The Pauli Exclusion Principle from the Point of View of Group Theory

7In short, the possibility of an objective equality (of a symmetry, of a congruence, and consequently of an invariant) in the sense of group theory implies a violation of the principle of indiscernibles. [7] Thus, to start with the simplest case, “the objective equality or the ‘indiscernibility’ of all the points of the Euclidean space” simply means this: “for any two 8 points p1 and p0 there is always an automorphism carrying p₀ in p₁” [8].

8That two atoms are interchangeable and nonetheless discernible in the space-time of classical physics implies two complementary idealizations, [9] which sub-tend the determinism and exactitude postulated by classical measurement theory. It is understandable that, under these conditions, classical mathematical physics made Euclidean space the principle of physico-mathematical individuation par excellence of elementary physical bodies (atoms) which are otherwise indiscernible. As Weyl admirably writes in very dense propositions, it is because in the sensible experience “the mere here is nothing by itself that might differ from any other here” that “the space is the principium individuationis” by rendering “possible the existence of numerically different things which are equal in all other aspects” [10].

9Physically speaking, it is considered that two atoms are sufficiently individuated if it is possible to assign them a single spatiotemporal place, [11] a place expressed by a quadruplet of numbers. Any measurement in the mathematical sense is an application and a restriction of the principle of complete determinability, which itself follows, as we have said, from the principle of sufficient reason. This violation of the principle of indiscernibles, since individuation occurs only extrinsically, is not less in the context of relativistic physics, despite a certain relaxation affecting the conditions of automorphism allowing the transport of point to point of a Riemannian manifold. Weyl expresses this situation by means of a famous analogy, comparing Euclidean space “to a crystal, constructed from unalterable uniform atoms, in an unchanging regular and rigid arrangement of grids; and Riemann space to a liquid, consisting of indiscernible, immutable atoms, whose arrangement and orientation, however, are mobile and subject to forces acting upon them”. Referring again implicitly to the Pauli Principle, Weyl asserts that quantum physics offers a better formulation although different from Riemann’s conception “when the quantities characterizing an electron (and its spin) must be adjusted to the theory of general relativity” [12].

10Weyl interprets the Pauli-Leibniz principle from the point of view of group theory. Clearly, this was still a minority perspective during that period. The PEP had indeed suggested that we were going to be able to get rid of the “plague of the groups” in physics. Against this “rumor”, Weyl endeavored, from 1928 and even more clearly in the 1931 edition of his essay, to defend the structural approach, without being locked into the probabilistic interpretation promoted by Max Born. Groups (groups of rotations and groups of Lorentz transformations, and groups of permutations) are at least useful for understanding spin. As Weyl specifies, “it seems that the PEP does not allow us to avoid it”: “the theory must adopt the representations of the group of permutations as a natural instrument to obtain an understanding of the relations caused by the introduction of the spin, as long as that its specific dynamic is neglected. I have followed the trend of the time, as far as it is justified, by presenting portions of group theory in as elementary a form as possible” [13]. He thus tackles the fundamental problem of atomic structure, and consequently of physical properties and of quantum laws, from the point of view of the “properties of symmetry”, and tries to show that they admit permutations “of right and left, past and future, and negative and positive electricity” [14].

The Status of the PEP Remains AmbiguouS

11The PEP is derived from “Bohr’s law of frequency”, whose explanatory value (fertility) does not need to be demonstrated and which can therefore be considered as confirmed experimentally: “This law gives the key to explaining the vast set of very precise observations amassed by specialists in spectroscopy on the subject of the emission of spectral lines, by irradiating atoms and molecules”. But it only becomes fully explanatory if we add the PEP to it: “we only get full confirmation if we add the hypothesis that two electrons cannot be in the same complete state (the Pauli exclusion principle).” Hence increased fertility since one can establish a quantum theory of chemical links and “mechanically” explain the periodic table of elements [15].

