COMPLEMENTARITY AS A GENERAL FRAMEWORK
JAIRO ROLDÁN CH.
Departamento de Física, Facultad de Ciencias Universidad del Valle, Cali, Colombia
Abstract: On the assumptions of completeness, conceptual economy, and universality
of quantum mechanics, and taking into account the developments linked to decoherence, one concludes that physical reality has only an epistemic nature. It is fundamental then to make a distinction between ontological reality, constituted by everything that does not depend at all on the collectivity of human beings, nor on their decisions or limitations, nor on their existence, and empirical reality, constituted by everything that not being ontological is, however, intersubjective. The fact that the direct presence of consciousness is not required to complete a quantum phenomenon implies that quantum phenomena are not mere “appearances to consciousness”. Empirical reality, as conceived in this paper, is not reducible to mere “appearances to consciousness”. Complementarity, that was proposed by Niels Bohr as a conceptual framework to answer the problems presented by the interpretation of quantum mechanics, is considered here as a way of talking about the empirical reality that goes beyond the simple description of experimental results, or better still, the observer’s perceptions of such results. It aims to describe not ontological reality, but empirical reality. It is proposed that the state vector represents, in general, infinite potentialities that are actualized when a quantum phenomenon is completed. Such potentialities, unlike the Aristotelian and the one suggested by Heisenberg, are epistemic and not ontological. The infinite potentialities of empirical reality are taken as an argument to claim that ontological reality is infinite, complementarity being the manner in which infinite ontological reality is manifested to the finite mind. The possibility that complementarity constitutes a general framework that can be applied to other areas of knowledge is based on an analysis of the relationship between theory and observation. Two possible examples of the use of complementarity in areas different from quantum mechanics are presented, one in thermodynamics and another in biology. Given that logic does not impose the mutual exclusion of two complementary concepts, there isn’t any inconsistency implied by complementarity from the point of view of classical or ordinary logic: one uses ordinary logic in each experimentally defined context. Consequently, there is no need for any new logic.
Keywords: quantum mechanics, Niels Bohr, indivisibility, complementarity, empirical
and ontological realities, potentiality, mind and matter.
A COMPLEMENTARIDADE COMO UM QUADRO GERAL
Resumo: Aceitando as suposições da completude, economia conceitual e
universalidade da mecânica quântica, e levando em conta os avanços relacionados à descoerência, conclui-se que a realidade física tem somente uma natureza epistêmica. É assim fundamental fazer uma distinção entre realidade ontológica, constituída por tudo que não depende na coletividade dos seres humanos, nem em suas decisões ou limitações, nem em sua existência, e a realidade empírica, constituída por tudo que, não sendo ontológico, é porém intersubjetivo. O fato de que a presença direta da consciência não é requerida para completar um fenômeno quântico implica que os fenômenos quânticos não são meras “aparências para a consciência”. A realidade empírica, conforme concebida no presente trabalho, não é redutível a meras “aparências para a consciência”. A complementaridade, que foi proposta por Niels Bohr como um quadro conceitual para responder aos problemas apresentados pela interpretação da mecânica quântica, é aqui considerada como uma maneira de falar sobre a realidade empírica que vai além da simples descrição dos resultados experimentais, ou melhor, das percepções do observador sobre esses resultados. Ela visa descrever não a realidade ontológica, mas a realidade empírica. Propõe-se que o vetor de estado representa, em geral, infinitas potencialidades que são atualizadas quando o fenômeno quântico se completa. Tais potencialidades, ao ocntrário das de Aristóteles ou das sugeridas por Heisenberg, são epistêmicas e não ontológicas. As infinitas potencialidades da realidade empírica são tomadas como um argumento a favor da infinitude da realidade ontológica, e a complementaridade é a maneira pela qual a realidade ontológica infinita se manifesta para a mente finita. A possibilidade de que a complementaridade constitua um quadro geral que possa ser aplicado a outras áreas do conhecimento é baseada emu ma análise da relação entre teoria e observação. Dois exemplos possíveis do uso da complementaridade em áreas diferentes da mecânica quântica são apresentados, um na termodinâmica e outro na biologia. Dado que a lógica não impõe a exclusão mútua de dois conceitos de complementaridade, nenhuma inconsistência é implicada pela complementaridade do ponto de vista da lógica clássica ou ordinária: utiliza-se a lógica ordinária em cada contexto experimentalmente definido. Consequentemente, não há a necessidade de qualquer lógica nova.
Palavras-chave: mecânica quântica, Niels Bohr, indivisibilidade, complementaridade,
1. Basic assumptions
1.1. Completeness and economy of thought
In science, it is customary to consider that if there are two explanations for the same phenomena, you should choose the one that introduces fewer entities, which is taken conceptually as the most economical and elegant. This suggests choosing orthodox quantum theory, but the economy of thought does not mean only accepting quantum theory just because it is the most economical. Alternative proposals such as the de Broglie-Bohm theory (DE BROGLIE,1927;BOHM,1952;BOHM &HILEY,1993), the general theories of hidden variable, and the GRW theory (GHIRARDI,RIMINI &WEBER, 1986), all have serious problems, particularly in the relativistic domain, as shown, e.g., by D’ESPAGNAT (1995, 2006). The additional fact that there are many of them also plays a significant role in the assumption that quantum mechanics is complete.
EVERETT’s (1957) theory accepts the completeness of the quantum algorithm. However, it has serious problems in addition to not being economical; it either introduces multiple worlds with a multiplication of systems, instruments and observers, or real superpositions of objects and minds.
1.2. Universality of quantum mechanics
The “macroscopic manifestations” of quantum mechanics like superfluidity, superconductivity, and Bose-Einstein condensates provide indirect experimental evidence for the universality of quantum mechanics (QM). It has also been shown that some quantum notions, such as quantum tunneling, are necessarily indispensable to explain data relating to systems whose size and complexity can be called macroscopic (LEGGETT, 1984, 1997). 1.2.1. The internal consistency of quantum theory and macroscopic
bodies
In some circumstances, the internal coherence of quantum theory demands that one treats a macroscopic body as a quantum object. While considering the double-slit thought experiment, Bohr, in answering Einstein’s
critiques about the internal consistency of quantum mechanics, considered that a macroscopic body must be submitted to the Heisenberg relations. We must remark that in the above-mentioned idealized thought experiments, the whole system is treated as isolated, and decoherence, which I will present later in this paper, is then avoided. The assumption of complete isolation was justified because it was under this assumption that Einstein formulated his criticism. It was a discussion on principles, so the idealization was allowed.
2. Ontological reality and empirical reality
A property, an argument or statement that does not refer at all to the collectivity of human beings, nor to their decisions or limitations, nor to their existence, we denote as ontological. Everything that is ontological constitutes Ontological Reality (OR).
