inequality in CHSH Form: A 16-D Problem
The theory of quantum mechanics motivates specification of probabilities for the four observable outcome possibilities of the polarization experiment as depending on the relative angle ( ) For efficiency in what follows, we shall denote the four probabilities appearing in Equations (1) by , , P P P ++ −− +− , and P −+ when the pertinent angle setting is evident.
These four probabilities surely sum to equal 1, because the sum of  (2) according to standard double angle formulas. It is worthwhile reminding right here that "the expected value of a probability distribution" is the "first moment" of the distribution. Geometrically, it is the point of balance of the probability mass function weights when they are positioned in space at the places where the possible observations to which they pertain might occur. It is a property of a probability distribution for the outcome of a specific single observable variable. A final peculiarity of Equation (2) to be useful far down the road in this explication is that the expectation value can also be represented as For the value of ( ) 2 * * sin , a b appearing in the final line of Equation (2) can also be written as This result codifies a touted feature of physical processes at quantum scales of magnitude, that the photon behaviours of particle pairs are understood to be entangled. Since the probability for the joint photon behaviour , the conditional distribution for either one of these events depends on the context of the conditioning behaviour: and ( ) , which is different still.
We have concluded what we need to say at the moment about the prescriptions of quantum theory relevant to quantum polarization behaviour of a single pair of prepared photons. Before proceeding to the specification of Bell's inequality, we need to address three issues: what quantum theory professes not to say on account of the uncertainty principle, the relevance of the principle of local realism, and a proposal regarding supplementary variables that may impinge on the experimental results.

The Uncertainty Principle: What Quantum Theory Disavows
The problem of quantum physics relevant to Bell , which codify our experimental measurement possibilities. It is a simple matter to determine algebraically that no two of these proposed matrices commute.
All this is to say that the technical manipulations of mathematical quantum theory instantiate formally just what we knew to begin with... that we cannot simultaneously perform the measurement observation of the polarization products at both angle settings ( ) , ′ ′ a b as well. Although quantum theoretical intrigue allows us to assert probabilities such as or whatever, it abstains from any prognostication of the form . This would be a probability assertion regarding simultaneous outcomes of a jointly unobservable pair of events.
Well, who would want to? We shall now find out. Although impossible to perform, we are surely permitted to think about what might happen if we could perform such simultaneous experiments. Enter the realm of a gedankenexperiment.

The Principle of Local Realism and Its Relevance to Bell
A feature that will be found crucial to the touted violation of Bell's inequality is that it pertains to experimental results supposedly conducted with a single photon pair at all four angle settings. This is very clear in the memorial article of Aspect [14] and many assessments of the inequality that properly recognize this.
An example would be the article of Adenier [15]. Many discussants do not. How did such a context for the experiment arise? When the probabilistic pronouncements of quantum theory were formalized, Einstein among others was puzzled by the fact that the conditional probability for the outcome of the experiment at station A depends on both the angle at which the experiment is conducted at station B and on the outcome of that experiment. This matter is codified by the conditional probabilities we have seen in Equations (5). This entanglement of seemingly unrelated physical processes was deemed by Einstein to be a matter of "spooky action at a distance". Along with Podolsky and Rosen [16] he proposed a solution to this enigma, positing that there must be some other factors relevant to what might be happening at the polarizer stations A and B which would account for photon detections that are found to arise. As yet unspecified in the theory, he considered such factors to identify unknown values of "supplementary variables". It was proposed that the probabilities inherent in the results of quantum theory must be representations of scientific uncertainty about the action of these other variables on the two photons at their respective stations. This was their proposed way of accounting for the spooky action at a distance: the "state" of a photon in a polarization experiment, along with the condition of its attendant supplementary variables, involves its disposition to respond to the experiment at any and every one of its relative angle settings.
However, there is one aspect of the matter upon which Einstein wanted to insist: this was termed "the principle of local realism". Although it is central to matters under consideration in this problem, the applicable formulation of this principle, its meaning, and its relevance to the CHSH formulation of Bell's inequality (which we are soon to address) have been matters of contention. In unembellished form, the locality principle merely asserts that physical mechanics engaging at some particular location are not influenced by physical conditions arising in another unconnected locale far removed in space. What this would mean precisely for a gedanken scenario such as we will be considering is a matter of published discussion, notably among Mermin [17], Hess and Philipp [18], and Mermin [19], though many more have been involved. The discussion concerns whether the principle of locality alone is sufficient to establish a factorization that is involved in the CHSH form of Bell's inequality, and how the principle might need to be extended.
Whatever the precise form of its motivation relative to locality, the defiance of Bell's inequality derives from a specific and precise mathematical condition that we will recognize in its development. Fair enough, quantum theory does stipulate the probability for photon behaviour at station A with its polarizer direction , say, then in this instance the measurement at A would have to be the same in any simultaneous gedankenobservation, no matter whether the direction setting at station B were b or ′ b .
That is to say, if the polarization observation ( ) 1 in a particular experiment on a pair of photons measure in the paired angle design ( ) Actually, our mathematical exposition of the probabilistic specifications of quantum theory has already deferred to such an understanding. We have been denoting the photon detection value at station A merely by ( ) A a rather than denoting it by ( ) , A a b , even before we have now introduced consideration of this principle of local realism. In the context of locality, the importance of such simplification of the notation was stressed explicitly by Aspect [14]. In fact we had no need to denote the paired direction at B in our notation for ( ) So quantum theory explicitly disavows addressing this matter directly, though it is surely a matter of relevance to the interpretation of quantum theoretic prescriptions.
We are ready to conclude this Section by proposing an experimental measurement that lies at the heart of Bell's inequality. We are not yet ready to assess it, nor to explain its relevance to the principle of local realism, but we shall merely air it now for viewing. Peculiar, it is considered to be the result of a gedankenexperiment.
Consider a pair of photons to be ejected toward stations A and B at which the pair of polarizers can be directed in any of the four relative angles we have described. According to the detection of whether the photons pass through the polarizers or are deflected by them, Bell's inequality pertains to an experimental Mathematically, we would refer to this quantity s as a linear combination of four polarization detection products. Any one of the four terms that determine the value of s could be observed in an experiment on a pair of prepared photons.

Einstein's Proposal of Hidden Variables Relevant to the Matter
The famous paper [16] which addressed these matters presented ideas that had been brewing for many years [20]. It is now widely known merely as the EPR proposal. The ideas were opposed to those of others who were proclaiming that the experimental and theoretical discoveries of QM support the view that at its fundamental level of particulate matter, the behaviour of Nature is random, and that quantum theory had identified its probabilistic structure. Convinced that "he (the old one) is not playing dice with the universe", EPR proposed that the formulation of quantum theory is incomplete, and quantum probabilities represent our uncertainty about the influence of unspecified supplementary variables. The article stimulated a fury of healthy discussion and argument that I shall not summarize here. Well documented both in the professional journals of physics and in literature of popular science, the discussion has featured considerations of the collapse of a quantum system when subject to observation that disturbs it, the non-locality of quantum processes, and esoteric formulations of the "many worlds" view of quantum theory. What matters for my presentation here is that Einstein's views were widely relegated as a quirky peculiar sideline, and the recognition of randomness as a fundamental feature of quantum activity came to the forefront of theoretical physics. Enter John Bell. Interested in a reconsideration of Einstein's view, he began his research with an idea to re-establish its validity as a contending interpretation of what we know. However, he was surprised to find this programme at an impasse when he discovered that if the principle of local realism is valid then the F. Lad probabilistic specifications of quantum theory which we have described above seem to defy a simple requirement of mathematical probabilities. In the context of a hidden variables interpretation of the matter, this seemed to require that the principle of local realism must be rejected. Reported in a pair of articles [8] [9], these results too stimulated a continuation of the flurry which has lasted through the 2015 publication in Nature of their apparently definitive substantiation by the research at the Delft University of Technology.
The specification of Bell's inequality can take many forms. The context in which it is addressed in the remainder of my exposition here was presented in an article by Clauser, Horne, Shimony, and Holt [13], commonly referred to as the CHSH formulation. This was the form that attracted still another principal investigator in this story, Alain Aspect. A young experimentalist, he wondered how could such a monumental result of quantum physics pertain only to a thought experiment, devoid of actual physical experimental confirmation. He thought to have devised an experimental method that could confirm or deny the defiance of Bell's inequality. My assessment of his empirical work follows directly from his clear and thoughtful explanation of the situation [14] reported to a conference organized to memorialize Bell's work. My notation is largely the same as Aspect's. I adjust only the notation for expectation of a random variable to the standard form of ( ) E X , replacing his notation of X which has become standard in mathematical physics in the context of bra-ket notation which I avoid. Here is how it works.