12However, the logical status of the PEP remains ambiguous. And its position despite its success remains affected by a certain circularity. Is this a fecund assumption or the “implication” of a rule of thumb? Stoner’s rule led to the PEP, and vice versa. This circularity can be found in recent introductory QM textbooks. [16] “Stoner’s rule led Pauli to postulate the exclusion of equivalent orbits: it is impossible for two electrons in an atom to be simultaneously in the same quantum state (n, l, j, m). This shows that ℜ^f is obviously not the space of the physical system 1-f in which f electrons revolve around a fixed nucleus, but that the reduction at {ℜ^f} takes place: nature has decided in favor of the reduction to the space of antisymmetric tensors, at least for the electrons. In view of the considerations of the previous paragraph, this reciprocally leads to Stoner’s rule” [17].

13The introduction of the fourth quantum number, the spin, thanks to the PEP, leads to a success that only makes its proper mathematical shaping more urgent. Weyl refers here “to the introduction of the intrinsic quantum number j in addition to the azimuthal number l, or the spin of the electron, on the one hand, and to the reduction of ℜ^f to {ℜ^f} by means of the Pauli exclusion principle, on the other hand. Millikan begins his American Philosophical Society review of “Recent Developments in Spectroscopy” with these words: “Never in the history of science has a subject suddenly passed from a state of utter obscurity and unintelligibility to that of full clarity and predictability as the field of spectroscopy since the year 1913”. Group theory offers an appropriate mathematical instrument for the description of the order thus achieved” [18].

14The problem, in the terms of group theory, is formulated as follows: find the finite symmetric group of permutations and its representations, then build on this basis the antisymmetric group corresponding to the application of the PEP. “The main problem that we propose to solve in this chapter is the group-theoretic classification of the spectral lines of an atom made up of an arbitrary number of electrons, say f, taking into account the reduction of the space ℜ^f to {ℜ^f} as required by the Pauli exclusion principle, and of the electron in rotation. For this, it is necessary to consider in detail the representations of the symmetric group, i.e. the group of all f! permutations of f things. These are more intimately linked to the representations of the group of all unitary transformations or of the group of all homogeneous linear transformations of a space ℜ_n” [19]. The process of individuation is thus reinterpreted within the framework of groups. The indiscernibility of particles (fermions and bosons) is nothing new compared to classical physics. This is how the indiscernibility of aggregates is maintained as long as they are of the same constitution (or of two individuals in the same “state” [20]). It only makes sense to restrict the notion of completeness to that discrete manifold that is the “system of complete states of an individual” whose wave function provides a “complete description”. To the series of predicates corresponds here the enumeration of the probabilities (of possible states) within this abstract space which is the “phase space”. On the other hand, it is fundamental for this transfer into physics of a metaphysical principle to be successful that the nature of the individuals thus described be experimentally verifiable. This is how the photons are posited, according to Weyl, as “individuals without identity”, while the free electrons or those bound to their nucleus constitute individuals endowed with a strong identity. This explains why the wave and particule aspects are inversely evident for bosons and fermions. [21] It is in this sense that we must understand the identification of Pauli’s principle with Leibniz’s metaphysical principle.

The Idea of a “Probabilistic Individuation”

15The result of this transfer is ambivalent to say the least: if on the one hand Weyl declares that “the consequence of all this is that the electrons satisfy Leibniz’s principium identitatis indiscernibilium, or that the electronic gas is a “monomial aggregate” (described by its Fermi-Dirac statistics), it is at the cost (1) of restricting the “Pauli-Leibniz exclusion principle” to electrons only (2) of its retrogradation to the rank of principle applied only to phenomena, and (3) finally of a denial, as in classical physics, of all individuality to the photon as well as to the electrons. “In a deep and precise sense, physics corroborates what the Mutakallimûn said: one cannot ascribe individuality to either the photon or the electron (positive and negative). As for the Leibniz-Pauli exclusion principle, it turns out that it only applies to electrons and not to photons” [22]. To put it in other words, fermions are “equal” or identical “specifically or generically”, not identical in Leibniz’s sense, i.e. individually. [23]