Everything that not being ontological is, however, intersubjective, which means it is true for everybody, constitutes Empirical Reality (ER). 3. Indivisibility
3.1. Indivisibility as contextuality
The Heisenberg relations illustrate that any pair of dynamical variables whose operators do not commute have a contextual nature: their values depend on the experimental context defined by the observer. Under the assumptions of completeness and universality, this contextuality of the dynamical variables implies that it is not possible to consider the observer and the observed as separated. One must rather view them as forming an indivisible whole. Because of this, it is not possible to attribute an ontological character to the dynamical properties.
3.2. Indivisibility as entanglement
The existence of entanglement leads to the conclusion that the state vector refers to the entire system. This means that the parts cannot be
considered as separate, no matter what is the spatiotemporal distance between them: what can be predicted about a subsystem will depend on what is measured in a correlated subsystem, even if a space-like interval separates them. 4. The physical existence of particles is not ontological
The dynamical properties of a system like an electron are contextual. Since the existence itself of the electron does not seem to have a contextual character, is it possible to attribute to it an ontological reality? In other words, is the existence the only ontological property of the particle?
If that were the case, the number of particles would be independent of the experimental context. In the relativistic domain, however, the operator representing the number of particles does not commute with some other operator, which means that the number of particles has a contextual character.
In Quantum Electrodynamics, for example, working in the Radiation or Coulomb Gauge, we have, on the one hand:
𝐴̂(𝑟⃗, 𝑡) = √4𝜋 𝑉∑ ∑ 𝑐√ ħ 2𝜔𝑘[𝑎̂𝑘,⃗⃗⃗⃗𝛼(𝑡)𝜖⃗ 𝛼𝑒𝑖𝑘⃗⃗∙𝑟⃗+ 𝑎̂ 𝑘 ⃗⃗,𝛼 + (𝑡)𝜖⃗𝛼𝑒−𝑖𝑘⃗⃗∙𝑟⃗] 𝛼 𝑘 ⃗⃗ ; 𝐸⃗⃗̂ = −1 𝑐 𝜕𝐴⃗̂ 𝜕𝑡 ; 𝐵⃗⃗̂ = ∇⃗⃗⃗ × 𝐴⃗̂ On the other hand, we have:
[𝑎̂𝑘⃗⃗,𝛼, 𝑁̂
𝑘´
⃗⃗⃗⃗,𝛼´] = 𝛿𝑘⃗⃗𝑘´⃗⃗⃗⃗𝛿𝛼𝛼´𝑎̂𝑘⃗⃗,𝛼 ,
[𝑎̂𝑘⃗⃗,𝛼+ , 𝑁̂𝑘´⃗⃗⃗⃗,𝛼´] = −𝛿𝑘⃗⃗𝑘´⃗⃗⃗⃗𝛿𝛼𝛼´𝑎̂𝑘⃗⃗,𝛼+ ,
where the eigenvectors of 𝑁̂𝑘⃗⃗,𝛼 are those that represent states with 𝑛𝑘⃗⃗,𝛼 particles in the state 𝑘⃗⃗, 𝛼.
The last equations mean that neither 𝑁̂
𝑘
⃗⃗,𝛼 nor 𝑁̂ = ∑ ∑ 𝑁⃗⃗⃗𝑘⃗⃗ 𝛼 𝑘⃗⃗𝛼 commute
with 𝐴⃗̂, 𝐸⃗⃗̂ and 𝐵⃗⃗̂. Therefore, Heisenberg type relations must be satisfied between
the quantities 𝑁̂𝑘⃗⃗,𝛼, 𝑁̂ and the fields 𝐴⃗̂, 𝐸⃗⃗̂ and 𝐵⃗⃗̂. In other words, the operator
representing the number of photons and the fields satisfy relations of Heisenberg.
One finds a similar situation in the theory of the “electronic” or “positronic” field. An experimental arrangement intended to measure the spatial distribution of charge implies an uncontrollable creation of electron-positron pairs. This means that the spatial distribution of charge and the number of particles are contextual variables.
The non-ontological nature of the existence of a particle leads to the conclusion that all constitutive properties like mass, charge, spin, and so on, are non-ontological.
5. Decoherence, measurement and macroscopic bodies
The formalism of QM says that any measurement involves an interaction between the object and a macroscopic object. As is well known (D’ESPAGNAT, 1995, 2006), this interaction produces, in general, a macroscopic quantum superposition which, however, is not observed. A well-known illustration of this is the thought experiment with the famous SCHRÖDINGER’s cat ([1935] 1980).
Under the universality assumption, the only essential difference between microscopic and macroscopic systems is that the latter strongly interacts with its environment: the quantum energy levels are extremely close even for tiny macroscopic bodies. Consequently, extremely weak fields can induce transitions. A detailed calculation shows that even a particle of dust in interstellar space cannot be considered as isolated over a sustained amount of time. It is then not realistic to consider a macroscopic system as isolated from its environment. The different models that take into account that interaction (decoherence model) show that, given the human limitations, for every human being the macroscopic quantum superposition cannot be observed.
The argument is as follows. Some measurements could be performed in principle because no law of physics forbids them and the sequence of operations by means of which they would be made can be precisely stated. In practice, they cannot, however, be done because they are tremendously complicated, and so in practice, one can regard them as impossible for a human being. Since the argument refers in an essential way to human possibilities, the conclusion is that decoherence solves the problem of measurement but in terms of ER: the reality of macroscopic bodies is only empirical. Decoherence shows how the classical
properties emerge in the macroscopic domain. The key to the solution that decoherence provides is the macroscopic nature of the instrument.
6. Non-ontological character of the state vector
The position that maintains an ontological interpretation of the state vector faces two serious problems: the ontological existence of macroscopic quantum superpositions and the violation of relativistic causality that happens when two systems separated by a space-like interval are in an entangled state.
Such violation constitutes a major problem for a 𝜓-ontological
position. In a 𝜓-epistemic position, the relevant fact is that the violation in question does not allow the exchange of information between human beings. If we use the word signal to indicate that exchange of information, we can say that relativistic causality must be understood in the sense that no signal can propagate faster than the speed of light. According to this interpretation, relativistic causality would have an epistemic and not an ontological sense. Interpreted in this way, we can say that the relativistic requirements are not violated. We have what Shimony calls a “peaceful coexistence” between quantum mechanics and relativity.
From an ontological point of view, which was the one adopted by Einstein, one should understand the notion of signal in an ontological sense. The word “signal” indicates any natural influence, not restricted by the possibility or not of being used to transmit information. One must state relativistic causality saying that no physical influence can propagate faster than light.
The violation of relativistic causality constitutes a major problem for a
𝜓-ontological position because it implies accepting the ontological existence of instantaneous influences. Such acceptance faces serious problems. See for example D’ESPAGNAT (1995, 2006).
