Tensorial quantum mechanics

Abstract

In July 1925, exactly one century ago, Werner Heisenberg created Quantum Mechanics (QM) as an invariant-operational formalism by taking as a standpoint the intensive patterns that had been already observed by experimentalists in the lab. One of the few drawbacks of this proposal was the impossibility to conceptualize it in classical terms. Attempting to restore a somewhat classical spatiotemporal representation, six months later, Erwin Schrödinger would present a formalism grounded on a wave equation in configuration space. Taking as a standpoint Schrödinger’s wave mechanics together with the methodological guide of Bohr and logical positivists, Dirac would present in 1930 an axiomatic re-formulation of the theory of quanta that would become known as the “standard” version of QM (SQM). This new version of the theory, that is still today taught in physics classrooms all around the world, would become accepted regardless of its inconsistent narrative as a “recipe” intended (but unable) to predict (binary) measurement outcomes. After pointing out some of the many problems of SQM, in this work we propose to go back to Heisenberg’s matrix mechanics (and also to his understanding of physical theories). We will argue that his matrix mechanics, as an operationally-invariant standpoint, opens the door to a tensorial enlargement of the mathematical formalism capable to account for new phenomena.

Introduction

Despite the many different “interpretational” layers added to the theory, there is a general consensus within the contemporary physics community on what constitutes the basic, “Standard” version of Quantum Mechanics (SQM), upon which further developments can be built. This is, in fact, what students are taught in Universities all around the world when following a course about QM. Indeed, when physicists talk about QM they actually have in mind either Schrödinger’s wave equation from 1926 (supplemented by Born’s probabilistic interpretation of the quantum wave function) or Paul Dirac’s vectorial re-formulation (heavily influenced by Bohrian and positivist ideas) presented in 1930 and rendered “more rigorous” in mathematical terms by John von Neumann in 1932 (see¹). Of course, there are many different textbooks with small variations and slightly different presentations, but the core of what is taught by physics Professors is always the same: a set of 8 or 9 postulates, that are common to all physics textbooks, which attempt to show how these axioms are capable to predict measurement outcomes. As it is well known, within this “standard” presentation, it is Heisenberg’s matrix mechanics which has been mostly left aside, and restored only partially and indirectly —with the rise of quantum information processing— in terms of density operators interpreted as statistical mixtures of pure states (see for a detailed analysis²).

In this work we critically examine how SQM was established through a series of controversial operations (many of which stem from Bohr’s contributions) and argue for the need not only to return to Heisenberg’s original formulation, but also to move beyond it toward a tensorial formulation. The paper is organized as follows. In section 1 we present the main ideas which allowed Heisenberg to develop QM. Section 2 addresses the replacement of matrix mechanics by Schrödinger’s differential wave equation. In section 3, we review the vectorial axiomatic re-formulation —already implicit in Schrödinger’s proposal— developed by Dirac in 1930. In section 4 we contrast some of Heisenberg’s and Einstein’s ideas with the current understanding of physical theories. In section 5 we argue for the need not only to return to Heisenberg’s matrix mechanics, but also to extend the mathematical formalism in tensorial terms. In section 6 we present Tensorial QM. In the last section we present some final remarks.

Heisenberg’s matrix quantum mechanics

After Max Planck postulated the discreteness of the quantum of action during the first year of the 20th Century, it took 25 more years for physicists to reach a closed mathematical formalism that would allow them to account in a quantitative fashion for the intensive line-spectra that were observed in the lab (see for a detailed analysis³, Sect. 8.5.1). It was Werner Heisenberg, a young physicist from Munich, who was finally able to create a consistent mathematical formalism capable to capture all observed quantum phenomena in an operationally-invariant manner⁴. In this respect —and although Heisenberg cannot be described as a positivist— one cannot stress enough the essential subversive role played, within Heisenberg’s thought, by Mach’s positivist critique of the a priori concepts of classical mechanics, and his observability principle —also essential within Einstein’s development of relativity—, as this allowed him to find a starting point less burdened by the presuppositions derived from classical physics. In the previous years Heisenberg had followed the lead of Niels Bohr, focusing on the problem (created by Bohr’s model) of describing the elusive trajectories of electrons inside the atom. Bohr, let us recall, trying to stick to the atomist representation familiar to physicists while incorporating the discreteness of the quantum of action, had supplemented his essentially inconsistent yet effective algorithmic model —capable to predict the spectral lines of a specific experiment— with the image of a “small planetary system” with microscopic particles traveling along discrete quantum orbits around a central nucleus. According to the Danish physicist, electrons would change orbits through an essentially irrepresentable process, a “quantum jump”. Even though none of this made much sense, nor could it be demonstrated or really explained, physicists were immediately captivated by the predictive capacity of Bohr’s model, they bought the whole package, and focused on trying to describe the real trajectories of these surreal quantum corpuscles. Heisenberg followed the lead of Bohr for some time, but the critical reaction of Arnold Sommerfeld and Wolfgang Pauli to Bohr’s ideas convinced him in 1925 that this path lead to a dead end and that he should forget about the attempt to describe the trajectories of particles (see for a detailed discussion^5,6). Following the same Machian principle which had helped Einstein to develop special relativity, Heisenberg would reframe the problem completely —escaping the classical images proposed by Bohr— starting solely from the intensive quantities (values between 0 and 1) that were actually described by experimentalists. As explained by Jan Hilgevoord and Jos Uffink:

“His leading idea was that only those quantities that are in principle observable should play a role in the theory, and that all attempts to form a picture of what goes on inside the atom should be avoided. In atomic physics the observational data were obtained from spectroscopy and associated with atomic transitions. Thus, Heisenberg was led to consider the ‘transition quantities’ as the basic ingredients of the theory.”⁷

That same year, emancipating himself completely from classical ideas, Heisenberg⁴ would present his groundbreaking results in the following manner: “In this paper an attempt will be made to obtain bases for a quantum-theoretical mechanics based exclusively on relations between quantities observable in principle.” Almost half a century later he would narrate how these events had actually unfolded:

“In the summer term of 1925, when I resumed my research work at the University of Göttingen —since July 1924 I had been Privatdozent at that university— I made a first attempt to guess what formulae would enable one to express the line intensities of the hydrogen spectrum, using more or less the same methods that had proved so fruitful in my work with Kramers in Copenhagen. This attempt lead me to a dead end —I found myself in an impenetrable morass of complicated mathematical equations, with no way out. But the work helped to convince me of one thing: that one ought to ignore the problem of electron orbits inside the atom, and treat the frequencies and amplitudes associated with the line intensities as perfectly good substitutes. In any case, these magnitudes could be observed directly, and as my friend Otto had pointed out when expounding on Einstein’s theory during our bicycle tour round Lake Walchensee, physicists must consider none but observable magnitudes when trying to solve the atomic puzzle.”⁶, p. 60

Max Born⁸, p. 160 would recall his surprise when his young student gave him a paper to publish in Zeitschrift für Physik which included a new non-commutative multiplication rule: “Heisenberg’s rule of multiplication left me no peace, and after a week of intensive thought and trial, I suddenly remembered an algebraic theory that I had learned from my teacher... in Breslau. Such quadratic arrays are quite familiar to mathematicians and are called matrices, in association with a definite rule of multiplication.” As Heisenberg would later explain:

“The equations of motion in Newtonian mechanics were replaced by similar equations between matrices; it was a strange experience to find that many of the old results of Newtonian mechanics, like conservation of energy, etc., could be derived also in the new scheme. Later the investigations of Born, Jordan and Dirac showed that the matrices representing position and momentum of the electron do not commute. This latter fact demonstrated clearly the essential difference between quantum mechanics and classical mechanics.”⁹, p. 4

Heisenberg’s theory, which had been developed exclusively from the tables of intensive patterns observed in the lab, was operationally grounded right from the start. Heisenberg was simply trying to provide an invariant mathematical scheme capable to account for the empirical tables of intensive data. Was there any consistency, any invariance to be found between these tables of (intensive) numbers collected through experimental procedures? The answer provided by Heisenberg in terms of a new mathematical scheme was of course a positive one. What is truly remarkable, let us add, is that in finding this invariance of intensities Heisenberg had rediscovered the theory of matrices. That same year, with the help of Born and Pascual Jordan, Quantum Mechanics (QM) would finally find its final form^10,11. But let us stop for a minute to take note of some important aspects of Heisenberg’s proposal. Firstly, that its success was the direct consequence of abandoning, at least for a moment, the atomist Bohrian approach of trying to describe the trajectories of presupposed corpuscles (a path he had attempted before but had lead him nowhere). Secondly, that his formalism was in fact invariant, something which permitted to discuss consistently the operational results found in the lab also when considering different reference frames or bases. Thirdly, that the quantities this formalism was pointing as its invariant elements were intensities (and not binary values), something which, of course, if taken seriously, imposed the need of a radical innovation: to develop a new originally intensive physical concept (we will come back to this point in section 5). And finally, that this formulation implied a mathematical theory to which physicists were not accustomed. In fact, the introduction of a new mathematical theory implied of course a radical shift in physics which, since classical mechanics, had been constrained to solving differential equations. Furthermore, the loss of a continuous representation implied for classically trained physicists a supposedly “higher level of abstraction” contrary to the “intuitive representation” provided by infinitesimal calculus. Heisenberg had no obvious picture he could apply in order to make sense of his new mathematical representation. As a consequence, physicists would observe the new theory with unease and discomfort. Matrix mechanics was accepted not without resentments to be the adequate theory for quantum phenomena. However, the immediate reaction was to interpret this theory, again, in atomistic terms, although, as Sin Itiro Tomonaga¹², p. 223] reports, as they did that, they still had many questions they could not answer: “[if the location of an electron and of other particles were to become such an abstract aggregate, or matrix, how can we explain in this theory the track of a particle observed commonly in a Wilson chamber? It was Lorentz who said: ‘Can you imagine me to be nothing but a matrix? It is hardly to believe that all this is real’.” Of course, in retrospect, it should have been clear not only that the atomist representation was not suitable for this theory of intensities, but also that the picture provided by differential calculus and Newton’s theory had also been, at the time of its creation —in contraposition to Aristotelian physics—, exactly the opposite of an “intuitive picture”. Physicists had simply become accustomed to the classical representation which they now regarded as commonsensical and self-evident. So even though Heisenberg’s formalism was capable of describing in invariant terms the intensive values observed in the lab, this did not seem to be enough. Physicists were simply not prepared to give up on the atomist spatiotemporal representation that had been established since modernity as the core of physical reality, even if the cost was as high as to accept the existence of Bohr’s strange and irrepresentable “quantum particles” and “jumps”. It is in this context that Schrödinger’s new differential formulation of wave mechanics would be received by the physics community with enormous relief.

Schrödinger’s wave functions and born’s interpretation

Heisenberg’s paper was sent in July 1925 and published in December that same year. But just one month later, in January 1926, Erwin Schrödinger would present —following the work of Louis de Broglie— a new “wave like” differential equation which was also able to compute the correct energies for the quantized hydrogen atom. Schrödinger would complete his formulation in a series of four papers published in Annalen der Physik, later on collected in a single work¹³. Of course, the choice of the physics community between Heisenberg’s “weird” matrices and Schrödinger’s good old comfortable differential equation was not really difficult. While Schrödinger’s proposal promised to restore a continuous space-time representation within a well known mathematical scheme, Heisenberg’s calculus seemed to lack any picture that would explain what was really going on with quanta, reinforcing in turn the unacceptable elements of discontinuity. In words of Max Jammer:

“After the conceptual cataclysm evoked by the latter [matrix mechanics] it seemed as if Schrödinger’s return to quasi-classical conceptions reinstated continuity. Those who in their yearning of continuity hated to renounce the classical maxim natura non facit saltus acclaimed Schrödinger as the herald of the new dawn. In fact, within a few brief months Schrödinger’s theory ‘captivated the world of physics’ because it seemed to promise ‘a fulfillment of that long-baffled and insuppressible desire. Einstein was ‘enthusiastic’ about it, Planck reportedly declared ‘I am reading it as a child reads a puzzle’, and Sommerfeld was exultant.”¹⁴, p. 269

However, this attempt to restore a somewhat classical representation of physical reality within the theory of quanta would immediately find evident limits for its consistent development. Schrödinger himself would recognize the difficulties: the domain of the wave equation he had constructed was not 3-dimensional but configuration space, and thus the possibility to regain a classical and intuitive spatial representation became far from obvious.

“The true mechanical process is realized or represented in a fitting way by the wave process in q-space, and not by the motion of image points in this space. [...] In this sense do I interpret the ‘phase waves’ which, according to de Broglie, accompany the path of the electron; in the sense, therefore, that no special meaning is to be attached to the electronic path itself (at any rate, in the interior of the atom), and still less to the position of the electron on its path. And in this sense I explain the conviction, increasingly evident to-day, firstly, that real meaning has to be denied to the phase of electronic motions in the atom; secondly, that we can never assert that the electron at a definite instant is to be found on any definite one of the quantum paths, specialized by the quantum conditions; and thirdly, that the true laws of quantum mechanics do not consist of definite rules for the single path, but that in these laws the elements of the whole manifold of paths of a system are bound together by equations, so that apparently certain reciprocal action exists between the different paths. [...] All these assertions systematically contribute to the relinquishing of the ideas of ‘place of the electron’ and ‘path of the electron’. If these are not given up, contradictions remain. This contradiction has been so strongly felt that it has even been doubted whether what goes on in the atom could ever be described within the scheme of space and time.”¹³, p. 25–26 (his emphasis).

Adding to the already complex situation, very soon the battle between wave and matrix mechanics would face an essential difficulty for both parties. After his second paper, and for the surprise of everyone in the physical community, Schrödinger demonstrated what he called “a formal, mathematical identity” between Heisenberg’s matrix mechanics and his own wave mechanics¹⁵. In the introduction he remarked that both approaches seemed in principle fundamentally different in mathematical methods, assumptions and presentation:

“Above all, however, the departure from classical mechanics in the two theories seem to occur in diametrically opposed directions. In Heisenberg’s work the classical continuous variables are replaced by systems of discrete numerical quantities (matrices), which depend on a pair of integral indices, and are defined by algebraic equations. The authors themselves describe the theory as a ‘true theory of a discontinuum’. On the other hand, wave mechanics shows just the reverse tendency; it is a step from classical point-mechanics towards a continuum-theory. In place of a process described in terms of a finite number of dependent variables occurring in a finite number of total differential equations, we have a continuous field-like process in configuration space, which is governed by a single partial differential equation derived from a principle of action.”¹⁵

What is important to notice, for our purposes, is that Schrödinger had only demonstrated that his wave mechanics implied the matrix formulation, but the converse was still missing. The reason, as shown in², was that wave mechanics was, in fact, included within Heisenberg’s matrix formulation. While Heisenberg’s mathematical theory referred to all matrices, of any rank, Schrödinger’s formulation restricted Heisenberg’s original operational space only to matrices of rank = 1. Unfortunately, adding to this confusion, Wolfgang Pauli, Carl Eckart and John von Neumann would give their own independent “proofs of the equivalence” giving rise to what Fred Muller termed “the equivalence myth”¹⁶. Useless to say, Heisenberg was not delighted with the equivalence and neither was Schrödinger who declared: “I was discouraged, if not repelled, by what appeared to me a rather difficult method of transcendental algebra, defying any visualization.”