16To borrow Weyl’s terms, here we are witnessing the return of the ghost of probability (as a modality) behind the PEP. In “classical mechanics”, the principle of complete determination postulates the existence of a “state of a punctual mass (or charge)” completely describable “by its position and its speed” and is articulated intimately with the causal principle of determining all successive states from an initial state. Quantum mechanics sees in the “state” of a particle “a superposition” of possible states and the only physically determinable and measurable state. The complete states of an individual (electron) form a discrete manifold (of possible states) of which the statistic (of Fermi-Dirac) simply proposes the enumeration. This background of possible superimposed states will justify the transfer to QM of the principle of indiscernibles, which Weyl calls for this very reason “Pauli-Leibniz principle”.

17Thus reformulated, the Pauli-Leibniz principle receives two major limitations: (1) it only applies to electrons assumed to be otherwise interchangeable; (2) it limits itself to characterizing the superposed wave states within the framework of a probabilistic theory where the principle of independence of possible states is denied. Mathematically, this negation is a consequence of the use of antisymmetric tensors. [24] So it is finally within the framework of the properties of formalism that this principle finds its true interpretation: “The Leibniz-Pauli exclusion principle, according to which no two electrons can be in the same state, becomes understandable in quantum physics, and this is a consequence of the law of antisymmetry”; “The permanent antisymmetry of the wave state thus explains the Pauli exclusion principle. The statistical independence of the quantum states of two electrons could not be denied in a more radical way than by this principle!” [25]

18The clarification of the notion of probabilistic dependence (relative or conditional probabilities) leads to a new dialectic of the objective and the subjective, and of the a priori and the a posteriori, if we compare it to the dialectic specific to the theory of relativity. [26] This is why the “primary probability” [27], “which has nothing to do with the knowledge or ignorance of the observer” but expresses “certain basic physical quantities and can in general only be determined on the basis of empirical laws governing these quantities” [28]. Instead of the probability being determined on the basis of an a priori grid of a space or of a prior distribution of “rigid” entities placed in a homogeneous, isotopic space is outlined, under the title of “play space” (Spielraum [29]), a completely different “space”, the mathematical essence of which still needs to be grasped.

19In its modern (set theoretical) approach, the axiomatization of probability provides an admirable framework for the construction of a “probability space”. In consonance with Husserl who, to prevent the most widespread interpretations, warns that “both probability and certainty are subjective expressions”, but without falling for all that into a subjectivist interpretation of probabilities (Laplace, De Finetti), Weyl reminds us that the determination of the type of entity, its “typification” so to speak, proceeds from an arbitrary subjective decision. However, the “choice” of the level of ontological division conditions (in a non-causal way) the delimitation and definition of the “play space” (Spielraum) in which we carry out the enumeration of the possibilities, the establishment of their dependencies or mutual independence is the measure of probability. The determination of what the elementary is cannot do without this residue of subjectivity that are choice and decision: it is “our decision to consider such and such things equal or different [which] influences the account of “different” cases on which the determination of probabilities is based “and which determines the nature of what is called, element, event, and consequently the physical meaning of what is designated as probability functions. Thus formulated, the “problem of individuation touches the roots of the calculus of probabilities”. However, the fundamental mathematical framework adequate to describe such a generative process is, according to Weyl, “the combinatorial theory of aggregates”, understood as a discrete manifold (finite or infinite) provided with a group structure. It is only in this way “that these things find their adequate mathematical interpretation, and one hardly finds another branch of knowledge where the relation of idea and mathematics is presented in a more transparent form” [30], idea being here a metonymy for philosophy.

