*Article* **Linear Superposition as a Core Theorem of Quantum Empiricism**

**Yurii V. Brezhnev**

> Department of Quantum Field Theory, Tomsk State University, 634050 Tomsk, Russia; brezhnev@phys.tsu.ru

**Abstract:** Clarifying the nature of the quantum state |**Ψ** is at the root of the problems with insight into counter-intuitive quantum postulates. We provide a direct—and math-axiom free—empirical derivation of this object as an element of a vector space. Establishing the linearity of this structure—quantum superposition—is based on a set-theoretic creation of ensemble formations and invokes the following three principia: (I) quantum statics, (II) doctrine of the number in the physical theory, and (III) mathematization of matching the two observations with each other (quantum covariance). All of the constructs rest upon a formalization of the minimal experimental entity—the registered micro-event, detector click. This is sufficient for producing the C -numbers, axioms of linear vector space (superposition principle), statistical mixtures of states, eigenstates and their spectra, and non-commutativity of observables. No use is required of the spatio-temporal concepts. As a result, the foundations of theory are liberated to a significant extent from the issues associated with physical interpretations, philosophical exegeses, and mathematical reconstruction of the entire quantum edifice.

**Keywords:** quantum foundations; non-axiomaticity; detector clicks; ensembles; superposition principle; arithmetic; numbers; vector space; abstracting; interpretations; self-referentiality

#### **1. Introduction and Summary**

. . . somewhat curious that, even after nearly a full century, physicists still do not quite agree on what the theory tells us . . .—G. 't Hooft ([1], p. 5)

It is almost a crying shame that we are nowhere close to that with quantum mechanics, given that it is over 70 years old now—C. Fuchs ([2], p. 32)

The contradiction between the fundamental nature of quantum theory (QT) and the phenomenological feature of its mathematics [3] is likely to never cease instigating the attempts to overcome it. As H. Putnam had said, "Human curiosity will not rest until . . . questions [of the nature of the QT-formalism] are answered".

The subject-matter and leitmotiv of what follows is that the linear superposition and theory's axioms have an origin—they are derivable, and it is entirely empirical. The theory is thereby demystified, and the interpretative challenges that accompany the exegeses of QT are a *nonexistent* problem coming from "a confusion of categories" ([4], p. 89), i.e., from the "semantic confusion" ([5], p. 10). A direct outgrowth of this ideology is not only a derivation of the superposition principle (page 35) but also the axiom-free production of the "chief" quantum formula—the Born rule *p* = |a|<sup>2</sup> [6].

#### *1.1. On the Foundations of Quantum Theory*

The debates concerning the foundations of quantum mechanics (QM) hitherto "show no sign of abating" ([7], p. 222, [8,9]), and despite widespread scepticism [10–15], it is generally acknowledged that the problem is a real one [9,16–19]; it is not something made up or "just a dispute over words" ([20], p. 5) and sometimes "has been regarded as a very serious one" ([15], p. 418, [21,22]). Say, R. Penrose has expressed (2004) an even more radical "conviction that present-day quantum mechanics has no credible ontology, so that it *must* be seriously modified".

**Citation:** Brezhnev, Y.V. Linear Superposition as a Core Theorem of Quantum Empiricism. *Universe* **2022**, *8*, 217. https://doi.org/10.3390/ universe8040217

Academic Editors: Steven Duplij and Michael L. Walker

Received: 8 February 2022 Accepted: 21 March 2022 Published: 28 March 2022

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**Copyright:** © 2022 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

In recent decades, the discussions have even worsened [2,23], and current research has intensified due to the tremendously increased and formerly inconceivable technological possibilities of operating with individual micro-objects and the urge to implement the idea of quantum computing [20,24].