Shortly after the acceptance of this equivalence between matrix and wave mechanics, Born began to study the scattering of particles by a spherically symmetric potential, developing what is now known as the ‘Born approximation’. He preferred to make the calculations by means of Schrödinger’s formulation due to its simplicity for this specific problem. To model the collision, Born proposed “to solve Schrödinger wave equation for the system-plus-atom subject to the boundary condition that the solution in a preselected direction of electron space goes over asymptotically into a plane wave with exactly this direction of propagation (the arriving electron). In a thus selected solution we are further interested principally in a behavior of the ‘scattered’ wave at infinity, for it describes the behavior of the system after the collision”. He solved the problem, showing that “the perturbation, analyzed at infinity, can be regarded as a superposition of solutions of the unperturbed problem” and proposed the reading that each coefficient “$\Phi _{n,m}(\alpha , \ \beta , \ \gamma )$ gives the probability for the electron, arriving from the z-direction, to be thrown out into the direction designated by the angles $\alpha , \ \beta , \ \gamma$ [...]”. Born corrected in the same paper, in a note added in proof, that the probability was proportional to the square of the coefficient instead to the coefficient itself. Ever since, this has been known as the Born Rule and his reading of the wave function as the probabilistic interpretation.

“Schrödinger’s quantum mechanics therefore gives quite a definite answer to the question of the effect of the collision, but there is no question of any causal description. One gets no answer to the question‘, ‘what is the state after the collision’, but only to the question, ‘how probable is a specified outcome of the collision’ [...]. Here the whole problem of determinism comes up. From the standpoint of our quantum mechanics there is no quantity which in any individual case causally fixes the consequence of the collision; but also experimentally we have so far no reason to believe that there are some inner properties of the atom which condition a definite outcome for the collision. [...] I myself am inclined to give up determinism in the world of the atoms. But that is a philosophical question for which physical arguments alone are not decisive.”¹⁷, p. 54

Born’s interpretation did not seem any less problematic, since it did not account for the interaction of these “probability waves”, nor the fact that these waves were represented in configuration space and the dimension changed with the number of wave functions considered in different situations. However, more or less in the same way that Bohr had already misdirected the problems with his atomic model, Born’s focus switched completely the attention from the intensive patterns considered by Heisenberg to a unilateral attention on the explanation of single outcomes. This, in fact, was due to the prevalence of the atomist presupposition that Bohr had successfully championed: the single outcome was taken, without any justification, as the effect of the presence (or absence) of a particle. And, following this line of reasoning, since we are talking here about a theory that refers to particles, we should concentrate on explaining their natural consequence as single outcomes. This, of course, since the original formalism talked about intensities, presented a consistency problem, an essential difference between what the theory described in formal terms and what it was supposed to deliver in terms of observation. So, in short, the probabilistic interpretation defended by Born connected implicitly the mathematical probability of the theory with the observation of a specific effect of a single particle. Let us add that this reference to elementary particles was not only completely ungrounded but had also no connection whatsoever to the consistent formal-conceptual definition of ‘particle’ that could be found in classical mechanics. In classical physics, a particle was mathematically represented in terms of well defined invariant properties —through Galilean transformations—, and in conceptual terms was conceived —following Aristotle’s and Kant’s categories— as an entity in the actual mode of existence —namely, as constrained and defined by specific logical and metaphysical principles such as those of non-contradiction and identity. In QM, on the contrary, regardless of the complete lack of a systematic conceptual and mathematical foundation that would support the reference to particles, the atomist narrative was inconsistently —and dogmatically— maintained. A way out of trouble has been sometimes to take this reference perhaps not as an ontological account of “reality-in-itself”, but at least as a “useful fiction”. But even if we thus play down the importance of the atomist presupposition by taking it as a mere “tool”, the deductions, methodological and operational steps that are implicit in the atomist representation, continue to function with the same force. As Faraday explained long ago: “the word atom, which can never be used without involving much that is purely hypothetical, is often intended to be used to express a simple fact; but good as the intention is, I have not yet found a mind that did habitually separate it from its accompanying temptations”¹⁸, p. 220. Even Schrödinger rephrased this idea for the quantum case: “We have taken over from previous theory the idea of a particle and all the technical language concerning it. This idea is inadequate. It constantly drives our mind to ask information which has obviously no significance”¹⁹, p. 188.

It is interesting to notice that after Born’s interpretation of the quantum wave function Schrödinger was invited to Copenhagen to discuss the existence or not of these new “quantum jumps”. In this important meeting Bohr would apply his rhetorical powers in order to win an important battle. As Heisenberg would recall the events in his autobiography, even though the many arguments that Schrödinger⁶, p. 73 had produced during the debate had allowed him to rationally conclude that “the whole idea of quantum jumps is sheer fantasy” the Danish physicist would simply invert the burden of proof turning things completely upside-down:

“What you say is absolutely correct. But it does not prove that there are no quantum jumps. It only proves that we cannot imagine them, that the representational concepts with which we describe events in daily life and experiments in classical physics are inadequate when it comes to describing quantum jumps. Nor should we be surprised to find it so, seeing that the processes involved are not the objects of direct experience.”⁶, p. 74

Bohr was asking Schrödinger either to grant him the existence of quantum jumps or prove their non-existence, something known in jurisprudence as the probatio diabolica: the legal requirement to achieve an impossible proof. Such Devil’s Proof is the logical dilemma that while evidence will prove the existence of something, the lack of evidence fails to disprove it. Immediately after the meeting Schödinger would recognize his defeat in a letter to his friend Wilhelm Wien:

“Bohr’s [...] approach to atomic problems [...] is really remarkable. He is completely convinced that any understanding in the usual sense of the word is impossible. Therefore the conversation is almost immediately driven into philosophical questions, and soon you no longer know whether you really take the position he is attacking, or whether you really must attack the position he is defending.”²⁰, p. 228

Before advancing let us note some of the conclusions of this section. Firstly, that Schrödinger’s formulation gained the acceptance of the physics community because of its promise of restoring a somewhat classical continuous representation. Secondly, that this preferred formulation, however, soon found some definitive obstacles —as the fact that the quantum wave function had to be considered within configuration space— that Schrödinger himself would recognize. Thirdly, the equivalence between matrix mechanics and Schrödinger’s differential equation was wrongly accepted —due to a series of incomplete demonstrations. And, finally, that Born’s probabilistic interpretation of Schrödinger’s wave function strongly supported by Bohr would, on the one hand, reinforce the presupposed reference to “quantum particles”, shifting the focus from intensive patterns (in Heisenberg’s formulation) to single outcomes, and, on the other hand, find a crucial obstacle when facing the difficulty to define the entirely non-classical “probability” that would unfold in this context.