20What is measured as probability are, in a very general sense, the degrees of freedom, in a physical sense as well as in a moral sense. These are the last words of Philosophy of Mathematics and Natural Sciences: “Indeed, the example of quantum mechanics has once again demonstrated how the possibilities with which our imagination plays before a problem is ripe for a solution are still largely out of date. Even so, the explanation of chemical bonding by the Pauli exclusion principle is perhaps a clue that the radical break with the classical scheme of statistical independence is an opening of the door as significant as the complementarity in question in quantum mechanics. [31]”

Philosophical Implications of PEP Before 1952

25De Broglie’s examination of Pauli’s principle aligns with Weyl’s analysis. Both the genesis and the applications of the Pauli principle signal a profound modification of the individuation principle prevailining until then in physics (including in the context of relativistic physics). The individuation of the body by a space—or a space-time—fades in favor of an individuation of a system and of the function Ψ associated with it.

26It is undoubtedly one of Broglie’s key theses that individuality and system are two complementary idealizations. [37] In classical mechanics, two particles of the same nature are identified by space or by their location in the space of permutations. Differences in the location of individual particles make for different systems. In quantum mechanics, particles lose this individuality in favor of a global characterization of systems and of the Ψ function associated with them. For any system comprising pairs of identical particles, there always exists a Ψ function which is symmetric or antisymmetric with respect to all the pairs of particles. The system will be symmetric or antisymmetric depending on whether the Ψ function is one or the other. It is impossible for it to be otherwise. However, while the Pauli exclusion principle is fully consistent with the other ingredients of quantum formalism, and has received a number of empirical confirmations, its “physical origin” remains mysterious according to de Broglie.

27De Broglie’s interpretation of the PEP in La Physique nouvelle et les quanta [38] calls for two remarks, which touch at the same time on the scope, the status, and the content of the PEP. Is the latter a real postulate or a rule of thumb? De Broglie asserts that it is analytically and a priori verifiable. But a validation is also given a posteriori, because of its fecundity: heuristic value and experimental confirmation.

28It is first of all a “postulate” which is susceptible to being a priori proved, i.e. mathematical and analytical. Thus, to establish a formulation equivalent to that of Pauli, de Broglie proposes a reductio ad absurdum: “Let us suppose that a system contains two electrons in the same individual state; if we admit, in accordance with the second statement, that the wave function is antisymmetric with respect to this pair of electrons, its sign must change if we swap the role of the two electrons, but since the two electrons are in identical individual states, this permutation can in no way modify the wave function: the wave function thus having to both not change and to change sign by the effect of the permutation, is necessarily identically zero and this vanishing of the function means in the new mechanics that the supposed state is non-existent.” Conclusion: “Therefore, there cannot be two electrons in the same individual state and we see that the second statement leads us to the first: the reciprocal is demonstrated just as easily”.

29But to grasp its physical meaning, it is also necessary to establish its a posteriori validity. What de Broglie does by showing its heuristic value and by exhibiting the experimental “facts” which confirm it. Among these facts, there are first those which motivated its formulation, in particular Stoner’s “semi-empirical rule” concerning the distribution of the particles according to the value of their spin. But above all, de Broglie deduces the existence of a new form of energy which he calls “exchange energy”, “a kind of interaction”. This leads to an interpretation and a physical clarification of the origin of PEP, [39] which breaks with classical representations.

30In Continu et discontinu en physique moderne, de Broglie endeavors to take the full measure of this rupture. From the comparison between classical physics and quantum mechanics, it emerges that, even in the first, the principle of individuation has a limited scope and that mass and location are interdependent. The dialectic between these two idealizations of the notions of “system” and “individual” is already at work in classical physics. [40] We approach the expression in terms of group with local transformations and the invariance of mass. We understand in what sense mass is a “being of reason”. The principle of individuation is therefore itself a priori: it is a priori impossible for two particles to be in the same place at the same time.