The reason for this state of affairs remains the same as it was before. Unlike the classical theories, e.g., thermodynamics or relativity theory, "Ma di assiomatizzazioni della teoria quantistica ce ne sono moltissime" ([17], p. 30) and the QM-axiomatics itself seems wholly divorced from human language [5,8,9,15,17,25–32]. Quantum postulates are not merely formal. They are phrased in terms of linear operators on a complex Hilbert space H [4,10,13,25,33–37] and, with that, literally not a single word here can be brought into conjunction with reality by means that have *at least some kind* of relationship with the classical description. What is more, it is very well known that the abstract character of these terms is required by the essence of the point (covariance) and, at the same time, that the attempt to link them with physical images is imposed by a decree and results in the famous paradoxes associated with concepts, such as causality, (non)locality, and realism [9,27,28,38–45]. All of that causes a problem with interpretations of QM.

It is well known that the theory has steadfastly resisted any unique ontological reading and, in particular, reconciliation between interpretations. This is reflected not only in the voluminousness of the literature. The differences in viewpoint are often based on points of principle [3,8,14,15,46–51], and even highly qualified publications face criticism [52–55]. Among other things, we encounter appeals [3,12,16,17,43,56–58] (there is even a manifesto(s) [50], (p. 990), [59], ), striking titles such as "scandal of quantum mechanics" [60,61], "QUANTUM OUTCOME: ALLAH WILLED IT?" ([62], p. 188; Wheeler), "the Oxford Questions . . . to two clouds" ([63], p. 6), "The Canon for Most of the Quantum Churches" ([50], p. 988), "Quantum mechanics for the Soviet naval officers" ([64], p. 161), "the patron saint of heretics in the One True Church of Copenhagen" (about D. Bohm), "A Feminist Approach to Teaching Quantum Physics" ([2], p. 182), "Church of the Larger Hilbert Space" (J. Smolin) [2,12], and also April Fools' [65] and the medical jokes about "the "state of health of the quantum patient'" ([66], p. vii), political parallels with "Marxism . . . [and] the Cold War" [67], and many more [3,9,27,68–71].

An interesting fact is that Cambridge University Press has published a 500-page-long book [2] containing an arresting electronic correspondence—D. Mermin called it "samizdat" (self-published)—between C. Fuchs and modern researchers and philosophers in the field of quantum foundations. This correspondence has continued ([23], over 2300 pages) and now covers 1995–2011. It characterizes the state of affairs in the field, and does not merely add to one's impression of the unending discussions about quantum matters (see introductory sections in [50] (!) and in [64]), it also represents, due to the lack of formality, a plentiful source of ideas and of valuable thoughts. Schlosshauer's very informative "quantum interviews" [16] pursue the same goals.

It is worth mentioning that the quantum challenges had led, quite a while ago, to attempts to revise, even formalizing, the logic of our thinking [72,73]—a very nice mathematical theory dating back to von Neuman in the 1930s ([25], Section III.5) termed quantum logic [74]. There are handbooks on that subject [75], and this topic is still under intensive investigation now. See also the last paragraph in Section 6.3.1.

The lack of transparent motivations for mathematics—a pressing requirement of physics—means that QM-formalism is hard to distinguish from a "cook book of procedures and rituals" (J. Nash), a "user-manual" ([32], p. 1690), [76], ([27], p. xiii), or from "a library of . . . tricks and intuitions" [21]. Therefore, the "dissatisfaction regarding comprehension" and the "need for interpretation that is alien to an exact science" ([77], pp. 7–8) lead to the fact that "we admit, be it willingly or not, that quantum mechanics is not a physical theory but a mathematical model" ([32], p. 1701) or that "nature imitates a mathematical scheme" ([78], p. 347; Heisenberg). De facto, QT "is in a sense like a traditional herbal medicine used by "witch doctors". We do not REALLY understand what is happening" (J. Nash) and "we have essentially *no* grasp on why the theory takes the precise structure that it does" ([2],

p. 32), which raises the suspicion that "something is wrong with the theory" (H. Putnam) and that "this quantum skyscraper is built on very shaky ground" ([64], p. 8). (Throughout the text, the *italic* and *slanted* type in "quotations" is original, unless otherwise indicated.)