Everett’s theory proposes to solve the problem of the violation of relativistic causality, without renouncing to the ontological interpretation of relativity and without invoking the existence of instantaneous influences. We mention, however, that the theory in question faces serious problems.
Two other theories consider the state vector ontological but also face serious problems: the theory of Bohm and the GWR theory.
All the difficulties that the theories that have a 𝜓-ontological position face lead me to the conclusion that the state vector is only epistemic. These difficulties reinforce my assumption about the completeness of quantum mechanics.
7. Non-ontological character of space-time
Under the assumptions of an ontological interpretation of physical reality, and in particular of space-time, locality and free choice of the observer, BELL (1987) derives certain inequalities between measurable physical quantities. Quantum mechanics predicts in certain cases a violation of Bell’s inequalities, and the experimental verdict confirms the violation. The difficulties that an ontological interpretation of space-time faces due to the existence of non-locality are similar to the difficulties faced by a 𝜓-ontological interpretation.
Along the lines of our above consideration, the violation of the Bell inequalities indicates that space and time are part of ER (D’ESPAGNAT, 1995, 2006). The fact that one cannot send instantaneous signals (what is violated is outcome independence and not parameter independence) reinforces the conclusion that space and time are not ontological: space-time as physics conceives it belongs to ER.
8. The quantum phenomena
The following argument shows the connection between measurement and irreversibility in quantum mechanics. Consider an interference experiment with an electron beam passing through two slits S1 and S2. Suppose now an experiment in which one adds a device D1 placed in front of one of the slits to know which electrons have passed through the slit. The interference is destroyed: the wave function seems to collapse to one of its components. The interference is the result of the phase difference between the wave functions associated with each one of the slits. Consider now a hypothetic experiment in which one adds in front of D1 another device D2 that could reverse things to the point of restoring the relative phase between the two waves that produce the interference. A device like this would allow one to observe the interference
and know through which slit each electron has passed. However, the irreversible nature of the process that takes place in the device D1 prohibits the possibility of device D2.
Figure 1. Two-slit experiment with two successive detectors at one slit.
Notice that the irreversibility linked to a measurement implies that the instrument must be macroscopic.
An additional argument can be found in D’ESPAGNAT (1995, Appendix 2), who analyzes the situation that would result if the measurement process were time reversed and shows that this possibility is not valid in quantum mechanics.
Since the quantum measurement is linked to irreversibility, an irreversible effect must be produced in the macroscopic instrument. When this irreversible effect takes place, the quantum phenomenon is complete. It is what Bohr in his terminology calls a phenomenon “brought to a close.”
9. Complementarity
9.1. The purpose of complementarity
Quantum phenomena are not mere “appearances to consciousness” because for completing a quantum phenomenon only the irreversible effect in the apparatus is required, not the direct presence of a conscience. It is necessary then to make a distinction between phenomena in the Kantian sense and quantum phenomena.
Quantum phenomena belong to ER, which means that ER, as defined in this paper, is not reducible to mere “appearances to consciousness.” The definition of ER as the set of phenomena, understood in the Kantian sense, implies that ER is “internal” to consciousness. This paper’s definition of ER implies on the contrary that ER is “external” to consciousness. “External” means here that even if it is dependent on consciousness, ER is not reduced to consciousness. There is no logical contradiction involved in the last phrase: a mirror image of something is dependent on but not identical to that something; a book is dependent on but not identical to its author.
Complementarity is a way to talk about ER that goes beyond the mere description of the experimental results, or better, the observer’s perceptions of such results. It aims to describe, but what it describes is not OR, but ER. 9.2. Precise definition of complementarity
The development presented here goes beyond what BOHR (1929, 1934, 1957, 1963) wrote about complementarity. For a full presentation of the precise definition, see ROLDÁN-CHARRIA (2014).
9.2.1. The meaning of concepts
Indivisibility opens the possibility that concepts valid in a given experimental context are not valid in a different one. To decide if one can or cannot use a particular concept to describe the information obtained in a given quantum experiment, one must keep in mind the entire experimental arrangement. Therefore, if a concept turns out to be adequate for one
experimental arrangement, one is not necessarily authorized to use it without restriction in another experimental arrangement. The experiment defines the use of the concept. Now, the meaning that is being given to the word use is here restrictive: if the experimental arrangement does not permit the use of one concept, this signifies that one cannot employ it in the reasoning. One must notice that what is being proposed is not a mere operational definition of the concepts. What is affirmed is that one is not authorized to use without restriction a concept, whose use is sanctioned by a concrete experimental arrangement, to account for another arbitrary experimental arrangement whatsoever.
9.2.2. Mutually exclusive experiments and concepts
Two concepts will be mutually exclusive if the possibility does not exist to define their meanings using only one experimental arrangement. One cannot then combine two mutually exclusive concepts to have a single conceptual image.
9.2.3. The instrument and the interior of the quantum phenomenon Indivisibility implies that a quantum phenomenon is an indivisible whole that includes the instrument of observation. In that totality, nevertheless, one should make a fundamental distinction between the instrument and the remainder of the phenomenon. At this point, it is necessary to recall the relation between irreversibility, physical measurement, and decoherence. Since one has to use the instruments and the data in a classical context, one should then describe one part of the totality constituted by the phenomenon in the classical context.
The proposal here is to call “the interior of the phenomenon” the remainder of the phenomenon that one cannot describe in the classical context. This interior is manifested in the classically described instrument through an irreversible effect.
9.2.4. The quantum object
When one goes from an experimental context to another mutually exclusive one, the interior and the instrument will change. The two slits experiment with electrons illustrates the idea. An experiment that permits one to observe the interference of electrons includes a fixed diaphragm that is part of the instrument. In a second experiment, mutually exclusive with the first one that allows one to know through which slit each electron passes, the diaphragm is not fixed, and it is no longer part of the instrument. Nevertheless, something of the interior stays constant: the beam of electrons. For a detailed analysis, see ROLDÁN-CHARRIA (2014).
The proposal is to call quantum object this something that remains constant. In other words, there are pairs of phenomena such that an aspect of their interior remains constant. This aspect will be called the quantum object, and one will say that the two phenomena have the same quantum object.
Another example presented in ROLDÁN-CHARRIA (2014) is the well-known Heisenberg microscope designed to perform two mutually exclusive experiments: one for measuring the position of an electron and the other for measuring the momentum of the electron. The quantum object in the two experiments is the electron and the photons.
9.2.5. Complementary phenomena
Two phenomena will be complementary if they are mutually exclusive, and they have the same quantum object.
9.2.6. Complementary concepts
We will call complementary concepts those defined by means of complementary experiments.
9.2.7. Comments
Indivisibility means that the quantum phenomena are indivisible wholes. Two complementary phenomena constitute then two complementary totalities. Why do these two totalities have a relationship? What is the reason
for them to be precisely complementary? They must have something in common. What is proposed here is to call “quantum object” the common aspect of two complementary phenomena, which makes them complementary.