Dirac’s vectorial axiomatization: collapses and contextuality

An essential aspect of Dirac’s 1930’s proposed axiomatization of QM is the complete replacement of matrices by vectors in accordance with his transformation theory. This replacement is already explicit within Dirac’s notation where a unit vector in a specific basis is written as a ket $|x\rangle$. Of course, as it is well known, a ket can be also seen as a rank one matrix through the following operation $|x\rangle \langle x|$. If H is a complex vector space and B(H) is the space of matrices, then we can relate the space of vectors H with the space of (rank 1) matrices B(H) through the map:

$$\begin{aligned} \nu :H\rightarrow B(H),\quad \nu (|x\rangle ):=|x\rangle \langle x|. \end{aligned}$$

Let us mention two relevant properties of $\nu$. The first one is that $\nu$ is not surjective. In fact, its image is equal to the set of rank one matrices. The second relevant property of $\nu$ is that it is injective. Hence, we can think of the space of vectors as a subset of the space of matrices. In other words, the vector space is much “smaller” than the matrix space (see²). As a matter of fact, this jump from matrices to vectors was already implicit in Schrödinger’s equation, which could be naturally read as a vectorial equation. Following Dirac’s notation, the equation that governs the time evolution of a vector $|v_t\rangle$ in a multi-dimensional vectorial space reads as follows:

$$\begin{aligned} i \hbar \frac{d}{dt}|v_t\rangle = {\hat{H}}|v_t\rangle . \end{aligned}$$

Dirac reformulated the mathematical content of the theory exclusively in terms of vectorial spaces. From a mathematical perspective, since vectors can re-generate the whole space of matrices (through the convex sum of rank 1 matrices), there seems to exist a complete equivalence with no loss of information. In fact, one can provide a definition of Hilbert spaces and dimension using sets and functions. However, this mathematical equivalence has led many to believe that there was also a physical equivalence, when this is not the case. This becomes clear once we recognize that in Heisenberg’s original formulation each and every matrix, of any rank, in each and every basis, possesses a clear operational content which can be directly linked to a particular experimental set up and consequently to a set of definite and observable intensive measurement results (see², Appendix A). The elimination of this infinitely many number of matrices implies also the elimination of an enormous amount of experimental data related within the theory to actual experimental situations. Why was that enormous amount of experimental phenomena not significant enough to be considered?

There has been an unacceptable reduction of the phenomena considered as physically meaningful which, later on, has been re-introduced within SQM in terms of the notion of mixture (i.e., rank $\ne 1$ matrices) and re-interpreted (in a mere epistemological, non-ontological manner) as statistical convex sums of pure states (i.e., matrices of rank = 1 or vectors) —showing the dependency of the whole formulation with respect to vectors represented by single kets (see², Appendix B). The idea that physical situations described by rank $\ne 1$ matrices can be replaced by completely different physical situations described by the convex sum of matrices of rank = 1 imposes an artificial discrimination which is not only alien to the mathematical formalism but is completely unjustified from an experimental perspective [The use and application of mixtures became truly relevant during the 1990 s when the operational definition of purity became to be regarded as “too limited”²¹, p. 419]. In short, a great multiplicity of phenomena, actually observable and captured by the formalism, was simply eliminated to focus only on a very small subset of situations described by pure states (rank 1 matrices represented by vectors), after which they were reintroduced as “mixed states”, as if they were in fact the effect of the sum of those pure states previously selected. But why such an interest on reframing everything only in terms of rank 1 matrices or vectors? The reason can be found in Dirac’s attempt to follow Bohr’s reference to a microscopic realm in combination with his logical positivist understanding of theories as describing observations.

Dirac²², p. 3] states his positivistic principles already in the first chapter of his book, where he emphasizes that “science is concerned only with observable things” and that “the main object of physical science is not the provision of pictures, but the formulation of laws governing phenomena and the application of these laws to the discovery of phenomena. If a picture exists, so much the better; but whether a picture exists or not is a matter of only secondary importance.” But, also right from the beginning, and without further justification, he assumes Bohr’s atomism too, as well as his complementarity (“quantum mechanics is able to effect a reconciliation of the wave and corpuscular properties of light” such that “[all kinds of particles are associated with waves in this way and conversely all wave motion is associated with particles. Thus all particles can be made to exhibit interference effects and all wave motion has its energy in the form of quanta”). Thus, that supposed intention of adhering to what is observed is in fact contaminated by the atomist presupposition, and this makes him impose a unilateral emphasis on single, unique, outcomes, as these can be interpreted as the presence (or absence) of those presupposed “particles”. And it is because rank 1 matrices are those that, when considering the diagonal basis, permit to predict single outcomes with (binary) certainty —something completely alien to the matrix formulation which provides an intensive quantification of phenomena— that these are taken as the “pure” representation of particles, and thus the main element of the formalism itself. Again, behind this move we find the atomist presupposition made by Bohr and followed by Dirac which determines a switch of the focus from the intensive values of matrix mechanics to a unilateral attention on single outcomes, which are then interpreted as the effects of “particles” —a move, as we have seen in the previous section, already implicit within Born’s interpretation. And this operation is done even at the price of an enormous loss of empirical (non-binary) information —which is later on understood as “uncertain” with respect to pure states. Furthermore, the incompatibility between superposed states and single measurement outcomes is “solved” by Dirac by making explicit the existence of “collapses”, creating in this way a new non-linear evolution triggered by measurement, incompatible with the already existent linear motion of the theory —encapsulated in Schrödinger’s wave equation.

In this respect it is essential to recognize the fact that ever since its establishment within the theory, the “collapse” of the quantum wave function has never been observed by experimentalists in the lab. As explained by Dennis Dieks:

“Collapses constitute a process of evolution that conflicts with the evolution governed by the Schrödinger equation. And this raises the question of exactly when during the measurement process such a collapse could take place or, in other words, of when the Schrödinger equation is suspended. This question has become very urgent in the last couple of decades, during which sophisticated experiments have clearly demonstrated that in interaction processes on the sub-microscopic, microscopic and mesoscopic scales collapses are never encountered.”²³, p. 120

Very recent experiments point exactly to the same conclusion^24,25,26.

Finally, let us remark another essential consequence of Dirac’s axiomatic reformulation of QM, namely, his contextual redefinition of the notion of (quantum) state. According to Bohr’s doctrine of classical concepts and complementarity approach²⁷, p. 96, a quantum entity requires different, mutually incompatible representations such as those of wave and particle: “We must, in general, be prepared to accept the fact that a complete elucidation of one and the same object may require diverse points of view which defy a unique description.” Now, given that in mathematical terms a basis would represent a particular experimental arrangement, the natural consequence implied that the choice of a basis would become a precondition for the very definition of the notion of quantum state itself. Dirac would argue that the same system could be represented in terms of different states, relative to the different bases:

“[E]ach state of a dynamical system at a particular time corresponds to a ket vector, the correspondence being such that if a state results from the superposition of certain other states, its corresponding ket vector is expressible linearity in terms of the corresponding ket vectors of the other states, and conversely. Thus the state R results from a superposition of the states A and B when the corresponding ket vectors are connected by $\vert R \rangle \ = c_1 | A \rangle \ + c_2 | B \rangle$.”²², p. 16

This is in fact the line of reasoning behind Bohr’s famous reply to the EPR paper in 1935:

“it is only the mutual exclusion of any two experimental procedures, permitting the unambiguous definition of complementary physical quantities, which provides room for new physical laws, the coexistence of which might at first sight appear irreconcilable with the basic principles of science. It is just this entirely new situation as regards the description of physical phenomena, that the notion of complementarity aims at characterizing.”²⁸, p. 700

Today, there are many interpretations of SQM which, following this path, have taken the relativist or contextual nature of quantum states as a standpoint in order to argue, like Carlo Rovelli²⁹, that “quantum states are relative states, namely states of a physical system relative to a second physical system” (see also^30,31).