Explicit Construction of s with Hidden Variables
Hidden variables theory proposes that the quantity s which we have introduced in Equation (6) should be considered to derive from a physical function of unobserved and unknown hidden variables, whose values might be codified by the vector λ , viz., for λ ∈ Λ . The variable designated by λ here could be a vector of any number of components identifying unknown features of the experimental setup that are relevant to the outcome of the experiment in any specific instantiation. The set designated by Λ is meant to represent the space of possible values of these hidden variables. The status of these variables in the context of any particular experiment is supposed only to depend on the state of the photon pair and its surrounds, independent of the angle setting ( ) * * , a b at which the polarizers are directed. According to the deterministic outlook underlying physical theory relying on hidden variables, if we could only know the values of these unspecified variables at the time of any experimental run and have a complete theoretical understanding of their relevance to the polarization behaviour of the photon pair, then we would know what would be the values of the polarization incidence detection of the photon pair at any one or all of the possible angle settings. Now the personalist subjective theory of probability (apparently subscribed to by Einstein, and surely by Bruno de Finetti and by me) specifies that any individual's uncertain knowledge of the values of observable but unknown quantities could be representable by a probability density over its space of possibilities. Aspect denotes such a density in this situation by ( ) ρ λ . For any proponent of quantum probabilities it might well be presumed to be "rotationally invariant" over the full 360˚ of angles at which the photon may be fluttering toward the polarizer. That is to say, the probabilities for the possible values of the supple- where expectation is assessed with respect to the density ( ) ρ λ , yielding then more simply as an expectation relative to the random polarization products at these various angles. Equation (8) follows directly from Equation (7) because a rule of probability says that the expectation of any linear combination of random quantities equals the same linear combination of their expectations. Fortunately, we have already reported in Equation (5) that the probabilities of quantum theory identify the expected value of any polarization product at the variable relative pola- So we are ready to proceed.

Finally, Bell's Inequality
We have now arrived at a place we can state precisely what Bell's inequality says.
There is just a little more specificity to detail before we soon will have it. However, I should alert you that there is a little tic in the understanding of Equation (8) to which we shall return after we learn how the inequality is currently understood to be defied by quantum theory. But on the face of it, the validity of Equation (8) is plain as day. Now re-examining Equation (7), it is apparent that it can be factored into a simplified form: It is important to notice that once again, in performing this simple factorization of the components ( ) , ′ a b . It is the principle of local realism or its extension, extraneous to any claims of quantum theory, that provides the observed value of ( ) , A λ a must be identical in these two conditions which are impossible to instantiate together. It is only under the condition of this assertion that we would be able to factor this term out of the two expressions. The same goes for the factorization of ( ) ; and furthermore, the second term in the factored representation would then be ( ) ( ) ( )

The Mistaken Violation of Bell's Inequality
It turns out that Bell's inequality is not deemed to be defied at every four-plex of possible experimental angle settings that we have characterised generically as At some paired directional settings of the polarizers it seems not to be defied at all. Among other pairings at which it seems to be defied, it is apparently defied more strongly at some pairings than at others. Aspect had thought that if we were to find experimental evidence of the defiance, we should try to find it at the angle pairings for which the theoretical defiance is the most extreme. It is a matter of simple calculus of extreme values to discover that the most extreme violation of the equality should occur at the angle settings ( ) = +  .) You may wish to examine our Figure 2 and notice that the angles between the various polarization directions we depicted there correspond to these relative angles. For the record, doubling these angles yields the values of 4 45 ± π = ±  and 4 135 − = − 3π  in these instances. And why does that matter?... (8)   The answer is seen most simply by constructing and then examining a matrix, which in the jargon of the operational subjective theory of probability is called "the realm matrix of possible observation values" that could result from the performance of the gedankenexperiment in CHSH form. I will display this entire matrix on the next page, in a partitioned form of its full extension as it pertains to every aspect of the problem we shall discuss. Then we shall discuss it, piece by piece. I should mention here that while the name "realm matrix of possibilities" has arisen from within the operational subjective construction of the theory of probability, the matrix itself is merely a well-defined matrix of numbers that can be understood and appreciated by any experimentalist, no matter what may be your personal views about the foundations of probability. In the jargon of quantum physics it might be called the ensemble matrix of possible observation vectors.

A Neglected Functional Dependence
In specifying the QM motivated expectation unique value for the fourth product. We now engage to substantiate this claim.

The Realm Matrix of Experimental Quantities
Consider the realm matrix of all quantities relevant to the observations that might be made in the proposed 4-ply gedankenexperiment on a pair of photons under investigation. On the left side of the realm equation is written the name ( ) R X , where X is a partitioned vector of names of every quantity that will be relevant to the outcome of the experiment and what quantum theory asserts about it. You will already recognize those appearing in the first two partitioned sections. On the right side of the realm equation appears a matrix whose columns exhaustively identify the values of these partitioned quantities that could possibly result from conducting the gedankenexperiment. We shall discuss them in turn.  1   1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1   1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1   1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1   1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1   1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1   1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1   1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3 1 1 1 1 1 3 1 1 3 The sixteen columns of four-dimensional vectors in the first partitioned block exhaustively list all the speculative 4 × 1 vectors of observation values that could possibly arise among the four experimental detections of photons at the four angles of polarizer pairings. In order to observe the detection products at the four surely have to observe each of the four multiplicands involved in their specifica- Since each of these observation values might equal only either −1 or +1, there are sixteen possibilities of the 4-dimensional result of the 4-ply experiment. There are no presumptions made about these prospective quantity values: neither whether they "exist" or not prior to the conduct of the experiment at all, nor even whether they exist in any form after Every other component quantity in the columns displayed in subsequent blocks of the realm matrix is computed via some function of these possibilities. Notice once again that the "exhaustiveness" of this list presupposes the principle of local realism, specifying for example that the value of ( ) A a identifying whether the photon passes through the polarizer at A or not, would be the same no matter whether the polarizer at which the paired photon engages station B is set at direction b or at ′ b .
To begin the completion of the realm matrix, the second block of components identifies the four designated products of the paired polarization indicators that yield the value of the quantity s as it is simply defined in Equation (6). The first row of this second block, identifying the product ( ) ( ) ±1, but it is logically independent of the product quantities appearing in the first three rows of block two. This is the quantity that Aspect/Bell think they are assessing when they freely specify the quantum expectations for all four angle settings as they do, seemingly defying Bell's inequality. We denote its name with calligraphic type to distinguish it from the actual polarization product b whose functional relation to the other three products we are now identifying. Peculiar, this singular component of the fourth partition block is not an "Alice and Bob" observation quantity, but rather an "Aspect/Bell" imagined quantity. It is logically independent of the first three "Alice and Bob" products. This is to say that whatever values these products may be, the value of  may equal +1 in the appropriate row among the first eight columns, or it may equal −1 in the corresponding column among the second eight.
However, it does not represent the photon detection product ( ) ( ) the four imagined experiments on a single photon pair.