In the example of the two slits experiment with electrons, the information is about the electrons that constitute the quantum object. In the example of the Heisenberg microscope, what is measured are the position and the momentum of the electron. The information is again about the electron; the quantum object is however composed of the electron and the photons. In conclusion, the information is about the quantum object or about part of the quantum object. The analysis of the possible uses of complementarity outside quantum mechanics will show if the information is about the object or part of the object.
To discuss why it seems reasonable that complementarity constitutes a general epistemological framework it is necessary to give some considerations to the nature of the state vector, the relation of Consciousness and ER, and the relation of the last two and OR.
10. On the nature of
I have presented arguments against a 𝜓-ontological interpretation of
the state vector that led me to the conclusion that the state vector is only epistemic. What aspect of ER does the state vector represent? My proposal is that it represents a potentiality.
In a quantum phenomenon, one can say something about the “beginning”: preparation of the system; and something about the “end”: effect of irreversibility in the instrument. What one cannot say is what happens in between, before the effect of irreversibility has taken place. One could say however that between the preparation of the system and the “bringing to a close” of the phenomenon there exists the potentiality of showing one of the possible results. The Aristotelian notions of potentiality and actuality can be applied here, but only in a non-ontological sense: the state vector, which is not considered here as ontological, would represent the potentiality, and the “actualization” would be context-dependent. HEISENBERG ([1958] 1970) introduces the notion of potentiality in QM, but with its original ontological sense. There was, however, a serious ambiguity in his treatment regarding when the potentiality
becomes “actual”: is it when the quantum phenomenon is registered in the instruments or when it is perceived by a consciousness (ROLDÁN-CHARRIA, 1991)? For a discussion of different proposals for bringing the notion of potentiality and actuality into quantum mechanics, see DA COSTA & DE RONDE (2013).
In the proposal that I am going to describe here, the “actualization” would take place when the phenomenon is registered in the instruments, which means when it is “brought to a close” by the effect of irreversibility.
Let us recall some aspect of the formalism. The dynamical variable considered here is the position 𝑟⃗ of an electron. An eigenvector of the operator representing 𝑟⃗ has as eigenvalues all the possible positions of the electron, which are continuous. We represent the state by its wave function
𝜑(𝑟⃗), whose square evaluated at a point 𝑟⃗ gives the probability of finding the electron at that point. In the proposal, the idea is that before one performs a position measurement all the eigenvalues are in potentia. Once the measurement takes place by the phenomenon being “brought to a close”, one of the potential positions is “actualized”: it passes from potentiality to actuality.
In the two slits experiment with electrons, there are two complementary experiments: Experiment 1 permits one to observe the interference of electrons, and Experiment 2 allows one to know through which slit each electron passes. A well-known fact referring to this experiment is the following: if Slit 2 is closed, one can say that each electron passes through Slit 1. In this situation, one assigns a wave function 𝜑1(𝑟⃗). If Slit 1 is closed, one can
say that each electron passes through Slit 2. In this situation, one assigns a wave function 𝜑2(𝑟⃗). If both slits are open, the wave function is the superposition
𝜓(𝑟⃗) = 𝜑1(𝑟⃗) + 𝜑2(𝑟⃗). In the proposal, 𝜓(𝑟⃗) = 𝜑1(𝑟⃗) + 𝜑2(𝑟⃗) represents the potentiality in Experiment 1. Before the experiment is “brought to a close”, the states represented by the two 𝜑𝑖(𝑟⃗) are in potentia. When the experiment is
“brought to a close”, one of them is “actualized”. In Experiment 2 the potentiality is transformed to one of the potentialities represented by one of the
𝜑𝑖(𝑟⃗), 𝑖 = (1,2).
The general idea of the proposal is as follows. For simplicity, I will consider only the discrete spectrum. Let {𝐴̂1, 𝐴̂2, … , 𝐴̂𝑖, … } be a complete set of
commuting operators with orthonormal eigenvectors |𝜓𝑎1𝑘,𝑎2𝑙,…,𝑎𝑖𝑚,…⟩:
One can express any |Ψ⟩ belonging to the Hilbert space in terms of the orthonormal basis {|𝜓𝑎1𝑘,𝑎2𝑙,…,𝑎𝑖𝑚,…⟩ } as:
|Ψ⟩ = 𝑐𝑎1𝑘,𝑎2𝑙,…,𝑎𝑖𝑚,…|𝜓𝑎1𝑘,𝑎2𝑙,…,𝑎𝑖𝑚,…⟩ .
We suppose that |Ψ⟩ is normalized.
In the proposal, |Ψ⟩ represents the potentiality that when a measurement of any of the operator of the set, for example, 𝐴̂𝑖 is performed,
one of its potential eigenvalues 𝑎𝑖𝑚 is actualized.
The fact that it is possible to express the state vector |Ψ⟩ in different
basis manifests the contextuality or indivisibility that precludes an ontological interpretation of the potentiality, as proposed by Heisenberg.
In effect, let {𝐵̂1, 𝐵̂2, … , 𝐵̂𝑗, … } be a second complete set of commuting
operators with orthonormal eigenvectors |𝜓𝑏1𝑘,𝑏2𝑙,…,𝑏𝑗𝑚,…⟩:
𝐵̂𝑗|𝜓𝑏1𝑘,𝑏2𝑙,…,𝑏𝑗𝑚,…⟩ = 𝑏𝑗𝑚|𝜓𝑏1𝑘,𝑏2𝑙,…,𝑏𝑗𝑚,…⟩ ,
where the 𝐴̂𝑖, 𝐵̂𝑗 are non-commuting: [𝐴̂𝑖, 𝐵̂𝑗]~ℏ ∀𝑖, 𝑗.
One can express the vector |Ψ⟩ as:
𝑑𝑏1𝑘,𝑏2𝑙,…,𝑏𝑗𝑚,…|𝜓𝑏1𝑘,𝑏2𝑙,…,𝑏𝑗𝑚,…⟩ .
|Ψ⟩ =
The vector |Ψ⟩ represents now the potentiality that when a measurement of any of the operator of the set, for example, 𝐵̂𝑗 is performed,
one of its potential eigenvalues 𝑏𝑗𝑚 is actualized.
Each expansion of |Ψ⟩ in one of the possible basis expresses a set of potentialities. Which one of this possible set is actualized depends on the experimental context. This fact means that the potentiality here proposed is only epistemic.
Although the writings of Bohr inspire my interpretation, and that is the reason I call it “Bohrian”, a crucial difference between him and me is the following: I do not consider the state vector as just a theoretical device for computing measurement results.