To conclude, what we obtain with SQM is an essentially inconsistent scheme which talks in a vague manner about particles and waves, makes reference to unobserved “collapses” that turn the formalism inconsistent, and introduces a contextual re-definition of the notion of (quantum) state which destroys the possibility to refer consistently to a physical state of affairs in invariant and objective terms.

Some comments about theory and experience

After its establishment during the early 1930 s, the physics community would mostly accept the Dirac-von Neumann “standard” formulation of QM, with its presupposed atomist narrative, and, at the same time, as essentially a mere instrumental algorithm capable to predict measurement outcomes —if not with certainty at least probabilistically. After World War II, the view that philosophical questions about the meaning of QM were simply irrelevant and should be put aside in favor of practical calculations became mainstream (and thus helped to cement the atomist presupposition by expelling attempts to critically address this assumed conceptual representation [Paradoxically, positivism, whose aim was to eliminate all metaphysical elements from scientific theories, ended up favoring —at least in physics— the installation of a dogmatic metaphysics as an assumed starting point. We came to believe that we proceeded only from observations, without metaphysical additions, while a presupposed atomistic metaphysics continued to function in the background]). It is a few decades later, during the 1960 s, that John Bell would return, during weekends and almost in secrecy, to an analysis of the EPR Gedankenexperiment, setting the conditions for a small though influential insurrection that would lead to important developments. First John Clauser in 1972 and then —one decade later— Alain Aspect and his group, would experimentally test the ideas of Einstein about quantum entanglement that Bohr and the physics community had dismissed almost half a century before. The positive result of the experiments, pointing to the existence of quantum entanglement, would then help to create two new fields of research. While “philosophy of QM” would be established in Europe during the 1980 s, the growing interest in quantum information processing, due to the possible technological application of quantum entanglement, would lead during the 1990 s to the establishment, within physics, of another field of research called “quantum foundations”. It is mainly philosophers of QM who would then return to the many open questions and problems raised by the theory of quanta and its representation of physical reality (e.g., the measurement problem, contextuality, non-individuality, non-separability, non-locality, etc.). However, this return would turn out to be partial, as their praxis remained essentially functional not only to the presupposed atomist narrative, but also —and perhaps more paradoxically— to the instrumentalist account of physics which had focused in expelling these questions and problems from physics itself. Instead of producing a critical reconsideration of the fundamental assumptions of SQM and the instrumentalist understanding of physical theories, philosophers of QM would remain focused on the construction of new narratives designed to supplement —but not transform— the pragmatic models already created by physicists. Physical theories would be for them formal models created by physicists capable of predicting measurement outcomes. These models could not be questioned by philosophers, whose permitted task is to inconsequentially “add” philosophy, “add” interpretations.

Today, while physicists continue to understand SQM as a mathematical algorithm capable to predict measurement outcomes, philosophers argue that what might be missing is an added story that tells us what the world is like if the theory is true³². As mentioned above, accepting SQM as a standpoint and, thus, the reference of the theory to measurement outcomes and the projection postulate, the effort of philosophers of physics has been centered in trying to solve “the measurement problem” (i.e., the gap created between quantum superpositions and single outcomes) through the introduction of —what philosophers have technically called— “interpretations”. Yet, regardless of their efforts, the work of philosophers has remained not only unnoticed but even despised by a physics community that sees no problem in applying SQM as a “tool” in the same way that was taught by Dirac and von Neumann almost a century ago. Philosophers, on the other hand, have not been even able to reach consensus regarding the main story that would allow us to understand what QM is really talking about. Instead, due to the complete lack of objective conditions that would limit the unconstrained creation of narratives, in just a few decades, the field has transformed itself into what Adán Cabello has termed an “interpretational map of madness”³³.

At this point, it becomes essential to recognize the set of presuppositions that have been assumed —not only by physicists but also by philosophers— as the main standpoints for their own research:

• Observations, as unproblematic givens of experience, are the basic standpoint of empirical science in general and of physical theories in particular.

• Physical theories are mathematical models with correspondence rules that allow us to make predictions about observational events.

• Conceptual interpretations can be later added to empirical models in order to describe what the theory really talks about, or how the world could possible be according to the theory (at least approximately).

As a consequence, while physicists have continued to apply their instrumentalist praxis focusing on technological applications and mathematical developments, the creation of interpretations, framed as a purely philosophical exercise, has remained for them a completely uninteresting subject [As Maximilian Schlosshauer³⁴, p. 59] has recently described: “It is no secret that a shut-up-and-calculate mentality pervades classrooms everywhere. How many physics students will ever hear their professor mention that there’s such a queer thing as different interpretations of the very theory they’re learning about? I have no representative data to answer this question, but I suspect the percentage of such students would hardly exceed the single-digit range”, even disconnected from empirical science [Indeed, as stressed by the philosopher of physics Roberto Torretti³⁵, p. 367] the interpretations of QM must be considered as “meta-physical ventures [...] for they view the meaning and scope of QM from standpoints outside empirical science.”

It is again Heisenberg’s work that —departing today’s mainstream approach— gives us some clues in order to open the way for a different understanding of physical theories. Furthermore, it also implies an original methodology and guidance, allowing to think of new formal and conceptual developments. Firstly, his account of scientific observation, not as an unproblematic given of experience but —instead— as a necessary consequence of the theory itself, is essential in order to become aware of the fundamental role of conceptual systems —wether they are recognized or not. This goes against empiricism and in line with Einstein, who told Heisenberg⁶ that “it is only the theory which tells you what can be observed.” More specifically, Einstein would argue:

“From Hume Kant had learned that there are concepts (as, for example, that of causal connection), which play a dominating role in our thinking, and which, nevertheless, can not be deduced by means of a logical process from the empirically given (a fact which several empiricists recognize, it is true, but seem always again to forget). What justifies the use of such concepts? Suppose he had replied in this sense: Thinking is necessary in order to understand the empirically given, and concepts and ‘categories’ are necessary as indispensable elements of thinking.”³⁶, p. 678 (emphasis in the original)

It might be noticed, to avoid confusions, that the conceptual development of a moment of unity should not be confused with the theory-ladenness of observation famously discussed by Hanson. While in the latter case, the idea is that a given object is interpreted and shaped by the theory, in the case we are discussing here the object, the moment of unity, is not a given of experience but can only be determined as a categorical construction from within the theory itself —and it is only through this determination that we can relate and make sense of what has been observed. In physics, conceptual systems are not interpretations or ways to observe an object but —as Einstein frequently emphasized— the very categorical preconditions to qualitatively define what is the moment of unity in the first place.