31But this principle is based on certain presuppositions as to the nature of time, in particular its continuity, and it forces us to adopt a distinction between two distinct principles of individuation, for cases where two particles are identical: spatio-temporal individuation and complete individuation. The latter brings us closer to Leibniz. [41] But, as we have said, interaction hinders individuation and forbids considering “too absolutely the individual autonomy of the particles.” To explain interaction, classical physics was led to introduce the idea of potential energy. Now this idea, which is “very clear from a mathematical point of view […] remains physically quite mysterious” [42]. Unlike kinetic energy, potential energy cannot be individuated (“distributed among the constituents of the system: it belongs to the whole system and is, as it were, pooled by its constituents” [43]).

32In a movement of audacious generalization and passage to the limit, de Broglie sees in the relationship between “individuality and interaction”, one of those complementarities “that M. Bohr has been led to consider in his interpretation of quantum theories” [44] and which are only fully intelligible as two instantiations of the two abstract idealizations that are the notions of individual and system.

33The PEP appears under these conditions as a particular case of interaction and an application of Heisenberg’s principle, attributable (1) to the absence of individuation of particles of the same nature and (2) to the impossibility “in general” of supplementing this by locating “in our framework of space the elementary physical entities” [45]. Hence the encroachment of the regions. [46] But this account conceals a form of hidden “paralogism”, since the location denied on one hand is restored on the other. Strictly speaking, “one cannot say that two particles, whose states of motion are assumed to be exactly known, are distantfrom each other: we can just as easily say that they are in contact since they occupy both, sort of potentially, the entire recipient. This subtle argument clearly shows us that exclusion is closely related to the non-localization of physical units in space. Its existence therefore shows us once again how questionable our traditional conceptions of space are. Moreover, we can consider exclusion as a new form of interaction that is specifically quantum and different from the exchange energy” [47].

34The dialectic here takes on a quasi-Kantian meaning. It is in fact due to the inevitable antinomy that results from the double idealization of the system and of the physical individual. The solution lies in a “compromise”, which falls between “two extreme idealizations”—a compromise illustrated in the case of classical physics by the notion of “potential energy”. It is such a compromise that wave mechanics offers, introducing a new form of energy: “exchange energy”.

35

In short, there is a certain antinomy between the idea of independent individuality and that of a system where all the parts interact. Reality, in all its domains, seems to be intermediate between these two extreme idealizations and, in order to represent it, we must seek to establish a kind of compromise between them. Physics did not escape this fate and, in its classical form, it tried to achieve the compromise thanks to the notion of potential energy of interaction between particles. Although on close examination this compromise appears to be quite bastard, it nevertheless allowed a large number of facts to be represented on a macroscopic scale and for a long time seemed sufficient.
The situation became much worse when Quantum Physics, studying facts on the microscopic scale, realized that elementary entities could no longer be exactly located in space. (1) This fact, so surprising at first glance, made it impossible to attribute to the particles an individuality that could be constantly monitored and recognized: we studied the resulting complications. (2) Moreover, the possibility for several particles to occupy simultaneously, at least in a potential way, the same region of space, causes the appearance of new forms of interactions ignored by classical physics: the exchange interaction and exclusion interaction. The existence of these interactions is now physically certain, their importance undoubtedly capital, but their interpretation still totally obscure. In quantum physics, the compromise to be made between individuality and interaction therefore appears much more difficult to conceive than in classical physics: it must account for complex and surprising facts for our habits of thought and it will certainly not be able to be developed as part of our old ideas about space. [48]

36This diagnosis appears clearly prophetic, if we consider the contrast between the power of the mathematical formalism of quantum mechanics and the remaining problems and puzzles of its interpretation (which is inseparably physical and metaphysical). To conclude with the words of de Broglie this overview of the problem of individuation as it arose for him before 1952, we can say that the “nature” and “the true meaning” of exclusion remain unknown, hidden, [49] and that the origin of these difficulties seems to lie in the “shortcomings of our conceptions about space and time” [50].

After 1952. Critique of Bohm and critical reflection on Pauli’s principle

37In La Théorie des particules de spin 1/2 (Électrons de Dirac) begins a critique and an attempt to eliminate what he now calls the “Pauli conditions”, by showing that the theory thus reformulated is equivalent to the theory integrating Pauli’s principle, that it has an advantage, in terms of prediction, and that it dispenses with a priori (unjustified) principles.