11. On the relation between consciousness and ER
We need first to analyze the spontaneous “closure” of a phenomenon, as in the examples presented by WHEELER (1984), where he considers “the flash of a zinc-sulfide crystal when it is struck by a cosmic ray” or “a heavy-nucleus cosmic ray that drives into the summit of a granite mountain in Brazil and stops”. One can call measurement-like interactions those spontaneous “closing” of a phenomenon. In a measurement-like interaction, the observer only observes the data, he or she does not choose the experimental context. Why then does one say that the measurement-like interactions or the spontaneously closed phenomena belong to the empirical reality? Why can’t one consider them as ontological?
The essential point is that every registered or recorded phenomenon is intrinsically susceptible to be perceived. Once registered, the phenomenon is intrinsically capable of being perceived. The difference between a spontaneously closed phenomenon and a planned phenomenon is that in the latter the observer can arrange things to obtain all the possible information.
It is necessary to explain the meaning of the expression “intrinsically susceptible to be perceived” or “intrinsically capable of being perceived”. Before the discovery of quantum phenomena, one could also say that every physical interaction was capable of being perceived. The physical interactions, however, could, without any experimental contradiction, be considered as ontological, which means that the capability of being observed was not essential or intrinsic. The existence of minds was not implied. The quantum phenomena, planned or spontaneous, imply, however, the existence of minds.
The argumentation is valid during the time interval, let us called it Period B, between the emergence of anatomically modern humans until now. It could also be from the emergence of the primitive humans until now. What about the situation in the period, let us called it Period A, during which, according to the present scientific description, material reality (stars, galaxies,
the Earth, the universe) existed long before the emergence of beings endowed with consciousness?
A Kantian or Berkeleyan idealist can respond that the claim that there was a time when the universe existed without the existence of consciences and all similar assertions only means that humans can organize their collective experience describing them with such assertions. Such answer means that, when all is said and done, the scientific assertions concerning Period A are merely allegories.
On one hand, neither Kantian nor Berkeleyan idealism is consistent with this our present interpretation of QM: a phenomenon does not have to be perceived by a mind to be a phenomenon. The formalism demands only the presence of a macroscopic body, not necessarily linked to a consciousness. On the other hand, is it consistent to consider that the scientific assertions during period B are not allegorical at all, but that the same descriptions during the period A are allegorical? To have in a time interval an allegorical explanation for the scientific description and a non-allegorical one for another time interval does not seem entirely consistent. One can say that it is even arbitrary. One feels here a theoretical “dissonance”, if not an inconsistency. Notice that such a dissonance does not arise in the idealistic position.
The essential difference between the two situations is that in one, period B, science asserts the existence of minds and on the other, period A, that existence is not affirmed. If one found a conceptual framework in which one could claim the existence of mind for all the time described by science, there would be no difference between the two situations.
I have presented the general sketch of such a conceptual framework in ROLDÁN-CHARRIA (2014). A gradual development of it will be the subject of forthcoming papers. My proposal makes use again of the Aristotelian concept of potentiality. As was explained before, the words potentiality and actuality are taken in a non-ontological sense.
The essential idea is that Empirical Reality (ER), which will be called Matter, and consciousness, which will be called Mind, have potential and actual existence.
Potential Existence of Matter. The most accepted scientific cosmological theory claims that the universe began 13.8 × 109 years ago. At that very
moment, all quantum phenomena had a potential existence. When stable macroscopic structures where irreversibility occurs did emerge, the quantum
phenomena started to pass from potentiality to actuality: the spontaneously closed phenomena began to appear. During the whole time interval from the very beginning until now, quantum phenomena have existed in potentiality or actuality.
Potential Existence of Mind. Let us first recall the WIGNER’s friend paradox (1961) that involves two observers: Bob and his friend Alice. We consider a laboratory with an instrument that has a bulb that turns on if an atom disintegrates and remains turned off if the atom does not disintegrate. Bob does not observe the situation in the laboratory directly. Alice makes the observation and sends Bob a video in which she appears telling him what she has observed. In this case, Bob faces a superposition of two states: one in which Alice has a conscious perception of the bulb on and the other in which she have a conscious perception of the bulb off. Bob asks Alice if, before the time when he sees her video telling him if the bulb is on or off, she felt herself in a superposition of two states of consciousness. Experience shows that the response of Alice will be that she never felt being in such a superposition. Decoherence explains why Bob does not see a macroscopic quantum superposition. Can decoherence also explain why Alice did not feel being in a superposition? Decoherence shows that Bob does not perceive a macroscopic quantum superposition because it is outside of his possibilities as a human. As that is true for every human being, then it is also true for Alice. Given human limitations, for a human being, there are no macroscopic quantum superpositions, including a possible superposition of states of consciousness. The situation does not change if one considers an inanimate instrument or a human being with consciousness.
The key to the solution that decoherence provides is the macroscopic nature of the instrument, be it an inanimate device or a conscious being. Because of their macroscopic nature, neither cats, if considered endowed with consciousness, will feel themselves in a superposition of conscious states.
At the very beginning of the universe, consciousness had a potential existence. Since Wigner’s friend paradox is solved for consciousness linked to macroscopic systems, this paper’s proposal considers that consciousness can be manifested only in macroscopic systems. It also means that the conditions for consciousness to have the possibility to pass from potentiality to actuality and for quantum phenomena to be perceived are fulfilled when stable macroscopic structures, where irreversibility can happen, did emerge. In other words, the
conditions for consciousness to have the possibility to pass from potentiality to actuality are met at the time when quantum phenomena pass from potentiality to actuality.
Matter is Mind and Ontological Reality dependent. Mind is Ontological Reality dependent. Even from a dualistic position, in which Mind can exist without Matter in a non-material world, one can say that to be manifested in the material world Mind is Matter dependent.
Since there is no time interval in this proposal where Mind has not existed, there is no essential difference between Period A and Period B.
Mind and Matter coexist: neither of the two generates the other. Mind and Matter coevolved.
12.2. Ontological Reality as the cause of mind and matter
Cause is understood here in the general Aristotelian sense: cause as that from which something, and the properties of that something, come into existence; cause as the ultimate explanation of something; cause as the ultimate reason for something.
Even if quantum mechanics shows that the values of the variables are context-dependent, the abstract structure that the equations represent is valid in every context. The validity of this structure in every context is crucial to be able to reach intersubjectivity; it is a prerequisite for intersubjectivity; for making sense of the concept of empirical reality. The argument is the following: all the argumentations that lead us to the conclusion about the empirical nature of physical reality are based on the formalism, on the abstract structure that the equations represent. The Empirical Reality is constituted by everything that is intersubjective, which means it is true for everybody. For the conclusion about the empirical nature of physical reality being true for everybody, the abstract structure that the equations represent must be valid in every context.