To Heisenberg a physical theory is not merely a predictive algorithm but provides instead a system that consistently relates a mathematical formalism and a conceptual scheme under the condition of operationality, and thus produces a representation of an evolving state of affairs —which, again, is not a given of experience. This implies that —contrary to the current account of theories— conceptual systems are as fundamental as mathematical formalisms when attempting to provide a qualitative and quantitative account of phenomena. ‘Particles’ in classical mechanics, or ‘electromagnetic waves’ in the theory of Maxwell are not observables but formal-conceptual creations capable to provide an objective-invariant understanding of what is the same within change. In this respect, Heisenberg argues:

“The history of physics is not only a sequence of experimental discoveries and observations, followed by their mathematical description; it is also a history of concepts. For an understanding of the phenomena the first condition is the introduction of adequate concepts. Only with the help of correct concepts can we really know what has been observed.”³⁷, p. 264

This means that the set of presuppositions shared not only by Heisenberg but also by Einstein and Pauli depart completely from those considered —in line with Bohr, Dirac and von Neumann— within contemporary physical and philosophical research.

• The understanding of what is observed presupposes adequate concepts that are able to constrain experience in terms of specific moments of unity.

• Theories are systems which, through the consistent and coherent interrelation between mathematical formalisms and conceptual schemes, are able to account in operational terms for specific fields of phenomena.

• A conceptual scheme is not just a narrative added to a mathematical model but —instead— a categorical system which allows to determine what is the same within change, regardless of the empirical perspective or reference frame.

This conceptual aspect, which to contemporary eyes seems to be mere speculation, is in fact a strictly necessary speculation, fundamental to physical theories. And this even if it is not recognized, as we saw in the case of atomistic metaphysics, which continues to determine a priori the sense we give to the observed as well as the way in which we read the mathematical formalism of quantum physics. Let us add that in the case of SQM the problem is not only that the atomist representation is not recognized in its fundamental role and consequences; it is also and above all the fact that this representation cannot be consistently related —as is demanded— to the mathematical formalism, and cannot produce an adequate understanding of what is observed; it cannot meet the criteria just discussed. Instead, it forces us to disregard the mathematical invariance of the theory and to take an enormous number of phenomena as secondary or —even— insignificant.

Back to Heisenberg and beyond

As we have shown, immediately after matrix mechanics, there was first a failed attempt to return to a spatiotemporal wave representation through the construction of a differential wave equation, followed by a probabilistic interpretation grounded on the reference to corpuscles, which made us replace the focus on intensities —which had allowed Heisenberg to produce an invariant formalism in the first place— by a unilateral attention to single (binary) outcomes. This would be presented in an inconsistent axiomatic formulation by Dirac. The result: an extreme impoverishment of the empirical phenomena captured by the theory, the need to introduce in a completely ad hoc manner unobserved “collapses” as a new non-linear evolution triggered by “measurement” and, last but not least, the contextual redefinition of the notion of (quantum) state precluding the invariant reference to a state of affairs (section 3). In contradistinction, our proposal is to go against these successive steps which have taken us away from matrix mechanics, to escape the enthronement of pure states³⁸, the unilateral focus on single outcomes, the atomist picture, the need of unjustified “collapses”, and thus return to the richness of intensive phenomena and the operational-invariance already captured by Heisenberg’s matrix formulation.

In the history of physics it has been in fact invariance what has always allowed to determine, in formal terms, a moment of unity consistent throughout the different reference frames and experimental situations, thus allowing for an intelligible representation detached from particular, individual perspectives. This has always been seen, by physicists such as Galileo, Newton, Maxwell, and of course by Einstein when developing relativity theory, as an obligation, as a necessary condition in order to construct a consistent physical theory. In fact, Einstein, when developing his special theory of relativity, was faced with the incompatibility among three different conditions if they were maintained simultaneously: between the principle of relativity (the requirement to consistently translate the experiments in one reference frame to another equivalent one), the experimental finding of the invariant speed of light, and Galilean transformations³⁹. In order to maintain the principle of relativity (and thus invariance) as well as the experimental evidence of the independence of the speed of light with respect to different reference frames, Einstein decided to abandon the Galilean transformations, and thus the “commonsensical” classical spatiotemporal representation, producing a conceptual innovation completely alien to classical physics. Fidelity to the irreducible systematic conditions of a physical theory seemed rightly more important to him than fidelity to the picture that had been created by classical mechanics just a few centuries before. As a consequence, while in classical mechanics the spatial and temporal values were considered absolute (independent of reference frames) and speed and position as relative (to reference frames), in relativity theory it is the speed of light that would become absolute (independent of reference frames) and spatial and temporal intervals relative (to each reference frame). Of course, it should be stressed that the relative aspects involved in both theories did not imply any inconsistency, since in each case the relative values of properties would be consistently considered in terms of a global transformation, namely, the Galilean transformation in the case of classical mechanics and the Lorentz transformation in the case of special relativity. And it is this global aspect provided by invariance that would allow in both cases to retain a consistent representation where all reference frames remain of course completely equivalent, this is, related consistently to the same objective state of affairs.

In the case of QM, a similar situation appears, and the price to pay to maintain invariance is to remain close to the (intensive) experimental evidence while abandoning the classical reference to corpuscles and the unilateral focus on binary outcomes. This entails, as in the case of Einstein, to produce a radical conceptual innovation, alien to classical physics, namely, the need to develop a new physical concept of an intensive nature which captures what is invariant according to the formalism, and thus becomes the natural referent of the theory. Of course, this idea is in opposition with the approach that Bohr championed, and that, in order to retain the concepts of classical physics, destroyed the invariance present in the formalism. The path proposed here has already been followed in several papers^40,41,42, resulting in the development of a conceptual representation centered around the notion of power of action. In this respect, there are several aspects that need to be made explicit. The power of action is an originally intensive physical concept, a moment of unity which cannot be understood as depending on supposedly more fundamental atomistic constituents. In line with Heisenberg’s notion of closed theory, with this scheme there is no “common sense” classical foundation that we need to reach or find as a limit^43,44.

“The transition in science from previously investigated fields of experience to new ones will never consist simply of the application of already known laws to these new fields. On the contrary, a really new field of experience will always lead to the crystallization of a new system of scientific concepts and laws.... The advance from the parts already completed to those newly discovered, or to be newly erected, demands each time an intellectual jump, which cannot be achieved through the simple development of already existing knowledge.”⁴⁵, p. 25

Thus, departing completely from waves, corpuscles and actual events, we can understand that QM is about the relation between powers of action determined by invariant values (that can be computed through Born’s rule) we term their intensity or potentia. This means that “quantum probability” is not “classical probability”, and has to be interpreted in terms of providing an intensive knowledge that is completely certain. “Quantum probability” does not talk about the ignorance regarding the finding of “single events” or “particles” —as explicitly stated by Born¹⁷. Once this measure (quantum probability) is understood as making reference to an intensity, there is no need to consider any binary valuation, and since the intensity of all powers is invariant under linear transformations, a global (intensive) valuation is always possible and the notion can be objectively considered. This is in fact a natural way of understanding the mathematical formalism in case we are willing to escape the dogmatic reference to particles. Thus, when considering powers of action, the need to refer to a preferred basis is completely unecesssary. Furthermore, these intensive physical elements can be perfectly measured in the lab without the need to break the causal evolution of the represented state of affairs. There are no “quantum jumps” or “collapses” required in order to make sense of single measurement outcomes simply because the theory never makes reference to them. There are no particles being destroyed through measurement. Single outcomes are just particular ways to account for intensive patterns through repetition (when we are dealing with experiments that produce a single outcome at a time), and consequently should be always regarded in the context of the theory as partial, incomplete expressions of something else, namely, intensive patterns. Some remarks go in order.