At the same time, well-founded opinions have long been known to the effect that "quantum theory needs no "interpretation'" [43], in refs. [3,12,60] or that "only consequences of the basic tenets of quantum mechanics can be verified by experiment, and not its basic laws" ([11], p. 16). In other words, the discrepancies between opinions are significant and often radical: from epithets such as "schizoid, . . . situation is desperate" ([15], p. 420), ([79], p. 424) to supporting the rationale for quantum computations [24] and whole books written on the subject [8]. Concerning the "schizoid", the case in point is the many-world conception by Everett–DeWitt. See also pages 158, 161, 176–179 in [80] regarding the "state of schizophrenia" and "explanations" as to why "schizophrenia cannot be blamed on quantum mechanics" ([80], p. 182).

In any case, the controversy between "the warring factions, ..., many [quantum] faiths, . . . and instrumentalist camps" ([16], pp. 60–61), ([81], Section 5.5), [30,33]—"[t]hey all declare to see the light, the ultimate light" ([50], p. 987)—cannot be considered as an acceptable state of affairs (see also Section 11.1) for the simple reason that the quantum philosophy issues turn into an "industry"of interpretations—an unhealthy state of affairs— while, at the same time, the very same philosophers call for its denial: "interpretation of QM emerged as a growth industry" ([82], p. 92).

#### *1.2. Formula of Superposition*

Conversely, the "dominant role of mathematics in constructing quantum mechanics" has led to the conclusion that mathematical "assumptions are usually considered to be physical" ([32], p. 1691). That is to say, "there has been a substitution of concepts" ([76], p. 295) and "one of the consequences of quantum revolution was the replacement of explanations of physical phenomena by their mathematical description" ([76], p. 296). These characteristics convey, in the best possible way, the dissatisfaction with the fact that quantum physics "was actually reduced to a physical interpretation of the Hilbert space theory" ([32], p. 1690). The H-space in itself is a fairly cumbersome mathematical structure and even determines a crucial principle: the superposition of states [26]. It is thus not surprising that this principle becomes "one of the vague points . . . the [Dirac] argumen<sup>t</sup> is difficult to consider as rational . . . the physical principle simply fits underneath it" (excerpt from the preface to the Russian edition of [83]).

The mathematics of the H-space contains three constituents: a vector space, the innerproduct add-on over it, and topology. The two latter ones invoke the first one, which is completely independent (algebra) and begins with the formula

$$
\langle \Psi \rangle = \mathfrak{a} \cdot |\mathfrak{q}\rangle + \mathfrak{b} \cdot |\mathfrak{x}\rangle \,. \tag{1}
$$

This is the pivotal expression of quantum theory. Comprehending its genesis is tantamount to comprehending the nature of the *linearity* of QM.

In Formula (1), there occur the complex numbers a, b ∈ C, symbols of operations · and +, and also vectors |*ψ* , |*ϕ* , |*χ* ∈ H. It is clear that until an empirical basis for all these devices is found, the interpretation of Abstraction (1) and questions of the kind "Quantum States: What the Hell Are They?" (55 times in [23]) will remain a problem. To all appearances, the problem is considered so difficult—"quantum states . . . cannot be "found out'" ([8], p. 428)—that the non-axiomatic meaning of these symbols was not even discussed in the literature. In the meantime, not only is the situation far from hopeless, but it also admits a solution. The present work is devoted to gradual progress towards an understanding of Formula (1). Stated differently, Equality (1) becomes an empirical "theorem" (p. 56).

• The main part of the challenge is not only to ascertain what is being added/multiplied in Formula (1), but also to realize primarily *what "to add/multiply" is*, and "Where Mathematics Comes from" [84] at all.