Specifying the Functional Form via Block Four
Quantities in the fourth block of the realm matrix are designated with the names The quantities Next to notice is that the fourth row of the second matrix block, correspond- When that entry is +1 or −3, the corresponding entry of the second block is −1.
What this recognition does is to identify the functional relation of the fourth polarization product to the first three polarization products, viz., Here and throughout this note I am using indicator notation in which parentheses surrounding a mathematical statement that might be true and might be false signifies the number 1 when the interior statement is true, and signifies 0 when it is false.
Some eyeball work is required to recognize the functional relationship (11) by examining the final row of block two and of block four together. It may take even more concentration to recognize that this very same functional rule identifies each of the other three polarization products as a function of the other three as well! The four product quantities ( ) ( ) . .

A B
are related by four symmetric functional relationships, each of them being calculable via the same functional It is surely true that  equals a linear combination of four expectations of polarization products, as specified in Equation (8). Moreover, if the definition of ( ) s λ in Equation (6)  would equal 2 2 as proposed by Aspect/Bell. This involves no violation of any probabilistic inequality at all, and there is no suggestion of mysterious activity of quantum mechanics.
However, when it is proposed that the paired polarization experiments at all four considered angles pertain to the same photon pair, then each of the products is restricted to equal the specified function value of the other three that we identified explicitly for ( ) ( ) In this context, Aspect's expected quantity would be representable equivalently by any of the following equations: , , , , The symmetries imposed on this problem would yield an identical result in each case, which would surely not yield 2 2 at all. This is the mathematical error of neglect to which the title of this current exposition alludes. What might the symmetries yield?
The functional relation we have exposed in Equation (11) is not linear. If it were, then the specification of an expectation for its arguments would imply the expectation value for the function value. As it is not, the specification of expectation values for the arguments only imply bounds on any cohering expectation value for the fourth. These numerical bounds can be computed using a theorem due to Bruno de Finetti which he first presented at his famous lectures at the Institute Henri Poincaré in 1935. He named it only in his swansong text [21]. It was first characterized in the form of a linear programming problem by Bruno and Gilio [22], and has appeared in various forms in recent decades. Among them are presentations in dual form by Whittle [23] [24] using standard formalist notation and objectivist concepts. We shall review the content of de Finetti's theorem shortly, and then examine its relevance to assessing the expectation of ( ) s λ motivated by considerations of quantum mechanics. We need first to air some further brief remarks about the final block of the realm matrix.

The Remaining Block of Quantities and Their Realm Components
The first row of block five of the realm matrix merely identifies the values of ( ) s λ associated with the polarization observation possibilities enumerated in the columns of block one. Each component of this row is computed from the corresponding column of block two according to the defining Equation (1). It is evident that every entry of this row is either −2 or +2. This corresponds to the argument we have made following the factorization Equation (9) in Section 5.
The second row of this block pertains to a quantity denoted as . Its value is defined similarly to Equation (6) The third row of block five is merely an accounting device, denoting that the "sure" quantity, 1, is equal to 1 no matter what the observed results of the four imagined optic experiments of Aspect/Bell might be. Its relevance will become apparent when the need arises to apply de Finetti's fundamental theorem to quantum assertions.
It is time for a rest and an interlude. It is a mathematical interlude whose complete understanding relies only on your knowledge of some basic methods of linear algebra. If you would like a slow didactic introduction to the subject, my best suggestion is to look at Chapter 2.10 of my book [25]. You may even wish to start in Section 2.7. Another purely computational presentation appears in the article of Capotorti et al. [26], Section 4. I will make another attempt here in a brief format, merely to keep this current exposition self-contained. What does the fundamental theorem of probability say?

The Relevance of the Fundamental Theorem of Probability
In brief, the fundamental theorem says that if you can specify expectation values for any vector of quantities whatsoever, then the rules of probability provide numerical bounds on a cohering expectation for any other quantity you would like to assess. These can be computed from the compilation of a linear programming routine. If the expectations you have specified are incoherent (meaning self-contradictory) among themselves, then the linear programming problems they motivate have no solution. This theorem is immediately relevant to our situation here in which we have identified quantum-theory-motivated expectations for any three of the four detection products that determine the value of s for the gedankenexperiment. We wish to find the bounds on the cohering expectation for the fourth detection product which is restricted to equal a function value determined by these three. A discursive pedagogical introduction is available in Lad [[25], Section 2.10]. In brief, here is how the theorem works.
, and call its realm matrix then In general it will look something like the realm matrix we have just constructed for various aspects of our gedankenexperiment. It will have 1 N + rows, and some number K columns. Just as an example, the realm matrix we have already constructed happens to have ( ) 1 16 N + = rows and 16 K = columns. (Mind you, we have not yet specified expectation for the first N components of the quantity vector to which this realm applies, but let's not let that deter us. I am merely suggesting here an example of a realm matrix that could be considered to have ( ) rows. Let's continue with the general abstract specification.) Now any such vector of quantities can be expressed as the product of its realm matrix with a particular vector of events. The matrix equation, displayed in a form that partitions the final row, would look like this: On the left of this equation is the column vector of the quantity observations under consideration. To the right of the equality comes firstly the ( ) realm matrix whose K columns list all the possible columns of numbers that could possibly result as the observation vector. These K columns, each of which has . K x . (The initial subscripted dot denotes that this is a whole column of numbers. The number that follows the dot denotes which of the columns of the matrix it is we are talking about.) This matrix is multiplied by the final 1 K × column vector of events that identify whether the quantity vector will equal 0. But we do not know which of them is the 1, because we do not know which column of possibilities in the realm matrix will be the one that represents the observed outcome of the vector of quantities We can represent this matrix equation more concisely and in a useful form by writing it in an abbreviated partitioned form: The payoff from constructing this matrix structure is that now every row of this partitioned equation has on its left-hand side the unknown value of a quantity, i X . On the right-hand side in that row appears a list of the possible values of that quantity, each multiplied in a linear combination with the events that denote whether each of them is indeed the value of this quantity (in the context of the observed values of the other quantities shown in that column as well). Each row of this equation specifies how a different one of the quantities under consideration equals a linear combination of events. We have heard of that before. The expectation of a linear combination equals the same linear combination of expectations for those events, which would be their probabilities if we could specify values for them. This tells us that we can evaluate an expectation operator on this partitioned equation to yield the result that Well, we have not mentioned anything about probability specifications appearing in the vector on the right-hand side of this equality. The only restrictions of probability are that these must be non-negative numbers that sum to 1, since the vector as required of the expectations that we have presumed to be specified, and where the components of K q must be non-negative and must sum to 1.
Such a computation is provided by the procedures of a linear programming problem. The "solutions" to these linear programming problems are the vectors min q and max q that yield these minimum and maximum values for ( ) 1 N E X + subject to these constraints. The final row vector identifying ( ) 1 N E X + whose extreme values we seek is called "the objective function" of the problems. Its coefficients are the partitioned final row of the general realm matrix we identify as . Notice that that X is not bold. It represents merely the final quan- Here are the specific details appropriate to our gedankenexperiment. 1 1 1 1 1 1 1  1 1 1 1 1 1 1 1   1 1 1 1 1 1 1 1  1 1 1 1 1 1 1 1  1 1 1 1 1 1 1 I have listed the order of the quantities in the vector at left to begin with the sure quantity, 1, which equals 1 no matter what happens in the gedankenexperiment. There follow the four summands of the CHSH quantity s, of which we have noticed that each one of them is restricted in the gedankenexperiment to equal a function value of the other three. That is why there are only eight columns in their realm matrix, as opposed to sixteen columns in the expansive realm matrix we have already examined. As to the components of the vector 8 q at the right of the right-hand side, notice that quantum theory says nothing at all about these, individually. Each of them would equal the probability that the 4-ply gedankenexperiment would yield detection products designated by a specific column of the realm matrix. However, these would involve the joint detection of photon products in four distinct measurements that are known to be incompatible. On account of the generalised uncertainty principle, quantum theory eschews specification of such probabilities. Nonetheless, for any individual photon detection product in a specific experimental design, denoted on the left-hand side of the equation, quantum theory does specifies an expectation value of either 1 2 or 1 2 − , as we have recognized. Since these four products are not all free to equal +1 or −1 at the same time, we may assert expectation values for any three of them, and use linear programming computations to find the cohering bounds on the expectation of the fourth that would accompany them, yielding bounds on the expectation Equation (13).