First, like in all physical theories the repetition of an experimental procedure which keeps referring to the same state of affairs independently of changing the reference frame or the experimental procedure is an essential pre-condition for physical representation. Of course, this condition is never fulfilled when making reference to “quantum particles” which are always supposedly destroyed with each single measurement. However, it makes perfect sense when discussing about intensive patterns, for the intensive state of affairs, the set of powers of action and their definite intensities, do not change when measured one or many times.

Second, the reference to powers of action has nothing in common with the propensities, dispositions and potentialities that have been discussed within the philosophical literature. These notions have been introduced in order to solve the measurement problem, namely, to account for the gap between the mathematical representation in terms of superpositions and single outcomes^46,47. They are thus determined right from the start by the atomist presupposition present in SQM. The propensities, dispositions, potentitalities that are discussed in the literature are elements dependent on the consideration of single outcomes, and thus on an atomistic state of affairs that would be more fundamental. The measurement problem does not arise within our proposal simply because we are talking about intensive patterns (representing powers) which do not supervene on anything else, and not about single binary events representing particles. Heisenberg himself was dragged into the measurement problem, which he then attempted to solve through the metaphysical consideration of potentialities (e.g⁹.,, chap. 10), by Bohr’s ability to convince him to discuss the theory of quanta in terms of particles (and waves). This influence by Bohr can be seen in all its strength when Heisenberg was able to derived his famous inequalities in 1927 following Einstein’s consideration regarding observability (see⁷).

In this respect, our approach, which restores the relevance of the matrix formulation, is perhaps closer to Heisenberg’s original methodology —closer also to that of Einstein— in two kernel aspects. First, with respect to the consideration of what is observable in the lab (i.e., intensities) beyond the classical presuppositions, and secondly with respect to the need to develop a new conceptual system which does not need to connect to the classical picture. We might say in this respect that a notion that captures the originally intensive nature of the invariant elements of the formalism, as that of power of action, could have really been a possible way forward for Heisenberg —and away from Bohr’s proposal.

Third, as discussed explicitly in⁴⁸, what is being proposed here is not, as argued by Raoni Arroyo and Jonas Arenhart⁴⁹, p. 907, “one further position in the already huge cart of options of quantum mechanics.” Following what has been previously said, what we are proposing should not be mistaken for another “interpretation”. We are not taking SQM for granted and supplementing it with a narrative. We are instead critically addressing that standard version in order to become aware of its implicit representation —and the paradoxes and obstacles that it produces—, which all “interpretations” share, and, recognizing the nature of theories as formal and conceptual systems, we are proposing a different conceptual representation that is conditioned by the formalism and by operationality (and not by a familiar concepts). In contrast with the empiricist view of physics, where observations are givens, we are saying that a conceptual representation is key to understand what has been observed —and can in fact provide access to new phenomena. As discussed above, this path, which goes back to the invariance of intensities already present in Heisenberg’s formulation, permits to escape contextuality, preferred bases, collapses, the projection postulate and —consequently— the measurement problem itself. Furthermore, contrary to the arbitrary and personal nature of interpretations as narratives, when describing the necessary systematic relations between the invariant formalism, the conceptual scheme, and operationality, we are also describing the objective conditions for QM itself. Let us discuss this last point a bit further, for it is of great importance. On the one hand, we have a view of the conceptual, “philosophical” task in relation to physics, where we simply accept our “best physical theories” and “add” an interpretation. In the case of SQM —one of these “best” theories— the reference to single outcomes and particles conflicts with the invariance of the formalism and with observations, and therefore produce problems and obstacles which are insurmountable under such conditions. The philosophical task, in this view, would be to try to supplement this dysfunctional scheme with a story that makes sense of contextuality, collapses, etc. However, such additional narratives —which are not part of the theory itself— are not limited by specific conditions, and therefore any interpretation is in principle valid —allowing, in turn, for their exponential multiplication. Now let us contrast this with the view of physical theories that we are trying to develop based on certain ideas of Heisenberg and Einstein. In such a context, a physical theory is both a mathematical formalism and a conceptual scheme, and only in this way can it make sense of what is observed and produce a meaningful representation that allow us to think about such experience. Therefore, the conceptual scheme is, on the one hand, something fundamental and not an inconsequential aggregate, and, on the other hand, it responds to conditions that guide and limit it: its systematic relationship with the formalism (specially with its invariance) and operationality. Let us consider these conditions in the case of QM. The invariance of the formalism, that which gives it coherence, which allows for a global valuation, which allows us to identify what is the same across different reference frames, as well as to predict what is actually observed, points to intensive quantities. Since all that coherence and predictive power is lost when intensities are redirected to binary values, we must consider such intensive values as basic, original, sufficient. Given this, a necessary condition for a conceptual scheme that is consistent with the formalism is, in principle and above all, to constitute a moment of conceptual unity —a reference that remains the same— that captures the moment of unity given in the formalism; that is, we must start from the concept of an originally intensive physical element, to which the theory would mainly refer. And given that intensities cannot be redirected to binary values, a further condition is that the intensive physical element should not be made dependent on an atomistic representation that would supposedly be more fundamental. Those conditions, for example, were our starting point when we focused the representation on the concept of power of action. The condition of operationality also says, of course, that the proposed concepts must be able to relate to what is observed, and in fact the intensities of the powers of action can always be observed perfectly, and always in accordance with what is indicated by the theory (in the case of experimental situations where we proceed with a single outcome at a time, this happens of course thanks to repetition). But, in any case, leaving aside for a moment our terminological proposal, the important thing is to be able to show that not only is the supposedly speculative conceptual task fundamental in physics, but also that there are conditions that allow such a task to be guided, and that the “interpretational map of madness” is not at all necessary, but rather the symptom of a fundamental confusion.

Let us elaborate a bit more on the non-contextual character of our proposal, of which we can in fact provide a theorem. When we stick to the intensive values of action, we are able in fact to refer to a state of affairs that is independent of the particular representation in a reference frame (or basis), escaping thus the relativism with which most accounts of QM have contented themselves. In contraposition to a Binary State of Affairs (BSA), defined classically in terms of a set of true definite valued binary properties —and called elsewhere Actual State of Affairs—, we propose to relate the reference of QM to an Intensive State of Affairs (ISA) —also called elsewhere Potential State of Affairs (PSA)⁴¹. What has been demonstrated is that by considering an intensive, rather than binary, state of affairs, it is possible to restore a consistent global valuation for all projection operators independently of the basis. Let us recall some results from⁴¹. While a Global Binary Valuation (GBV) is a function from a graph to the set $\{0,1\}$, a Global Intensive Valuation (GIV) is a function from a graph to the closed interval [0, 1]. We term projection operators as intensive powers [For a detailed introduction, analysis and discussion of the notion of ‘intensive power’ we refer the interested reader to⁵⁰, and more specifically⁴¹,, Sect. 8 and⁴⁰, Sect. 3]. Let H be a Hilbert space and let ${\mathcal {G}}={\mathcal {G}}(H)$ be the set of observables. We give to ${\mathcal {G}}$ a graph structure by assigning an edge between observables P and Q if and only if $[P,Q]=0$. We call this graph, the graph of powers. Among all global intensive valuations we are interested in the particular class of ISA.