"What does it mean, physically, to "add" things?" [2], (p. 178; D. Darling). More than that, aside from the symbols {<sup>a</sup>, b, |*ψ* , |*ϕ* , |*χ* , ·, <sup>+</sup>}, Expression (1) contains the sign of equality = (see also [85] (pp. 29, 30 (!), [86]), and, surprising as it may seem, it conceals one of the key points—the third principium of quantum theory (III, p. 29).

The guiding observation is based on the fact that the only thing that we have access to are the microscopic events, and therefore, "we have little to begin with other than what an experimental physicist would call experiments with a single microsystem" ([87], p. 5).

"[W]e must recognize that the focusing on individual elements whatever these may be is absolutely indispensable for all our thinking. . . . What may be regarded as an individual event?"

#### R. Haag ([88], p. 302)

Consequently, we must begin from individual events and from collecting them into ensemble formations. It is precisely in this context that we will use the word empiricism— quantum empiricism of micro-acts of perception—and it is in this respect that QT has a statistical nature. As Einstein put it, "It may be a correct theory of statistical laws, but an inadequate conception of individual elementary processes" ([30], p. 156; Einstein); see also ([25], ([30], Chs. 7–8), [70], [89], pp. 38–40), ([89], p. 40). Such a viewpoint has been long championed by L. Ballentine [90] and H. Groenewold ([91], p. 468) and justified in detail by G. Ludwig [87,92–94]. A. Leggett proposes accordingly "extreme statistical interpretation"[16] (p. 79) , [95], in the sense that "to seek any further "meaning" in the formalism is pointless and can only generate pseudoquestions". With that, he overtly applies such characteristics as "complete gibberish" ([95], p. 70) and "verbal window dressing" ([16], p. 79).

The difficulty is, of course, in creating the object |*ψ* itself. A step-by-step characterization of this procedure (Sections 3–8) and key words to what follows have been reflected in the (sub)section titles listed in the Contents.

#### *1.3. Physics Mathematics; Doctrine of Numbers*

Thus, the situation appears to be one whereby the physics itself faces inconsistencies in its foundations and the mathematical superstructures are difficult to reconcile with its motivations (physical principles) [96]. However, on the other hand, attempts to axiomatize an interface between them [97] only conceal a deeper insight [22]. M. Born had called attention to the fact that "probable refinements of mathematical methods will not suffice to produce a satisfactory theory, but that somewhere in our doctrine is hidden a concept" and T. Maudlin was more definite: "physicists have been misled by the mathematical language they use to represent the physical world".

In other words, we observe an overemphasis on the role of the ready-made mathstructures—algebras, spaces, and the like—and an under-evaluation of "seemingly naïve" empirical aspects voiced in the ordinary language [98]. The situation is no different from that which H. Weil had characterized in the introductory section to ([99], p. 10) as follows.

"All beginnings are obscure. Inasmuch as the mathematician operates with their conceptions along strict and formal lines, he, above all, must be reminded from time to time that the origins of things lie in greater depths than those to which their methods enable them to descend".

The "origins" are expressible of course only in the natural language; Section 2 is devoted to this.

What we propose below is an implementation of the idea that the postulational view must be abandoned and replaced by a negation of the prior existence of both the physical "preconceived notions" [92] (p. 328) and the mathematical structures. Physics and mathematics should be created "from scratch". Paul Benioff calls this idea "a coherent theory of physics and mathematics" [2] (p. 33; P. Benioff), [96] (p. 639), Then, due to the initial absence of mathematics, introducing mathematical structures is almost ruled out, proofs must be replaced by an empirical inference, and semantics of physics—the language of physical reasoning—is initially under a linguistic ban. It cannot exist a priori. That is to say, even the natural-language conjunction of mathematical terms with physical adjectives (and verbs [100] (p. 3102; "to happen, to be, to exist")) becomes far from being free, as with the classical description's language (Sections 2.1, 5.4, 6.4 and 6.5). R. Haag, on the first page of the work [101], emphasizes:

•"we should not consider ["vocabulary of Quantum Theory"] as sacrosanct. . . . every word in the vocabulary is subject to criticism".