The Result: Quantum Theory Identifies Restrictions on the Valuation of q 8
This is what we find. The columns of the matrix below display the computed results of the paired min q and max q vectors corresponding to four linear programming problems. Each of them determines a bound on an expected function value that appears in one of the four forms of the expectation value  which we displayed in Equation (12). The first pair of columns, for example, identify the fifth row of the matrix in Equation (13) as the objective function,   max  min  max  min  max  min  max   1   2   3   4   5 , , , Each of these column vectors resides in 8-dimensional space, providing a coherent assessment of probabilities for the constituent event vector without specifying precise probabilities for any of them. In fact, quantum theory denies itself the capability of identifying such probabilities precisely. We will discuss this feature further, below. However the results of the linear programming computations can and do specify possibilities for what might be specified in a way that would cohere with what quantum theory can and does tell us. The columns of this matrix identify some of them. In fact, these columns display extreme values of what are possible. Any convex (linear) combination of them would cohere with quantum theory as well. Thus, geometrically the columns constitute vertices of a polytope of quantum-theory-supported possibilities for This polytope is called "the convex hull" of these vectors. However, although we have found eight of them, the rank of the matrix of all of them is only four! That is, these eight-dimensional vectors all reside within a four-dimensional subspace of a unit-simplex. Why is quantum theory not more specific in specifying the expectation of Bell's quantity ( ) In any of these columns appear three values of ( ) ( ) tions supported by quantum theory, and a fourth value which is either a lower For each of these would amount to claims regarding the joint outcomes of in- view Equation (13) while reading the following remarks. They concern assertions that quantum theory does allow us to make, and those that it doesn't. Recall that we are considering a linear programming problem in which quantum expectations are asserted for the polarization products at the angle settings ( ) Examining the corresponding columns of the realm matrix seen in (13), it is evident that these involve assertions regarding the outcomes of ( ) ( ) ( ) irrespective of the values of ( ) ( )

Transforming the Expectation Polytope into Quantum Probabilities
The expected photon detection products displayed in Section 7.2 can be transformed into P ++ probabilities by applying the transformation of Equation (3) to the eight vertices. This yields the vertices of another polytope in the space of the probability vector .4268

And Now Viewing It !... as It Passes through Our Space
The convex hull of the 4-D column vectors shown in Section 7.4 can be visualized through a sequence of 3-D intersections it affords with slices perpendicular to any one of its axes. The symmetry of the configuration implies that slices along the other axes would create identical intersection sequences.

What to Make of Aspect's and Subsequent Empiricism
Taken in by the alluring derivation of Section 5.3 which ignores the symmetric functional relations among the polarization products of the gedankenexperiment, Aspect and followers were convinced that Bell's inequality has been defied, and that the theory of hidden variables must be rejected. This conclusion would support the assertion that quantum theory has identified the structure of randomness which supposedly inheres in Nature at its finest resolution. The behaviour of the photons is considered to be governed purely by a probability distribution. It remained only to devise some physical experiments that could verify the defiance of the inequality.
According to the tenets of objective probability theory and its statistical programme, probabilities are not observable quantities. What are observable are  outcomes of random variables which are generated by them. It is a matter of statistical theory to devise methods for estimating the unobservable probabilities and their implied expectations from carefully observed outcomes of the random variables they generate. Understood in this way, Equation (8) which I repeat here constitutes a structure requiring estimation if the violation of Bell's inequality is to be verified: Resorting to long respected statistical procedures, the unobservable expectations of detection products on the right-hand-side of this equation can be estimated by the generally applicable non-parametric method of moments. Supported by the probabilistic law of large numbers, its validity as an estimating which is common to all of these random experiments. A similar programme would be followed in estimating the other three components of ( ) E s λ     . Using the notation of Aspect [14] we would conduct N repetitions of the CHSH/Bell experiment with the relative polarizing angles set at ( ) where the component estimator with a similar specification for the components of ( ) The momentous results were published by Aspect et al. [11] [12], confirming the apparent defiance of Bell's inequality to several decimal places. Through the following three decades the experimental setup was embellished so to account for a variety of various possible loopholes tendered as an explanation.

Examining and Reassessing Aspect's Empirical Results
What are we to make of Aspect's and subsequent empirical results?
Aspect [[14], p.15], and [12] reports the estimation  from experimental data, using the method of moments as defined in Equations (14) and (15). Of course actually, it is impossible to conduct an experiment on a single pair of photons at all four angle settings, much less conduct a sequence of such experiments. Instead, experimental sequences of observations using different photon pairs were generated at each of four angle settings. These were presumed to provide independent estimates of the four expectations as they appear in Equation (15). These independent estimates were then inserted into Equation (14), yielding Aspect's touted estimate ( ) E s λ     near to 2 2 . Although experimentation protocols have subsequently been improved to account for the challenges of possible loopholes during the following thirty years, the estimation procedures using the improved data have been the same. Results

Exposition by Simulation
Because Aspect's experimental observation data is not available in full, a method F. Lad for correcting his estimation procedure shall now be displayed using simulated data based on quantum theoretic specifications, along with a presentation of its numerical implications. To begin, four columns of one million (10 6 ) pseudo random numbers, uniform on [ ] 0,1 , were generated with a MATLAB routine.
These were then transformed into simulated observations of paired photon polarization experiments at the four relative angles we have been studying. These transformations were performed using the QM probabilities based on calculations of ( )  ′  a b a b a b a b . We shall refer to this matrix of simulated polarization products below as the SIMPROD matrix.
Aspect's estimation Equation (15) was applied to each of these columns, yielding estimates of the expected polarization product pertinent to that column, . These appear in the first row of Table 1. These four estimates were then inserted into Equation (14)  , as was Aspect's reported empirical estimate, proposed as an evidential violation of Bell's inequality. As we now know, the problem is that when the product observations are supposed to apply to the same photon pair, the observed value of the polarization product at any angle is required to be related to the product at the other three angles via the functional equation we specified in our Equation (11). The four of them may not all range freely in a gedankenexperiment, as they may in real experiments on different pairs of photons. Rather, they are required to be bound by the symmetric functional relation ( )

, , G
to each choice of three components of the rows of the SIMPROD matrix. Each result was entered into the same row of a companion matrix of the ". These display estimates of ( ) . . .
, , E G     required for estimation of the four alternative expectation equations (12). In this way we can be considered to have generated 4 times 10 6 simulated versions of the Aspect/Bell gedankenexperiment. Their component results can be taken to be any three simulation results from a row of SIMPROD along with the fourth result being the functionally generated result found in the same row and the appropriate fourth column of SIMGEN. Finally, the last row of Table 1 presents the estimated values of ( )  deriving from these simulated experiments. They appear as "corrected estimates", column by column, for each of which the ( ) is the one appropriate to that column while the other three expected polarization products are those appropriate to the other three columns of row 1 of the Table. The elements of this row display corrected estimates of ( )  as they should be calculated with the simulated Aspect data. Each of these four estimates is slightly different from the others. Averaging them over the four ways of generating a column of polarization products from the other three columns of simulated products would yield a "Corrected estimate" of ( ) Based on Aspect's report of his experimental data, I feel quite sure that applying this same estimation procedure to his experimental data, considered as a simulation of the impossible gedankenexperiment, would yield a similar result.