The same applies to the non-contingent properties of particles: they are a prerequisite for intersubjectivity, for making sense of the concept of empirical reality. The argument is the following: first, the number of particles is contextual and implies that the existence of the particles is contextual. However, the non-contingent or constitutive properties of the particles are contextual. This contextuality does not mean however that the
contingent properties are ontological since the existence of the particles is non-ontological. Second, to have the possibility of reaching an intersubjective agreement about the particles, some properties of them must be context-independent. These properties are the constitutive or non-contingent properties.
Is it possible to consider the abstract structure of the equations as ontological? I have concluded that entities as the state vector that appear in the equations are not ontological. The only possibility would be to consider the relation between the entities as ontological. However, the relation itself is just a form whose content, its matter, are the entities that appear in the relation. One would have here an ontological form whose matter is non-ontological. Because the equations themselves require the conjunction of form and matter, one concludes that such equations are not ontological.
One could assert that the ontological forms when “embodied” in ER give rise to the equations of physics that give rise to physics. This evokes the ideas of Plato vaguely. D’ESPAGNAT (1995) introduces the idea of structural “extended causes”, that would be nothing less than the very structures of OR and would constitute “the ultimate explanation of the very fact that the laws – that is, physics – exist”. I think these possible ontological forms are closely related to d’Espagnat’s “extended causes”.
The relevant point here is that the answer to the question about the possible ontological nature of the equations is negative. The possibility of exploring the idea that the form of the equations is ontological remains however open. Structural realism is relevant at this point.
The above-presented analysis constitutes an argument for saying that the cause (understood as we said before in the Aristotelian sense) of the structures represented by the equations and the non-contingent properties of particles is Ontological Reality. It is claimed here that Ontological Reality is the rationale, or the cause, of Matter, of Mind and the relationship between Matter and Mind. To avoid an infinite regress of causes, one must consider that Ontological Reality has no cause.
For Aristotle, the concept of cause is ontological. In the present proposal, the cause is indeed considered as belonging to Ontological Reality. The Aristotelian concept of cause evoked here is similar to the “extended cause” of D’ESPAGNAT (1995, 2006).
Why do I use the Aristotelian meaning of cause? The interpretation of space-time as epistemic implies that relativistic causality is epistemic. As relativistic causality is a refinement of the usual general idea of causality, the latter is then only epistemic. The usual idea of cause is further subjected to the powerful criticism of Hume that led Kant to the conclusion that causation is only a human way of organizing phenomena. Therefore, causality is for him epistemological. The idea of cause I am proposing is ontological. For that reason, I chose the Aristotelian causality.
My proposal responds the question about the rationale, the ratio essendi of Matter, Mind and the relationship between them. There are several possible answers:
i) This question is a metaphysical question devoid of sense. This position implies that the concept of OR is devoid of sense. That conclusion is not consistent with my acceptance of the concept of OR as meaningful.
ii) The cause of Mind and the relationship between them is Matter. Matter would be then the ultimate reality. This is the materialistic position. In my interpretation, Matter is Mind dependent. Materialism, therefore, is not consistent with my interpretation.
iii) The cause of Matter and the relationship between them is Mind. Mind would be then the ultimate reality. This is the idealistic position. In my interpretation, Mind is Matter dependent. Idealism, therefore, is not consistent with my interpretation.
iv) The cause of Matter, Mind and the relationship between them is OR. It is the only possibility that remains.
12.3. Unicity of Ontological Reality
It does not seem reasonable to attribute multiplicity to Ontological Reality in the sense of conceiving the existence of several ORs. Which one of them would be the cause of Mind and Matter? How to choose between them? 12.4. Ontological Reality is Infinite
We have said that the state vector |Ψ⟩ belongs to ER and represents the potentiality that when a measurement of any operator of a complete set of commuting operators is performed, one of its potential eigenvalues is
actualized. Since in general the number of eigenvalues is infinite, this means that the potentialities of Matter are infinite. In the present proposal, there is a symmetrical position of Mind and Matter: Matter is Mind dependent; Mind is Matter dependent; they both coexist: neither of the two generates the other. This symmetry leads to the conclusion that the potentialities of Mind are also infinite.
Another conclusion of this infinity of potentialities of Mind and Matter is that OR, being the cause of both Mind and Matter, must be infinite. The meaning of the infinity of OR is as follows: OR is One, but it is manifested to Mind in the infinity of potentialities of Matter and Mind. The finitude of Mind is manifested in QM by the impossibility to describe physical phenomena in a single context. They are indeed describable by human language but not in a single context. One can describe them only in linguistic contexts that are mutually exclusive but refer to a same object. In other words, in complementary linguistic contexts.
13. General conception of complementarity
Is seems reasonable to expect that the expression of the finitude of Mind manifested in QM by the use of Complementarity is general. One can expect then a use of Complementary outside QM. One must remember however that what in QM determines the character of mutual exclusion is not logic but the mutual exclusion of the experimental contexts. The question about how to extend complementarity outside the scope of QM, in which one clearly defines the experimental contexts, is then valid and arises naturally. Besides, the question becomes more important if one considers fields where scientific observations are not made. A first approach to this issue was presented in ROLDÁN-CHARRIA (2007).
Concerning this question, it is necessary to take into account the indissoluble relation that exists between observation and theory. Two complementary, and, therefore, mutually exclusive concepts, cannot be used in the same reasoning. The just mentioned indissoluble relation between observation and theory implies then that analysis, questions, inferences that involve one complementary concept exclude analysis, questions, inferences that involve the other.
13.1. The indissoluble relation between observation and theory
Let us analyze some facts that have to do with the sense of sight. The following facts show that there is a lot more to a sight that what arrives at the eyes. If to see, or rather to make visual observations, one eye and a brain in proper conditions are enough, a group of visual stimuli will produce the same perception in every mind. There are various examples in this respect (KUHN, 1970;CHALMERS,1999) that show that this is not the case. I will consider only one. Let us think about a radiograph (CHALMERS, 1999). If there is not more to sight than what reaches the eye, all of us should see the same, independently of whether we have or not all the concepts and theoretical framework of a doctor. Besides, many of us who do not have these concepts when looking at a radiograph, do not see more that shadows where the doctor sees some maybe dangerous disease. The doctor has something that we lack, and this something is a theoretical framework, learned with effort through long years of study and guided experience. There are other examples similarly presented by KUHN (1970).
A fact that shows in a dramatic way the mutual dependence between theory and observation is that narrated by Oliver SACKS in his essay “To see and not to see”, included in his book An anthropologist on Mars (1996). Sacks describes the case of a person that has been blind almost all his life, because he became blind at infancy due to wrongly diagnosed cataracts. When he is close to fifty, the person is correctly diagnosed as having cataracts, and he undergoes an operation to remove them, adapting some glasses and recovering his sight. Contrary to what one might naturally think, maybe considering it as obvious, the process of starting to see is not just a mere adaptation to light or something like that, but rather the painful process for a person in his fifties to start constructing the fundamental concept of space, without which visual perception is not possible. The person in effect does not distinguish, in the beginning, his dog from his cat using his recently restored sight. He can only do it through touch. That is because his world has always been extended in time but not in space. His cat and his dog have not been from the sensorial point of view anything more than a series of successive tactile sensations. With the use of his eyes, the same happens at the beginning. Again, his cat and his dog are a group of partial images that he cannot integrate into one instant global image.