Returning to the ensemble formations, it is only they that have to come to the fore, and argumentation should be subordinated only to the low-level microscopic empiricism. The predominance of the empirical over the theoretical will then immediately touch on the closest creature of the latter—the notion of a number—since numbers do not come "from the sky", and the theory will have to be a quantitative one.

Despite the overflow of abstracta in QT, the *doctrine of number* —,number × unit-, to be precise (Sections 7 and 9)—has, it seems, not ye<sup>t</sup> entered foundational discussions [102]. Consequently, the numbers turn into a kind of "problem of numbers" (principium II), and we are thus led to the necessity of revising the take on the foundations themselves:

,quantum fundamentals- ,the problem/doctrine of the number- .

This paradigm shift is a unique trait of the quantal (not the classical) view of things and a substantial part of the following is devoted specifically to that.

In the outline of the present work, the workflow will constitute re-creating the structure of a linear vector space. More precisely, producing an *a priori unknown* mathematics, which *will be* an algebra of such a space with a complex conjugation. As a matter of fact, we provide an answer to Haag's question "How do we translate the description of an experimental arrangemen<sup>t</sup> into mathematical symbols?" in the context of their own "idea of basing the interpretation of quantum theory on the concept of "events" which may be considered as facts independent of the consciousness of an observer" ([88], p. 295).

The main point to be immediately emphasized is that the mathematization of the discrete micro-acts of observations is quite a nontrivial procedure (105), and further, the strategy, along with the structure of this article, can be schematized as follows.

This box-diagram cannot be reduced or restructured. For example,

•*Superposition foregoes numbers*, and measurement and physical properties follow *strictly after* the |ket -vectors have been created.

By and large, the aforesaid ideology is supported by the common belief—often certainty even [12]—that QM is *not* perturbative, its linearity is *not* associated with linear approximation of something else, and, in general, it is *not* extensible (ultimate [103] and non-deformable) and must be free of interpretations [12,43]. All of these concerns, in one way or another, are directly related to the derivation of Formula (1).

#### **2. Points of Departure**

In the Beginning was the Word—A. Zeilinger ([39], 01:0547)

Most of the time the apparatus is empty and sometimes you have a photon coming through—A. Zeilinger ([39], 1239)

Since empiricism is in essence supra-mathematical [104], i.e., it is concerned with *meta*mathematics [105,106], its mathematization, i.e., theory construction, should begin not with postulates and definitions, but rather with the semantic formation of an object language (of "the Quanta") of vocabulary that "may only be described by "words" and not by a theory" [58], [87] ( p. 106), [93]. As A. Peterson and K. Popper had observed, "Math can never be used in phys until have words" ([107], p. 209), i.e., "we cannot construct theories without using words" ([108], p. 12). Therefore, relying on the established understanding of the underlying causes for the quantum eye on physics [13,25,27,33], up until the end of this section, we will adopt the natural-language meaning of the words observation, system, state, numbers (!), plus and to divide, physical influence, transition, large/small, micro/macro, etc. Their contents will later be defined more precisely or entirely changed. For instance, the sense of the word "state" will be drastically transformed, to which we are drawing attention in advance. Accordingly, a degree of informality—it has been clarified in Remark 10—is inevitable here, but "the lack of precision . . . is a necessity" ([109], p. 48) at the moment.

#### *2.1. Variations as Micro-Level Transitions*

We will (and "must" [105] (Ch. 3)) first view the concept of a system at an intuitive level ([110], Section 1.1)—there is what is referred to as an isolated system S.

S Let us tentatively (a priori) relate the concept of a *state* to the associated context describable by the words "the system S can *vary*, *be different*, or *in different states*". That is to say, system S is always in a certain state Ψ belonging to the set T = {<sup>Ψ</sup>, Φ, ...}, each element of which is admissible for S, and all of them are different from each other: Ψ = Φ.