A Comment on Empirical Work and Statistical Estimation
While Aspect's conception of statistical estimates appropriate to the photon de-F. Lad tection problem is understandable, and corrections can be made to improve its relevance to the Aspect/Bell problem, developments of statistical theory and practice during the past fifty years have surely generated superior methods for evaluating the physical theory of quantum behavior. These rely on the subjective theory of probability which, under the leadership of Bruno de Finetti and researchers adhering to his viewpoint, has gained substantial credibility from the past half-century of research in the foundations of probability and statistics. There are even some prominent physicists among its proponents, though not many. Proclaimers of inherent randomness in the physics of quantum behaviour have won the day for now, largely on the basis of the mistaken violation of Bell's inequality that we have debunked in this article. In the very least, it is apparent that calls for open access to raw data [28] from several well-known research programs that publish summary results, usually in the form of discredited p-values, need to be heeded.

Concluding Comments
The As to the characterization of the theory of hidden variables, this is another endeavour that has been misconstrued in accepted literature, largely on the basis of the mistaken understanding of the defiance of Bell's inequality which we have corrected here. I have examined this matter in a separate manuscript entitled "Resurrection of the principle of local realism and the prospects for supplementary variables." Along with a manuscript on my reassessment of Mermin's "quantum mysteries" [29], it is currently available only on my ResearchGate Discussions of related issues proceeding henceforth will need to begin with this new recognition. Interestingly, this resolution was suspected in some way by Bell himself, though not the analytical detail. This was clearly evident in his musings on the hidden variables question in Bell [10] which he himself had reprinted in a collection of his publications, Bell [30]. My discovery of the functional relations involved among the components of the 4-ply gedanken quantity S and the 4-D polytope of their cohering quantum theoretic distributions is truly novel.
A final reference relevant to this analysis is the article of Romano Scozzafava [31] on the role of probability in statistical physics. He discusses several issues that clarify fundamental matters in the context of the constructive mathematics of Bruno de Finetti's operational subjective statistical method.
programming the 3-D slices of the QM-motivated coherent prevision polytope using MATLAB, and for discussions concerning the linear algebraic structure of the problem. The University of Canterbury provided computing and research facilities. Thanks to Paul Brouwers, Steve Gourdie, and Allen Witt for IT service, and to Giuseppe Sanfilippo for format consultation.

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.

Introduction
Most physicists are familiar with the Schrödinger equation, which describes the movement of a spin-0 particle with mass (m) moving at speeds much smaller than the speed of light (c) [ [7]. The one-dimensional Klein-Gordon equation corresponding to a free particle is given by the following expression [6] [7]: In Equation (2), KG ψ is also a scalar wavefunction. Equation (2) In Equation (3) is not a scalar wavefunction but the two-component spinor wavefunction: . , Journal of Modern Physics Clearly, a price in mathematical complexity is paid for improving the relativistic description of quantum particles. Consequently, from a purely pedagogical point of view, it would be convenient to be able to have a Schrödinger-like equation capable to describe quantum-particles at relativistic energies. Unfortunately, this is not in general possible [6] [7]. Nevertheless, it was recently found a Schrödinger-like equation capable to describe quantum-particles at quasi-relativistic energies [11] [13] [14] [15] [16].
Rigorously, the number of particles may not be constant in a fully relativistic quantum theory [6] [7]. This is because when the sum of the kinetic and the potential (U) energy of a particle with mass m doubles the energy associate to the mass of the particle, i.e., 2 2 E K U mc ′ = + = , then a pair particle-antiparticle could be created from E′ [2] [6] [7]. Consequently, the number of particles is constant at quasi-relativistic energies, i.e., when . At quasi-relativistic energies close to 2 mc , the Schrödinger equation does not provide a good description of the states of the quantum particle because it assumes that 2 2 K p m = , while at relativistic speeds the correct relation between K, p, and the square of the velocity of the particle ( 2 v ) is given by the following equation A free spin-0 particle can be (approximately) described by the following qua- Clearly, Equation (10)  This includes a free particle [11], confinement of a quantum particle in box [11] [14] [15], reflection by a sharp quantum potential [15], tunnel effect [15], and the quasi-relativistic description of Hydrogen-like atoms [14] [15] [16]. Therefore Equation (10) (10) is a Schrödinger-like equation, it permits to calculate probabilities like it is done for Equation (1) [11].
Moreover, Equation (10) allows for a quasi-relativistic description of multi-particle systems where the number of particles is constant [17]. This includes all problems in Chemistry where the number of electrons is constant and The energy of the most energetic electrons in heavy elements is quasi-relativistic. Therefore, often their description either involves a perturbative theory based on the Schrödinger equation [2] [4] [5], or a more precise but much more complicate quantum electrodynamic description [18]. The quasi-relativistic wave equation potentially represents a novel non-perturbative approach for tackling such problems without having to pay a heavy price in mathematical complexity, thus helping to grasp the essence of the consequences of introducing the ideas and concepts of spatial theory of relativity in quantum mechanics.
In this work, first, for completeness, the connection between Equation (10) and the Klein-Gordon equation will be summarized. Then, for the first time, a quasi-relativistic version of Equation (3) will be directly obtained from the Dirac equation. Finally, also for the first time, an equation giving the quasi-relativistic energies of the bound states of the electron in Hydrogen-like atoms will be obtained using a perturbative approach based on the quasi-relativistic wave equation. The quasi-relativistic energies calculated in this way have a much better correspondence, with the energies calculated using the Dirac equation, than the energies calculated using a perturbative theory based on the Schrödinger equation.

Relationship between the Klein-Gordon and the Quasi-Relativistic Wave Equations
From the following well-known relativistic equations [8] [9] [15]: And: One can formally obtain Equation (2) by substituting E and p in Equation (11) by the following energy and momentum quantum operators [ The factor ( ) 2 E mc + in Equation (11) is always different than zero for 0 E > ; consequently, Equation (11)  Then from Equations (13) and (14) follow the following differential equation [13] [14] [15]: A simple substitution in Equations (2) and (15) shows that the following plane wave is a solution of both equations for 0 E > : Moreover, the following wavefunction is a solution of Equation (10): Therefore, Equation (17) allows finding a solution of Equation (2) with 0 E > from a solution of Equation (10). This is the relationship between the freeparticle Klein-Gordon and quasi-relativistic wave equations. This relationship is also valid when the particle is moving through a potential U [11] [13] [14] [15] [19]. For instance, the quasi-relativistic wave equation for a particle moving at quasi-relativistic energies through piecewise constant potentials is given by the following equation [15]: Looking for a solution of Equation (18) of the form: It is obtained the time-independent quasi-relativistic wave equation [15]: At low velocities, when ~1 Equations (12) and (14) follows that [15] [17]: [3] [4]. It is worth noting that Equations (10) and (18) (20)), respectively corresponding to different kinetic energies K 1 and K 2 , then the following wavefunction is a solution of the Klein-Gordon equation:

The Quasi-Relativistic Wave Equation of a Free Electron
The wavefunction in Equations (1) Substituting Equation (23) in Equation (6), and considering that for a free electron 2 E K mc = + , allows for rewriting Equation (6) as the following system of two time-independent spinor equations [2]: In Equation (24) Therefore, when 0 E > , each one of the two components of ϕ exactly satisfies the same time-independent quasi-relativistic wave equation, which corresponds to a free spin-0 particle with kinetic energy K . Consequently, when 0 E > , the three-dimensional version of Equation (10)