38The steps are as follows: first, give a more precise form to the uncertainty relations. [51] Then a reformulation of the PEP, using the Wentzel, Kramers and Brillouin (W.K.B.) method. He then establishes the equivalence of the two theories, even the superiority of the new one over that of Pauli, “because our result proves that, even in Pauli’s theory, the action of the electromagnetic field on the proper moments intervenes in the expression of functions C₁ and C₂, therefore, in that of formula b_k⁽⁰⁾, that is to say to the zeroth-order approximation”—which Pauli seems to exclude. The “first advantage” over that of Pauli is that it translates into a new notion, that of “group speed” corresponding to the phase of the Jacobi function S = S’₀ + ∫ U dt”. The second advantage is that it leads to different predictions, which allows us to hope to decide experimentally between the two theories. [52] Finally, it makes it possible to downgrade this principle of a priori postulate to the rank of particular case. [53]

39When this fascicle appeared in September 1952, David Bohm published in the Physical Review two articles promoting a theory, opposed to orthodox theory, and whose starting point was de Broglie’s pilot wave theory of 1927. These two publications woke de Broglie from his sleepy compromise and convinced him to revert to his old unpublished theory, under the title of double solution theory.

40The 1956 essay opens with a critique of both the orthodox interpretation and that of Bohm, who wrongly admits that the Ψ function expresses a physical reality, and therefore considers renormalization, which is called “wave function collapse”, as a physical process produced by observation. Such a hypothesis is “inadmissible” for de Broglie. Contrary to the probabilistic (“orthodox”) interpretation and Bohm’s theory, the theory of the double solution is based on a generalization of the wave-particle duality and the coupling of the abstract wave (expressed by the Ψ function) to a physical wave expressed by (u), the first being subjective and non-relativistic, and the second being objective and relativistic.

41The reasons why de Broglie rejects the orthodox interpretation, which tends to transform a pure “formal” (fictitious and abstract) entity into physical reality, are essentially negative. For it to have a physical meaning, it would be necessary to be able to assign to the “states” represented by the values of the function a system of coordinates. However, the coordinates have no meaning for the Ψ function. It is therefore succumbing to a strange illusion to endow the Ψ function (and the wave it represents) with an objective physical meaning. To understand this, de Broglie traces the mathematical genesis of this function (Lagrange, Jacobi, Hamilton) which is inseparable from the construction of this abstract space which is called “configuration space”. The latter has as many dimensions as the physical system considered comprises free particles (i.e. 3N coordinates for N particles), coordinates whose meaning is “fictitious”, since they correspond to possible simultaneous locations, combined with dynamic parameters (such as electric charge, etc.). A trajectory of a point representing a particle corresponds, in this context, to a state of the system. The interpretation of this function as representing a physical process in three-dimensional space is therefore surprising to say the least. It is certainly not a priori excluded that the two may coincide, as is the case in classical physics. If recourse to configuration space is a necessity within the framework of quantum mechanics, de Broglie nonetheless indicates a horizon which is also a limit: the need to go beyond our usual concepts of physical space, that of particle, and consequently that of individual entity in favor of more adequate conceptions.

42It is within this new perspective that Pauli’s principle must be interpreted. This principle has a mathematical sense in configuration space, that of the probabilistic wave function. "Let us consider two particles of the same nature, two electrons for example, they are so similar that it is impossible to attribute an individuality to them: it is one of the essential results of quantum physics to have brought to light this "indistinguishability" of particles of the same nature”. The following develops the consequences on the measure (i.e. on the "probability amplitude").