In other words, the concept of a quantum system may not have a precise/axiomatical definition at the moment. Otherwise, if it comes to that, the system is what is being constantly varied when observed, and "varied" is the key word here.

The statement "states are different" does not require a consideration when Ψ and Φ, referred to as state, are the abstract elements of an abstract set {<sup>Ψ</sup>, Φ, ...}. However, in order to tie its elements to reality, we have to introduce the criteria of coincidence/distinguishability of one from the other. Criteria may not come from observational procedures, without which it is impossible to either detect states or claim that they differ, coincide, or that they are, if any.

On the other hand, the nature of micro-phenomena shows that observations are always associated with an irreducible intervention in the system, manifesting in what is known as *transition* Ψ Ψ (or destruction). As an example, observations at accelerators are literally the destructions, and bulk at that. Due to a lack of criteria, there is no sense in attributing to this concept the adjectives small/large, (in)significant/partial, or collocations such as "comparison of destructions at instants *t*1, *t*2". Let us proceed from the idea that initially there is nothing but the transition. Transitions may actually occur without destruction Ψ Ψ, however.

Two different Ψ, Φ may be destroyed into new Ψ , Φ, as well as into the combinations of the old/new. Thus, strictly speaking, the sense of words "different, new, ..." eludes us in this case, which is why even the identification of Ψ-elements and the T itself, as a set, becomes questionable. Therefore, besides the formal writings Ψ = Φ and Ψ = Φ for Ψ, Φ ∈ T, the *physical* distinguishability/equivalence (recognizability ≈/≈) needs to be established. As to the identification (and to the identity) in this regard, see von Neumann's reasoning: "One might object against II . . . " on page 302 of their book [25]. The sole thing that distinguishability may rely on is the transition acts. In turn, variation is a key element in transitions, which is why we will begin constructing with distinguishability.

Let us take the still virtually unlimited way A of intervening A in S and attempt to introduce distinguishability Ψ ≈ Φ as A -distinguishability. Due to the fact that microtransition Ψ A Ψ is not pre-determined, initial states Ψ undergo arbitrarily free changes. Next time, the results will be different and absolutely arbitrary (the term "different" is understood to mean =), and each act is indiscernible from a case in which it contains ones similar to itself within itself. It would be natural to associate such a case to the absurd, which is unrelated to the meaning of the words "physical observation", and to discard the given A .

Non-meaninglessness arises only if we impose the negation of random combinations of = and = in transitions, at least for a part of T, i.e., introduce the preservation acts Ψ A Ψ. The "preservation" should be read here as indestructibility of state, i.e., as a (=)-coincidence under the secondary act Ψ Ψ Ψ. Otherwise, the vanishing difference between "preservation" and "variation" leads to linguistic chaos ([111], p. 232). This means that the destruction Ψ Ψ may not be considered as a one-fold one. State Ψ on the right should be examined for changeability and transform into the left part of the subsequent transition: Ψ Ψ Ψ. Thereby, the structure Ψ Ψ with the *binate* entity "before/after" or "on the left/right" becomes the key one, and we consider it an initial object in subsequent constructs. The preserved states are, by definition, those that pass the reproducibility test.