The Pauli-Like Quasi-Relativistic Wave Equation
The Schrödinger-like Pauli equation given by Equation (3) can be obtained from the Dirac equation for an electron interacting with an external electromagnetic field [2]. Following the same procedure, a quasi-relativistic version of Equation (3) can be obtained. When an external electromagnetic field interact with the electron, Equation (24) should be modified in the following way [2]: In Equation (27), The Substituting Equation (29) in the first equation of Equation (27) For a free electron moving through a constant magnetic field, with magnitude B ext pointing in the z direction, Equation (30) can be approximated as: Which is the time-independent Pauli-equation corresponding to Equation (3).
However, if one assumed that and Equation (28) can be approximated by the following expression: Substituting Equation (32) For a free electron moving through a constant magnetic field, with magnitude ext B pointing in the z direction, Equation (33) can be approximated as: (34) Equation (34) is the quasi-relativistic version of Equation (31). When the electron moves slowly, 1 v γ  , thus Equation (34) coincides with Equation (31).
Equation (34) includes two corrections to Equation (3). First, includes the correct relativistic relation between K and p. Second, as shown in Figure 1, the energy difference corresponding to the two components of ϕ is not independent of K, as suggested by Equation (31), but decreases by a factor of twice

Relativistic Corrections to the Energies of the Bounded States in Hydrogen-Like Atoms
For Hydrogen-like atoms, we can assume the vector potential in Equation (27)   atom with a nucleus of mass n m , and e m is the electron mass [2]. Often the following approximation to Equation (36), which is valid when obtained using a perturbative approach based in the Schrödinger equation [2] [12]: In Equation (37) is the relativistic correction to the kinetic energy, which is given by is the so-called the Darwin correction, which is only not null when In some cases, for heavy Hydrogen-like atoms with 1 Z  , the term inside the square root in Equation (45) could be negative; in these cases, the approximation to the square root included in Equation (45) should be used. As should be expected, [15]. It is worth noting that QR E is identical to the positive energies calculated for the Hydrogen atom using the Klein-Gordon equation [19].
is the new Darwin correction, which also is only not null when 0 l = : is the new spin-orbit correction, which also is only not null when 0 l ≠ :  Table 1.
More importantly, Equation (42) provides a better approximation than Equation (37) to the values of L E ∆ calculated using Equation (36). This is confirmed by the plots shown in Figure 2 showing the dependence on Z of Clearly, as expected, at quasi-relativistic energies ( 1 Z  ), Equation (42) (dashed blue curve) provides a much better approximation than Equation (37) (dot-dashed black curve) to the values of L E ∆ calculated using the exact Dirac's energies (continuous red curve).

Conclusion
It was shown that the time dependent Equations (1) and (18) Equations (37) and (42) can both be obtained from Equation (27) with a null vectorial potential (A) and o eA given by Equation (35). For obtaining Equation (37), Equation (28) should be approximated in the following way [2]: Then, substituting Equation (A1) in the first equation of Equation (27) results [2]: Or: The  [5]; therefore, the relativistic corrections to the energies calculated using the Schrödinger equation are contained in this term [2]. However, if Equation (28) is approximated in the following way: Then, substituting Equation (A4) in the first equation of Equation (27) results: Or: The time-independent quasi-relativistic wave equation for Hydrogen-like atoms is equal to Equation (A6) after excluding the term between curls in the left size of Equation (A6) [12]; therefore, the relativistic corrections to the energies calculated using the quasi-relativistic wave equation are contained in this term.
In Equation (A3), the term between curls produces three relativistic corrections to the energy, which are given by Equations (39) to (41) [2]. It can be shown, following the same procedure [2], but using the wavefunctions satisfying the quasi-relativistic wave equation for Hydrogen-like atoms [14] [16], that the term between curls in Equation (A6) produces two relativistic corrections to the energy, which are given by Equations (46) and (47).  Equation (2) should be changed, but any other equations can be unchanged.
tiparallel spins, i.e., spin magnetic quantum numbers m s of 1/2 and −1/2. An orbital filled by an electron couple has s = 0 and bosonic character. The multiplicity of a reactant is defined as |2(S)| + 1 where S is the total spin quantum number. The Wigner spin conservation rules state that multiplicity is conserved. The transmission coefficient κ of absolute reaction rate theory also indicates the necessity for spin conservation. Burning is fermionic combustion that occurs when sufficient energy is applied to a bosonic molecule to cause homolytic bond cleavage yielding fermionic products capable of reaction with the bifermionic frontier orbitals of triplet multiplicity O 2 . Neutrophil leucocytes kill microorganisms by bosonic combustion and employ two mechanisms for changing the multiplicity of O 2 from triplet to singlet. Microorganisms, composed of bosonic singlet multiplicity molecules, do not directly react with bifermionic O 2 , but are highly susceptible to electrophilic attack by bosonic electronically excited singlet molecular oxygen ( 1 2 O * ). Hydride ion (H − ) transfer is the common mode of cytoplasmic redox metabolism. Bosonic transfer of an orbital electron couple protects from damage by obviating fermionic reaction with bifermionic O 2 . Bosonic coupled electron transfer raises the consideration that quantum tunneling might be involved in facilitating such redox transfer.

Introduction and Background
A wavefunction (ψ) defines a quantum system. An orbital is described by , , l n l m ψ where n is the principle quantum number, l is the azimuthal or angular momentum quantum number, and m l is the magnetic quantum number. An electron occupying an orbital is described by the wave function

Fermionic and Bosonic Orbitals
Fermions can combine to yield a wavefunction with bosonic character. An alpha particle made up of four fermions is bosonic [3]. An electron is a fermion. As such, an orbital filled by a single electron has an s = 1/2 and fermionic character Reaction chemistry can be approached from a fermionic-bosonic orbital perspective.
Consistent with the fermion nature of electrons, Pauli's exclusion principle limits an orbital to a maximum of two antiparallel electrons, i.e., m s of 1/2 (↑) and −1/2 (↓). In Figure 1, note that the lower energy 1s and 2s orbitals of atomic N each contain two antiparallel electrons, i.e., an orbital couple with s = 0. These orbitals are closed to reaction chemistry. The frontier orbitals of atomic N include the three 2p orbitals. These 2p orbitals are degenerate, i.e., each orbital has the same energy. Each 2p orbital contains a single fermionic electron. Hund's maximum multiplicity rule states that the electrons in degenerate singly occupied orbitals will have parallel spins [7]. As such, each of the three 2p frontier orbitals of N have an s = 1/2 and the S of N is 3(1/2). The spin multiplicity, i.e., |2(S)| + 1, for N is thus |2(3/2)| + 1= 4. Stated differently, atomic nitrogen is a triradical with quartet spin multiplicity. Each 2P orbital of N is a SOAO, and as such, atomic N is trifermionic. As depicted in Figure 1 and stated in Table 1, the product of reacting two quartet multiplicity N atoms is singlet multiplicity N 2 . The lower energy 1s and 2s orbitals of N all contain coupled antiparallel electrons with s = 0. These non-frontier bosonic orbitals do not participate in reaction. Likewise, the sigma bonding (σ) and antibonding (σ * ) orbitals of N 2 , derived from the 1s and 2s orbitals of the atomic N's, are bosonic and closed to reaction chemistry. The frontier π bonding orbitals of N 2 are both filled by an electron couple with s = 0 and have bosonic character. In its ground state, N 2 is singlet multiplicity, triple bonded and bosonic.