This type of instant global image is the essence of the concept of space that he must construct. All the drama involved in the construction of this concept that Sack narrates in his essay, combined with the fact that blind people start losing the concept of space little by little, indicates that theory impregnates all observation.
Facts are not something wholly independent of the theoretical framework, something whose observation the proper use of the senses is enough, where this proper use means the total detachment from any theory. Facts do not exist independently of the theoretical framework, which will imply that the senses observe them better as the mind detaches from theory. If, as it seems, the facts and the theory form an indivisible wholeness, if all facts are impregnated or tainted by theory, does it mean that the physical reality is subjective and wholly arbitrary? The answer is negative, in the case of the blind man that begins to see, once he creates the concept of space, he will see the same as the rest of us. The conclusion is that the scientific observations are intersubjective. Intersubjectivity is a form of objectivity in the sense of independence from the particular subject, although not from the subject in general. Therefore, the empirical, the phenomenal is objective.
The indissoluble relation that exists between experience and theory implies that theory impregnates all experience, and the latter is as precise and fallible as the theoretical framework that sustains it. Lastly, an experience is a network of concepts, and for that reason, if two experiences are mutually exclusive this means that the networks of concepts linked to them are mutually exclusive.
The previous conclusion allows us to hope that it is possible to extent complementarity to fields different from physics. If two networks of concepts are mutually exclusive, then so is all the reasoning based on these networks of concepts. Nevertheless, because they are complementary, both networks of concepts are required to exhaust all that one can know about the same object. Therefore, we cannot hope to exhaust all the descriptions of the object with a single network of concepts. Moreover, neither can we hope for an ontological description of the properties of the object, because all of them appear to us as contextual, relative to a network of concepts that is not the only one possible concerning the behavior of the object.
Complementarity opposes conceptual or philosophical monism because it sustains that a unique conceptual network does not exist to contain
all phenomena. The world shows up as multi-contextual, in a variety of necessary but mutually exclusive contexts. Now, although complementarity sustains that each experience is relative to a context, it should be remembered that once we find the context, the use of the language is once and for all fixed.
From where does the possibility of mutually exclusive concepts arise? In QM, what opens this possibility is the indivisibility of the phenomena that includes the instrument of observation. One can expect then that joined to any eventual use of complementarity there is always a type of indivisibility or totality.
14. Methodology to validate a proposal of complementarity
The general methodology to validate a proposal of complementarity between two conceptual networks is the following:
a) Identify the characteristics of each conceptual network.
b) Show that they are indeed mutually exclusive, which means that questions and analysis that are valid in the context defined by one conceptual framework are not valid in the other one.
c) Show that the two conceptual networks have the same object. The common object justifies the relationship between two mutually exclusive conceptual networks. Once the common object is identified, it is clear that both conceptual networks are required to exhaust all available information about it. As noted earlier, the analysis of the possible uses of complementarity outside quantum mechanics will show if the information is about the object or part of the object.
15. Applications of complementarity
There are several proposals for possible uses of complementarity in different areas of knowledge. To discuss all of them would make this paper unduly long. I will thus present here only two that I thing are more relevant to the general argument.
15.1. Complementarity between dynamics and thermodynamics
The difficulty of reconciling irreversible phenomena observed in the macroscopic domain with the reversible dynamical laws is well known. For simplicity, we will make the analysis within the context of classical mechanics.
In thermodynamics one express irreversibility by the entropy and the second law of thermodynamics. In order to reduce thermodynamics to dynamics, one would have to express the entropy, which is a macroscopic variable, in terms of the state variables of the atoms that make up the macroscopic body. Poincaré proved by a mathematical theorem that this is not possible. It becomes necessary to use, in addition to the mechanical, other concepts in order to find an explanation to thermodynamic irreversibility. The second law of thermodynamics is then irreducible to the dynamics.
It is generally accepted that such a reduction is achieved by statistical mechanics, which uses the concepts of probability. In the usual formulation, the idea of a statistical ensemble is introduced because it is considered in practice impossible to know the exact dynamic state of a macroscopic system. The use of probability comes because of the ignorance of the observer. It would seem that irreversibility is the result of our ignorance, something wholly subjective. This situation results in a dispute between the objectivist and subjectivist interpretations of irreversibility, which have to do with the way one should interpret the notion of probability.
BOHR (1932, 1957) suggested a purely epistemological interpretation of irreversibility, proposing a complementarity between the dynamical and the thermodynamical concepts. One can understand Bohr’s idea in the sense of mutual exclusion between the experimental contexts that define the dynamical concepts and those that defines the thermodynamical concepts. However, both types of concepts are required to exhaust all possible information about a macroscopic body. In other words, both types of concepts refer to the same object: a macroscopic body. This proposal of Bohr implies a kind of indivisibility related to a macroscopic body.
One can argue that the formalism of statistical thermodynamics indicates that Bohr’s argumentation is reasonable. Gibbs’s definition of entropy is 𝑆 = −𝑘 ∑ 𝑝𝑖 𝑖ln 𝑝𝑖, where 𝑝𝑖 is the probability that the system is in the dynamic state 𝑖. The temperature is expressed as 1
𝑇
⁄ = 𝑘(𝜕 ln Ω 𝜕𝐸⁄ )
𝑉.
Context A: This context gives us the precise knowledge of the dynamical state of the system; such knowledge in classical dynamics is possible in principle. That knowledge is equivalent to knowing that the probability of one of the dynamical states is 1, and the probability of the others is zero. In this case, Gibbs expression reduces to a 𝑆 = −𝑘 ln 1 = 0. One cannot define either entropy or temperature, or irreversibility. Indeed, if it is not possible to define
𝑆, there is no second law of thermodynamics, which is the expression of
irreversibility.
Context B: This context allows us to define the thermodynamical variables. In this context, the precise dynamic state of the system is unknown, which means that no dynamical state has probability 1.
Both experimental contexts and their corresponding concepts, the dynamical and the thermodynamical, are mutually exclusive, but refer to the same object: the thermodynamical system. They are therefore complementary.
The solution proposed by Bohr is, of course, valid also for the quantum domain. What is crucial to remark is that the proposal is an application of complementarity to classical physics.
15.2. Complementarity between physics and biology
Throughout history, there has existed, and there still exists, a conflict between two positions regarding the relation between biology and physics. One of them considers that a living organism is nothing more than an enormous and complex set of atoms from whose interactions the laws of life arise; in other words, biology would be reduced physics. The other position does not accept such a reduction and insists that to understand how an organism works one must first know its position and function in the body; in other words, biology would have special laws not reducible to those of physics.