Thus, logic requires beginning with the transition compositions

$$
\overline{\underline{\underline{h}}} \quad \neg \overline{\underline{\underline{\underline{\underline{\tau}}}}} \quad \overline{\underline{\underline{\underline{\underline{\tau}}}}} \quad \neg \overline{\underline{\underline{\underline{\tau}}}} \quad \overline{\underline{\underline{\underline{\underline{\tau}}}}} \quad \neg \overline{\underline{\underline{\underline{\tau}}}} \quad \cdots \quad \neg \overline{\underline{\underline{\underline{\tau}}}}
$$

wherein the cases such as

$$\cdots \quad \quad \overline{\Psi}\_{\prime\prime} \stackrel{\cdots \sim \varphi \sim}{\neg \neg \varphi} \stackrel{\cdots}{\Psi}\_{\prime\prime} \stackrel{\cdots \sim \varphi \sim}{\neg \Psi}\_{\prime\prime} \stackrel{\cdots}{\tag{2}} \tag{2}$$

are ruled out (a ban on changing of what has been unchanged), and the never-ending sequence

$$
\overline{\Psi} \quad \overline{-}\overline{\gamma}\stackrel{\sim}{\pi'} \quad \overline{\Psi}' \quad \overline{-}\overline{\gamma}\stackrel{\sim}{\pi'} \quad \cdots \quad \overline{-}\overline{\gamma}\stackrel{\sim}{\pi'} \quad \overline{\Psi}\_{\theta} \quad \overline{-}\overline{\gamma}\stackrel{\sim}{\pi'} \quad \cdots \tag{3}
$$

(non-recognisability of states) must be terminated

$$
\overline{\Psi} \quad \overline{-}\overline{\gamma}\_{\overline{n}'} \stackrel{\sim}{\Phi} \overline{h}\_{\wedge} \quad \overline{-}\overline{\gamma}\_{\overline{n}'} \stackrel{\sim}{\rightsquigarrow} \dots \quad \overline{-}\overline{\gamma}\_{\overline{n}'} \stackrel{\sim}{\Phi} \overline{h}\_{\wedge} \quad \overline{-}\overline{\gamma}\_{\overline{n}'} \stackrel{\sim}{\rightsquigarrow} \overline{\Psi} \tag{4}
$$

yielding a "finiteness" (= realisticness) and the concept of conserved/distinctive *α*-states. The terminology *α*-event [12] could be used instead.

Freedom of elements in Sequence (4), including the choice of *α*-states, is not limited by anything besides the ban on Equation (2). Therefore, this arbitrariness, which is physically never recognizable, curtails the generic chain from Sequence (4) into the shortened one

$$
\overline{\Psi} \dashv \stackrel{\scriptstyle}{\dashv} \sim \overline{\lnot \, \overline{\llcorner} \, \overline{\llcorner} \, \cdots \sim \overline{\ll'} \, \overline{\ll'}} \stackrel{\scriptstyle}{\dashv} \sim \overline{\ll'} \stackrel{\scriptstyle}{\dashv} \cdots \stackrel{\scriptstyle}{\dashv} \stackrel{\scriptstyle}{\dashv} \tag{5}
$$

which is identical to the scheme

$$\overbrace{\overline{\cdots \cdots \overline{\overline{\mathbf{h}} \cdots \cdots}}}^{\overline{\cdots}} \cdots \overbrace{\overline{\cdots} \overbrace{\cdots}}^{\overline{\cdots} \overbrace{\overline{\mathbf{h}}}^{\overline{\mathbf{h}}}} \tag{6}$$

with certain *α* ∈ T.

Discussions on "what happens . . . [and] "how" ([25], p. 217) at the very microscopic level are extremely widespread in the literature [18,41,44,45,53,56,77] (see [19,33,40,112] for the exhaustive references), although it is not difficult to predict the fact that the attempts to understand the inner structure of Box (6) will only lead back to an identical box; so, the "turtles all the way down" (ascribed to W. James), followed by the grea<sup>t</sup> Wheeler's slogan "No tower of turtles" (1989).

Indeed, the uncontrollability of micro-changes is universally known, ye<sup>t</sup> describing them as a process in time *t* %→ *t* + *ε* will start employing language terminology—functions, arithmetic operations, the physical words, etc.—that has not ye<sup>t</sup> been created even for the fixed instants *t*1, *t*2. However, what may be associated with fixed time are only nontemporal entities, for which we have nothing but transitions (Equation (5)). The attempt to manage them, i.e., to control intervention in S, results in looping or "measuring the measurement", in addition to the ambiguity of this term itself.