Transmission Coefficient of Absolute Reaction Rate Theory
Absolute reaction rate theory states that the rate of a chemical reaction requires that reactants first combine to form an activated complex, where k is the rate, κ is the transmission coefficient, kT/h has the dimensions of frequency and K * is the equilibrium constant for the activated complex. The transmission coefficient, κ, for typical reactions approximates unity, i.e., each activated complex yields product, but not every activated complex at the potential-energy barrier will cross over to product [8]. The value of κ decreases by several orders of magnitude in reactions involving change in spin state [8].

Wigner Spin Conservation from a Fermionic-Bosonic Perspective
The Wigner spin conservation rules state that a reacting system resists any change in spin angular momentum, i.e., multiplicity [9] [10]. The total spin number, S, of an atom or molecule defines its multiplicity; i.e., |2S| + 1 = multiplicity. When S = 0, the multiplicity is singlet, when S = 1/2, the multiplicity is doublet, when S = 1/2 + 1/2, the multiplicity is triplet, et cetera. Reactions involving change in multiplicity have transmission coefficient, κ, values of less than 10 −4 . The spin states or multiplicities of the reactants determine the spin state or multiplicity of the activated complex, and are conserved in the spin states or multiplicities of the resulting product or products. For example, if the impossibility of orbital overlap is ignored and reaction is assumed to involve a bosonic singlet multiplicity molecule and a bifermionic triplet multiplicity molecule, then the activated complex must have a bifermionic triplet multiplicity, and bifermionic triplet multiplicity must be conserved in the product or products. These and other possibilities are described in Table 1. Journal of Modern Physics

Combustion
Combustion, defined as an act or instance of burning, requires fuel and molecular oxygen, and produces heat and light. The organic molecules that serve as fuel are of singlet multiplicity and present bosonic frontier orbitals that are unreactive with the bifermionic frontier orbitals of 3 O 2 . Consistent with absolute reaction rate theory and the spin conservation rules, such reactions are not spontaneous.

Fermionic Combustion
To initiate burning, a sufficient amount of energy, e.g., a flame, must be applied to cause homolytic bond cleavage of the singlet multiplicity fuel molecule. Each homolytic cleavage yields two doublet multiplicity SOMO products. These fermionic products can directly react with the bifermionic frontier orbitals of 3 O 2 .

Bosonic Combustion
The neutrophil leukocyte, a phagocytic white blood cell, is tasked with defending the host animal against a vast variety of pathogenic microorganisms [17]. Fifty years ago, I pondered the possibility that phagocytic leukocytes kill microbes by changing the multiplicity of molecular oxygen from triplet to singlet [18].

Neutrophil Combustive Microbicidal Metabolism
Neutrophil reduced nicotinamide adenine dinucleotide phosphate (NADPH) oxidase controls HMP metabolism by accepting two reducing equivalents from NADPH thus liberating the oxidized NADP + that is required for glucose-6-phosphate (G-6-P) dehydrogenase metabolism of glucose. Biochemical dehydrogenations involve hydride (H − ) transfer. The bosonic character of such redox exchange will be considered subsequently. The riboflavin prosthetic group of NADPH oxidase facilitates decoupling of the bosonic electron pair. Riboflavin mediated separation allows fermionic expression of the separated electrons and results in reactive electron capture by bifermionic 3 O 2 [14]. The product of such univalent reduction is the doublet multiplicity hydroperoxyl radical ( 2 HO 2 ). O − is a radical-radical annihilation yielding bosonic singlet multiplicity hydrogen peroxide ( 1 H 2 O 2 ) and bosonic electronically excited singlet multiplicity molecular oxygen ( 1 2 O * ) [19]. As described in Table 1, reactions of fermions yield bosonic products.
Neutrophils contain abundant myeloperoxidase (MPO). The haloperoxidase action of MPO provides an additional mechanism for generation of bosonic  [20]. MPO can catalyze classical peroxidase activity involving radials, but such activity is distinct from the acid haloperoxidase action involved in microbicidal action [17].
Generation of 1 2 O * violates Hund's maximum multiplicity rule; i.e., the electronic configuration with highest multiplicity has the lowest energy. The greater the number of wave functions possible for a system, the lower the energy. Higher multiplicity states produce greater nuclear-electron attraction and are of lower energy [11]. As such, 1 2 O * is metastable with a lifetime of about a microsecond. This lifetime restricts its potent electrophilic reactivity to within a radius of about 0.2 microns (µm) [12]. Upon phagocytosis, the microbe becomes the locus of neutrophil microbe killing. Generation of the bosonic reactant 1 2 O * within the phagolysosome space of the neutrophil directly focuses its potent electrophilic reactivity to the target microbe and minimizes collateral damage. Purified MPO selectively binds all gram-negative bacteria tested and can bind and inactivate endotoxin even in the absence of haloperoxidase function [21]. Selective MPO binding to microbes correlates with selective MPO-mediated microbicidal action. Bosonic combustion is limited by the lifetime of 1 2 O * . Such reactive restrictions have the advantage of selectively focusing and confining combustive action to the microbe while avoiding bystander injury to host cells [22].

Bosonic Transfer of Reducing Equivalents
Cytoplasmic redox transfers, i.e., pre-cytochrome electron transfers, typically involve the movement of two reducing equivalents from one singlet multiplicity molecule to another, and is described as H − transfer. Such hydride transfer involves the movement of a proton plus an orbital couple of antiparallel electrons.
The orbital couple has a s = 0, and as such, transfer is singlet multiplicity and bosonic.
Biological systems are exposed to an atmosphere with abundant O 2 . The bosonic character of biochemical systems provides protection against direct reaction with bifermionic O 2 . As previously considered, any biologic transfer involving a single fermionic electron would open the possibility for direct fermionic reaction with O 2 . The resulting fermionic-bifermionic reaction would produce a fermionic product and the possibility for further fermionic-bifermionic propagation.
Redox transfer of a bosonic orbital electron couple might offer additional advantage. The bosonic nature of the alpha particle facilitates quantum tunneling from the nucleus [23]. The bosonic nature of a Cooper pair of electrons facilitates superconductivity [24]. Alpha particle radiation and Cooper pairing in superconductivity are very different from each other, and both phenomena are very different from biochemical redox electron transfer. However, the commo-Journal of Modern Physics nality of bosonic pairing in quantum tunneling raises suppositions with regard to a possible role in facilitating biological redox transfer.