43We must therefore admit that any observable quantity, such as [Ψ]², must be insensitive to any permutation of the role of the particles. This leads to restricting the possible shape of the wave functions. As interactions of particles are always symmetric functions of their coordinates, if we have found a solution Ψ(x₁, y₁, z₁,… x_i, y_i, z_i,…, x_k, y_k, z_k,…, x_N, y_N, z_N, t) of the wave equation, the function that we obtain by permuting the role of the corpuscles i and k, that is to say Ψ(x₁, y₁, z₁,… x_k, y_k, z_k,…, x_i, y_i, z_i,…, x_N,y_N,z_N, t), is still a solution, as well as any linear combination of the two solutions thus obtained of the form a Ψ(x₁, y₁, z₁,… x_i, y_i, z_i,…, x_k, y_k, z_k,…, x_N, y_N, z_N, t) + b Ψ(x₁, y₁, z₁,… x_k, y_k, z_k,…, x_i, y_i, z_i,…, x_N, y_N, z_N, t) [54]

44This indiscernibility results in the insensitivity of “linear combinations” to permutations of signs expressing the permutation of particles (i and k). The general formula follows: “The wave function Ψ of a system which contains particles of the same nature must be either symmetric or antisymmetric with respect to all of its constituents”, depending on whether the permutation produces a change or not of the sign “without changing absolute value” [55].

45The Ψ function has a purely mathematical and symbolic function, and has a subjective or imaginary meaning. It is in the passage from this abstract identification to the location of the physical entity that the Pauli principle intervenes, and the division of “physical entities” into two categories of apparently tight states if we follow the two statistics which correspond to them: Fermi-Dirac and Bose-Einstein. [56] The fact remains that it is in this transition and through the application of this principle that an experimental verification becomes possible and thinkable.

46However, Pauli’s postulate and Fermi-Dirac’s statistics represent a challenge for the double solution theory, because the latter is led to a manifestly contradictory postulate: the “upholding of the notion of trajectory” [57]. The difficulty is overcome if we admit that the “wave trains u overlap partially” to form a “single wave” that can be expressed by a formula which preserves the distinction between two distinct mobile regions”, and nevertheless a possibility of fusion, which would explain the behavior of the bosons. De Broglie limits himself here to showing that his theory is compatible with the PEP, and not to offering a critique. So he concludes by indicating what is missing for his theory to be equivalent to the orthodox version. It remains to justify the division between symmetric and antisymmetric (with bosons grouping together in clusters or herds, and fermions always “appearing” in isolation), and in order to do so, to introduce the notion of spin. As it is, the wave equation only expresses zero spin particles. [58] But he warns that this complement, by extension to “particles of spin other than zero” and in particular “to the wave mechanics of the Dirac electron” (“what we will do only in chapter XVI”) will not suffice.

47It was not until the end of Chapter XIX that a turnaround took shape, even if it remained largely programmatic. It will no longer be a question of simply finding the analogue of PEP, but of initiating criticism, and in particular that of the separation between the two categories of particles and their associated statistics. Thus begins the demotion of the PEP to the rank of “simple recipe of calculation” just like the other principles of the QM. However, caution is in order: “These problems are certainly very difficult and it seems premature to tackle them in the current state of the double solution theory. But difficult does not mean impossible and what is unsolvable today can be solved tomorrow” [59]. That was in 1952.

48What has become of the dialectic of the individual and the system which reached its peak in quantum physics within the framework of the double solution theory? [60] This dialectic affirms on a philosophical and logical level what we have named above a dependence on the context: “engaged in a system, a physical unit loses to a large extent its individuality, the latter coming to merge in the more extensive individuality of the system. This is particularly clear in the case of particles of the same nature and results in completely unforeseen consequences to which classical ideas could never have led, but which are in perfect agreement with a large number of experimental facts (new statistics, exclusion principle, etc.)” [61].

49After the turning point of 1952, de Broglie’s philosophical and epistemological interpretation found itself inserted into a new mathematical and physical framework. What was interpreted as idealization is now expressed in terms of a measure of probability, itself understood as a mathematical expression of an imaginary and subjective component of the physical interpretation.