"[I]t is not meaningful to speak of a measurement "at time ˜*t*." . . . the real physical meaning of the time parameter . . . has nothing to do with the notion "time of measurement"". "[T]he description of the measurement process in quantum mechanics in terms of "pre-theories" is not possible"

> G. Ludwig ([87], p. 288), ([92], p. 340)

See also [58] (p. 100), [92] (p. 365), [94] (p. 150), [113] (pp. 644–646), and [114]. Just as before, the physical assessments such as "abrupt", "(ir)reversible", "(non)simultaneous", "immediately following ..." [25] (pp. 231, 410), or the "weak/nondemolishing" (measurements [53]), etc.are unacceptable here. No temporal process may be present in the foundations of the theory (([87], Sections VII.4, 6), ([92], Chs. III, XVII), [93]) since it is immediately not clear: "Furthermore, what exactly are we having at instants *t*1 or *t*2?". In the reverse direction—,time measurement-—the situation is also rather indefinite since the ""Time" is not an entity to which the operations of measurement, direct or indirect, apply" ([114], p. 5).

**Remark 1.** *All the information stated above means that attempts to deduce QM dynamically ([16], 10 · Reconstructions) are beforehand doomed to vicious circles "round the boxes" and time t, such as attempts to dynamically "vindicate" Lorenz's contraction instead of kinematic postulates of the relativity theory [69]. A consistent theory must rest either on "irreducible" elements* (6) *or upon "boxes" of a different kind. In the latter case, the theory becomes a particular* model with interpretation*; e.g., the Lindblad equations [115,116], decoherence [112,117–119], stochastic dynamics,*

*and other statistic-dynamical models [40,77]. Anyway, an ability to model and understanding are not the same thing, and this point was repeatedly emphasized in the literature ( [16], ([17], Section I.2), [32]) with regard to QT.*

That said, if theory is built *as a fundamental one rather than as a model* ([16], p. 144), with a primary entity *changeability* <sup>A</sup> , Box (6) may only be involved in it as the initial starting point and as an *indescribable* object with the absolute rather than with a relative sense. Elements of reality, in whatever understanding—say, Bell's "*be*ables" [28]—may not exist before/after/inside/outside of the box. It can only be the *structureless abstractio*. Accordingly, the notions of preparation, of measurement, of "interaction with", and of a physical process are meaningless without the construction of Box (6).

These statements are clearly in agreemen<sup>t</sup> with the fact that any reasoning must not contradict the formal logical rules [105]; hence, there must exist [96,106,120] the empirically undefinable logical atoms. A. Peres writes ([121], p. 173): "While quantum theory can in principle describe *anything*, a quantum description cannot include *everything*. In every physical situation *something* must remain unanalyzed". Moreover, as Pauli put it, "Like the ultimate fact without any cause, the individual outcome of a measurement is . . . not comprehended by laws". Specifically, the set T and transitions arrows <sup>A</sup> are also the atoms. It is a ". . . preexisting concept . . . We cannot formulate the theory without this concept", concludes B. Englert ([12], p. 2). From the aforesaid, we may formulate the following tenet.

IQuantum statics should forego quantum dynamics.

#### (*The first principium of quantum theory*)

The rationales do not end here and will be later amplified once we begin to exploit the terminology that is usually taken for granted from the outset, viz., the quantitative descriptions ([2], p. 178). If they arise not as numerical interpretations of something but out of an experiment, then observation should be the beginning, and the "manufacture of numbers'—the end. In other words, the model "theory with boxes" other than Boxes (5) and (6) implicitly implies the logical sequence ,model of process- ,numerical interpretation-, in which empiricism holds a role other than primary. It is clear that, regardless of the model, such a situation will always remain unsatisfactory in the physical respect.