I am not going to discuss the reductionist and non-reductionist positions in biology, nor talk about the ideas concerning this subject that the work of Prigogine on dissipative structures can imply. Neither will I discuss the validity of the properly biological considerations of Bohr. All this would exceed the aim of this article. My interest is to see if the sense in which Bohr proposes to use complementarity in biology is the same as the sense in which he uses the notion in quantum physics. More generally, I want to analyze whether the
proposed use corresponds to the precise definition of complementarity that I presented above.
BOHR (1957) proposed an original solution using the ideas of complementarity and indivisibility that may be expressed as follows: living organisms constitute an indivisible wholeness. This indivisibility means that questions about them, formulated to inquire whether their behavior is reducible to the physical laws of their atoms or if they have their own laws irreducible to physics, are questions that one can formulate only by specifying an experimental context.
Let us consider an experimental context suitable for the analysis of the atoms of an organism, with the same methods and the same accuracy as used in an experimental context suitable for the analysis of the atoms of an inert body. The result will be that the atoms of the organism obey indeed the physical laws to which all atoms are subjected. The network of concepts for the description and explanation of the experiment will be those of physics. In such experimental circumstances, however, the organism will be deprived of its life.
If we suppose now another experimental context, in which one renounces to analyze the atoms of the organism with the same methods and the same accuracy as used in an experimental context suitable for the analysis of the atoms of an inert body, the living properties of the organism will be manifested. The network of concepts for the description and explanation of the experiment will be those of biology.
The two experimental contexts are mutually exclusive: the state of an inert body is mutually exclusive in relation to the state of an organism. The concepts defined in both types of experimental contexts are required to exhaust all information about an organism. In other words, they have the same object: the organism.
Behind each use of complementarity, there seems to be a type of totality or indivisibility. This is a point that is necessary to investigate in each use of complementarity. In quantum physics, one confronts the indivisibility of the quantum phenomenon that constitutes a totality that includes the instrument of observation. In thermodynamics, there also seems to be an indivisibility that would be behind the complementarity between dynamics and thermodynamics. In biology, the issue would be a matter of the totality of the organism.
16. Complementarity and logic
Given that logic does not define the mutual exclusion of two complementary concepts, there isn’t any inconsistency implied by complementarity from the point of view of classical or ordinary logic. One can consider complementarity as a new logical relationship that is, however, not radically different from ordinary logic, as is the case with quantum logic, because one uses ordinary logic in each experimentally defined context. Consequently, there is no need for any new logic in order to insert complementarity. On the relation between complementarity and logic, see DA COSTA &KRAUSE (2004).
Submitted: October 30, 2015; accepted: June 30, 2016. Bibliographical references
BELL,J.S. (1987). Speakable and unspeakable in quantum mechanics: collected papers on quantum philosophy. Cambridge (GB): Cambridge University Press. BOHM,D. (1952). A suggested interpretation of the quantum theory in terms of
hidden variables. Parts I. II. Physical Review 85: 165-79, 180-93.
BOHM,D.&HILEY,B. (1993). The undivided universe: an ontological interpretation of quantum mechanics. London: Routledge & Kegan Paul.
BOHR, N. (1929). The quantum postulate and the recent development of atomic theory. Nature 121: 580-90.
–––––– (1932). Chemistry and the quantum theory of atomic constitution. Faraday Lecture. Journal of the Chemical Society 1932: 349-84.
–––––– (1934). Atomic theory and the description of nature. Cambridge(GB) : Cambridge University Press.
–––––– (1957). Physique atomique et connaisance humaine. Paris: Gauthiers-Villars. –––––– (1963). Essays 1958–1962 on atomic physics and human knowledge. New
York: Interscience.
BROGLIE,L. (1927). La mécanique ondulatoire et la structure atomique de la matière et du rayonnement. Journal de Physique et le Radium 8(5): 225-41. CHALMERS,A.F. (1999). What is this thing called science? An assessment of the nature
and status of science and its methods. Buckingham (GB): Open University Press.
DA COSTA, N.C.A. & DE RONDE, C. (2013). The paraconsistent logic of quantum superpositions. Foundations of Physics 43(7): 845-58.
DA COSTA, N.C.A. & KRAUSE, D. (2004). Complementarity and paraconsistency. In: Rahman, S.; Symons, J.; Gabbay, D.M. & van Bendegem, J.P. (eds.). Logic, epistemology, and the unity of science. Dordrecht: Springer, pp. 557-68.
D’ESPAGNAT,B. (1995). Veiled reality: an analysis of present-day quantum mechanical concepts. Reading (MA): Addison-Wesley.
–––––– (2006). On physics and philosophy. Princeton: Princeton University Press. EVERETT III, H. (1957). “Relative state” formulation of quantum mechanics.
Reviews of Modern Physics 29: 454-62.
GHIRARDI, G.C.; RIMINI, A. & WEBER, T. (1986). Unified dynamics for microscopic and macroscopic systems. Physical Review D 34: 470-91. HEISENBERG,W. ([1958] 1970). Physique et philosophie. Paris: Albin Michel. KUHN,T.S. (1970). The structure of scientific revolutions. 2nd ed. Chicago: University
LEGGETT, A.J. (1984). Schrodinger’s cat and her laboratory cousins. Contemporary Physics 25: 583-98.
–––––– (1997). The current status of quantum mechanics at the macroscopic level. In: Nakajima, S.; Murayama, Y. & Tonomura, A. (eds.). Foundations of quantum mechanics in the light of new technology. Advanced Series in Applied Physics, vol. 4. Singapore: World Scientific, p. 257-66. ROLDAN-CHARRIA, J. (1991). Language, mécanique quantique et réalité. Thèse de
Doctorat. Paris: Université de Paris 1.
–––––– (2007). Complementarity, knowledge and reality. Presentation in the conference Transdisciplinarity and the Unity of Knowledge. Philadelphia: Metanexus Institute.
–––––– (2014). Indivisibility, complementarity and ontology. A Bohrian interpretation of quantum mechanics. Foundations of Physics 44: 1336-56. SACKS,O.W.(1996). An anthropologist on Mars: seven paradoxical tales. New York:
Vintage.
SCHRÖDINGER,E. ([1935] 1980). The present situation in quantum mechanics: a translation of Schrödinger’s ‘cat paradox’ paper. Proceedings of the American Philosophical Society 124: 323-38.
WHEELER,J.A. (1984). Bits, quanta and meaning in problems. In: Giovannini, A.; Mancini, F. & Marinaro, M. (eds.). Problems in theoretical physics. Salerno: University of Salerno Press, pp. 121-41.
WIGNER,E. (1961). Remarks in the mind-body question. In: Good, I.J. (ed.). The scientist speculates. London: Heinemann, pp. 284-302.