Summary and Conclusion
Reaction chemistry involves frontier orbital interactions. An orbital is fermionic if occupied by a single electron, and bosonic if occupied by an electron pair.
With regard to orbital reactivity, bosonic orbitals react with bosonic orbitals generating bosonic products, fermionic orbitals react with fermionic orbitals generating bosonic products, and fermionic orbitals react with bifermionic molecules generating less fermionic products. Fermionic-bosonic reactions are improbable, but the products of any such reaction must conserve the fermionic character of the reaction complex. As a general observation, all reactions favor bosonic products. Burning or fermionic combustion is initiated by homolytic bond cleavage producing fermionic products that react with bifermionic triplet The Law of Universal Gravitation represents a milestone of scientific knowledge to the main thread of this study is represented by the calculation of the Gravitational Constant G, starting from the interaction between electron and positron. The calculation of G was carried out on an atomic scale, instead of considering the large masses predicted by Newton's equation. We wonder if this constant can represent a common denominator between the behavior of particles, the structure of matter and the dynamics of celestial masses and if, at theoretical level, the constant G can be considered a "mediator" between General Relativity, the forces of the Standard Model of particles and String Theory. The Law of Universal Gravitation represents a milestone of scientific knowledge to interpret celestial mechanics and a cornerstone of predictive science. In Newton's formulation and in the field equation of general relativity appears the proportionality coefficient G, independent of the physical location and masses used to determine it experimentally. In the last two and a half centuries, there have been experiments for the approximate calculation of G, from the eighteenth century to the present with increasingly refined methods and instruments [1] [2]. Recently, a team of quantum physicists from the University of Vienna and the Austrian Academy of Sciences realized for the first time in the laboratory a miniature version of the Cavendish experiment using millimeter-order masses. A result that opens new perspectives for the possible connection between gravitational and quantum physics [3] [4] [5] [6]. In this study, contrary to the techniques used in the past, such as the torsion balance or based on the principle of the pendulum, is proposed the calculation of the constant of gravitation G starting from the charge and the electron mass. The calculation, expressed in theoretical form, was conducted by the Belgian physicist Fernand Léon Van Rutten, and presented posthumously, having disappeared in 2016. This is a written memoir that Belgian Physicist left to his daughter as a scientific testament. The calculation of the constant of G, in the Van Rutten point of view, originates from the "bricks" of the matter rather than start her big masses of the celestial bodies, according to the Newtonian concept. A universal constant being of G, his estimated value, leaving from the electron positron interaction with respect to the big masses, therefore represents a point of connection among the concept of "micro" and "macro" cosmos, and nominates himself as unifying element be-Journal of Modern Physics tween the fine matter and the gravity. The costing of gravitational or constant universal gravitation strength of G, is in fact the same one for all the bodies equipped with mass, be they as big as the stars or as little as sand grains. And, in the universe everything reduces himself to particles. Studying the world at smaller scales, on the order of the Angstrom, provides an opportunity to understand what we observe at larger scales. Of course, for each of these scales the behavior of matter is different.
The idea of a "hierarchy of universes" is not new. It, in fact, was already alive in Democritus of Abdera, understood as "scale factors", while new concepts were taken up in 1761 by J.H. Lambert and gradually developed until today, through H. Alfvén, O. Klein, D.D. Ivanenko and others [7]. Over the centuries, the need of physicists to find a formula or mechanism that brings together the four forces that interact on matter, gravity, electromagnetism, strong interaction and weak interaction, has been a common thread and an ambitious goal in the world of Physics. Recently, the study of gravity has been extended to include antimatter [8]. A holistic approach that associates physical structures, apparently different as gravity and electromagnetism, had been studied in the beginning by Michael Faraday (1849-1950) and then resumed, after about half a century by Weyl (1918) [9] and from the '20s by Albert Einstein with the "Unified Field Theory". But, after the innovations of the late 1800s and early 1900s, the search for a universal theory that encompassed the four forces that interact on matter became an insistent goal in the scientific world and among Physicists. The goal was, and is, to conceive a new theory, the "Theory of Everything", initially coined by J. Ellis (1986) [10] and pursued by Stephen Hawking [11]. Among the best-known empirical observations, the relationship between the gravitational universe and the universe of elementary particles stands out, the result of which concurs to hypothesize the existence of a similarity in a geometric and physical sense between macro universes and strong micro universes [12]. The adjective "strong" must be however contextualized in the scale physics, where the strong nuclear force that helps to keep together the matter is far superior to the other three fundamental forces: gravity, electromagnetism and weak nuclear forces. The calculation of G, leaving from the electron-positron analysis proposed in this study, reaches the surprising result to combine the electromagnetism with the gravity, Fine Structure with the String Theory, through a deterministic physical principle and not a mathematical formulation with the limit already highlighted by the Gödel's Theorem of Incompleteness [13]. A question, that of unifying the fundamental interactions of physics, which does not cease to arouse interest in research, also discussed in recent publications [14] [15] [16].

Constants and Variables
Etymologically and conceptually, the term "constant" ensures that some quantities remain so over time. However, in spite of their current use, the origin of to now. In this study, the value of the Constant G has been deduced using other physical constants, such as the speed of light, mass, the fine structure constant [17] [18], Plank's constant, the inverse of the fine structure constant multiplied by 0.75, and fundamental quantities such as the electron charge. Unlike the constants used for the calculation of this work, as known validated by observation and experiment, the Gravitational Constant G derived from the interaction between electron-positron, is instead theoretical in nature. Its value, however, was found to be very close to both G of Newton's gravity, and to that recently calculated in the laboratories.
The variables in the calculation, however, are represented by the gyromagnetic orientation of the dipole electron-positron, which exerts pressure on matter when it intercepts it, and the distance electron-positron, which in this case must be much greater than the wavelength. From the outcome of this study, the value of the constant G can be considered for the different scales of mass magnitude, from the subatomic to the cosmic universe and vice versa and, in perspective, contribute to a better definition of the value of the constant G in the International System.

Method
The Method used to calculate the G constant is based on the concept of the "circumstantial paradigm" [19] associated with the deductive method, i.e., guiding the logical procedure from hypothesis to conclusion. In this case, the "clue" coincides with the hypothesis that there exists a particle responsible for gravity, produced due to the interaction between electron-positron and their formation and destruction processes over time.
The procedure to realize the calculation of the constant G involves the use of other physical constants and is divided into two phases. The first considers an electromagnetic interaction in the electron-positron pairs; the second phase concatenates, through the gyromagnetic ratio of dipole, electromagnetic interactions with Newtonian dynamics, from which it is possible to obtain the value of gravitational constant G.

Discussion
In this study, we will try to show that the gravitational constant of G could be the result of a relationship between other physics constants. To explain the Universal Attraction Law, it was often assumed that a particle called graviton exists [20] [21] [22] [23]. In this study, we will show that a particle responsible for the gravity exists indeed. The space is not completely empty, it contains neutrinos, electromagnetic waves and fields, like the electromagnetic field and the gravitational field. The importing thing more, according to the cosmologists, is that the space contains most of the energy of the cosmos [24]. In this study we will show, as he says A.V. Rykov [25], that this energy could be formed by virtual couples of the V. Straser E, has the dimensions of an electric field. This field may be due to the virtual electrons and positrons surrounding the mass M, and we will not make any other assumptions about this field.
Following the reaction action principle, for two particles at a distance r, on the invisible faces of the other particles, the force that tends to bring them together will be in Equation (12) On the other faces, that is, those of the other mass, γ 0 of the dipole coming from this other mass undergoes an average orientation perpendicular to the surface of this other mass.
Their orientation is no longer anisotropic and because, in relation to their size, these masses are very distant from each other. In first approximation, γ 0 is affected by the other particle, it can simply be multiplied by cosϑ.
So, we'll have for the force that tries to repel these particles, described in Equation (13): ( ) − (14) in Equation (15) as 1 e α −  we can neglect it to the denominator and doing f 2 -f 3 , we get the force both between the two masses, and for the gravitational constant of G: If we take e 2 /m 2 = 2.7801987 E32 and α = 102.7770278 we obtain the gravitational constant of G = 6.678532 × 10 −11 m 3 ·kg −1 ·s −2 .
This gravitational constant of G is very close value the experimental values obtained in these last decades. In a conference about the value of G in the 1998,

Conclusions
We conclude that the calculation of G is revealed to be compatible with other experimental measures of the gravitational constant of G, obtained by other authors with theoretical and experimental methods.
We advanced various hypotheses with this work: • The Strings are virtual electrons positrons dipoles.
• These dipoles pressing the matter.
• The validity to add the three quarters of the constant of the Fine Structure in the projection calculation of ƴ on axis.
The hypothesis formulated in this study has our permission to make a first fine approach to calculate the constant of gravitation G to leave only from the charge and the mass of the electron and from the Fine Structure constant. The calculation of G, made leaving from the mass and from the electron charge, avoids making instrumental mistakes for his determination. This study clearly shows the relationship between gravity and the electromagnetism and the Fine Structure, besides to offer, at hypothesis level, also a reflection on the antimatter and add the new pieces to the complex mosaic of "Theory of Everything".