Definition 1

Let H be a Hilbert space. An Intensive State of Affairs is a global intensive valuation $\Psi :{\mathcal {G}}(H)\rightarrow [0,1]$ from the graph of powers ${\mathcal {G}}(H)$ such that $\Psi (I)=1$ and

$$\begin{aligned} \Psi (\sum _{i=1}^{\infty } P_i)= \sum _{i=1}^\infty \Psi (P_i)\end{aligned}$$

for any piecewise orthogonal projections $\{P_i\}_{i=1}^{\infty }$. The numbers $\Psi (P) \in [0,1]$, are called intensities or potentia and the nodes P are called powers. Hence, an ISA assigns a potentia to each power.

Intuitively, we can picture an ISA as a table,

$$\begin{aligned} \Psi :{\mathcal {G}}(H)\rightarrow [0,1],\quad \Psi : \left\{ \begin{array}{rcl} P_1 & \rightarrow & p_1\\ P_2 & \rightarrow & p_2\\ P_3 & \rightarrow & p_3\\ & \vdots & \end{array} \right. \end{aligned}$$

Theorem 1

Let H be a separable Hilbert space, $\dim (H)>2$ and let ${\mathcal {G}}$ be the graph of powers with the commuting relation given by QM.

• Any positive semi-definite self-adjoint operator of the trace class $\rho$ determines in a bijective way an ISA $\Psi :{\mathcal {G}}\rightarrow [0,1]$.

• Any GIV determines univocally a set of powers that are considered as truly existent.

Proof

• 1. Using Born’s rule, we can assign to each observable $P\in {\mathcal {G}}$ the value $\text{ Tr }(\rho P)\in [0,1]$. Hence, we get an ISA $\Psi :{\mathcal {G}}\rightarrow [0,1]$. Let us prove that this assignment is bijective. Let $\Psi :{\mathcal {G}}\rightarrow [0,1]$ be an ISA. By Gleason’s theorem there exists a unique positive semi-definite self-adjoint operator of the trace class $\rho$ such that $\Psi$ is given by the Born rule with respect to $\rho$. [As remarked in⁵¹: “Prior to the Bell and Kochen-Specker theorems, Gleason’s theorem demonstrated that, for any quantum system of dimension at least three, the unique way to assign probabilities to the outcomes of projective measurements is via the Born rule. In particular, Gleason’s theorem excludes any deterministic probability rule given by a {0, 1}-valued assignment of probabilities to all the self-adjoint projections on the system’s Hilbert space.”]

• 2. Consider the function $\tau :[0,1]\rightarrow \{0,1\}$, where $\tau (t)=0$ if and only if $t=0$. Now, given a GIV $\Psi :{\mathcal {G}}\rightarrow [0,1]$, the map $\tau \Psi :{\mathcal {G}}\rightarrow \{0,1\}$ is a well-defined map.

$\square$

Definition 2

Let ${\mathcal {G}}$ be a graph. We define a context as a complete subgraph (or aggregate) inside ${\mathcal {G}}$. For example, let $P_1,P_2$ be two elements of ${\mathcal {G}}$. Then, $\{P_1, P_2\}$ is a contexts if $P_1$ is related to $P_2$, $P_1\sim P_2$. Saying it differently, if there exists an edge between $P_1$ and $P_2$. In general, a collection of elements $\{P_i\}_{i\in I}\subseteq {\mathcal {G}}$ determine a context if $P_i\sim P_j$ for all $i,j\in I$. Equivalently, if the subgraph with nodes $\{P_i\}_{i\in I}$ is complete. A maximal context is a context not contained properly in another context. If we do not indicate the opposite, when we refer to contexts we will be implying maximal contexts.

For the graph of powers, the notion of context coincides with the usual one; a complete set of commuting operators. However, all projection operators can be assigned a consistent value bypassing in this way the famous Kochen-Specker theorem,

Theorem 2

(Intensive Non-Contextuality Theorem) Given any Hilbert space H, then an ISA is possible over H.

Proof

See⁴¹. $\square$

Thus, contrary to SQM’s reference to systems with properties (which impose a binary valuation), our conceptual representation of quantum physical reality is not relative to any particular context, it is global and essentially intensive. We refer the reader to^40,41 for a detailed discussion and analysis.

Last but not least, it is important to mention that there are good examples of the capacity of this proposal to provide some other interesting results, such as the objective account of bases and factorizations⁵², the consistent definition of entanglement beyond contextual relativism⁴², the production of a meaningful notion of measure (for quantum entanglement)⁵³ as well as a consistent account of multi-partite entanglement⁵⁴.

Perhaps it may also be of interest to consider, before we advance, an unexpected conclusion emerging from what we have discussed. The orthodox understanding of current physics is largely justified on an empiricist foundation, wherein the theoretical work of physics must always start from observations, considered as uncontaminated data, stripped of any theoretical presupposition. And it is on this foundation, on the supposedly theoretically innocent nature of observations, that the objectivity of physics is somehow justified. Beyond the problems behind this idea of physics (especially in its account of observation), the truth is that what has been said so far seems to suggest that this condition may not have been met, as commonly believed. One could argue that the only one who remained firmly rooted in what is actually observed, the only radical empiricist, the only one who managed to shed inadequate assumptions and consider anew what was actually observed (and, from that, develop the formalism of quantum mechanics), was Werner Heisenberg. He managed to free himself from the atomistic narrative that imposed a series of dogmatic presuppositions and could thus confront what was observed directly, concluding in the invariance of intensities suggested by his matrix mechanics. From all of this, it becomes apparent that what happened afterward was, in large part, a gradual departure from the radical attention to what is actually observed. By reinstating the atomistic dogma and reintroducing the fundamentals of classical concepts, invariance was destroyed, and a series of presuppositions were imposed on observations that were not only foreign to them but, even worse, were incompatible with them. The history of quantum mechanics seems to describe, after Heisenberg’s matrix mechanics, an increasing departure from the focus on what is observed. Less empiricism, not more —at least if we decide to take it seriously.

We want now to advance another argument for the return to matrix mechanics, an argument this time related to the impoverishment of the theory caused by the radical restriction —imposed by the vectorial proposal of Dirac— of the matrices that can be considered as physically meaningful. As shown in², an enormous amount of empirical information, of phenomena that can be observed, captured and predicted by the theory, was left aside or taken as secondary simply because it did not comply to the expectations of an atomist binary representation. The physical situations captured by the original matrix mechanics are enormously superior than what can be considered in a vectorial formulation that focuses on “pure states” —a notion which is in itself inconsistently defined^38,55. This, we believe, is an argument in favor of matrix mechanics that cannot be left aside. However, what we propose here is not just a return to Heisenberg’s matrix mechanics but to put forward its natural extension to tensors.

Tensorial quantum mechanics

As we have said, a representation of physical reality consists of a system that articulates an invariant formalism and a conceptual scheme. This system must in turn be operational, that is, the existence or non-existence of what the representation in question describes must be proved through the experimental observation in the lab. We have proposed such a physical representation, that is, a system that articulates Heisenberg’s invariant formalism with a conceptual scheme anchored in that invariance (rather than destroying it), and which can be related without conflict to the observations predicted by that formalism, which can capture in its entirety the phenomenal field proper to quantum theory. When describing its operationality, the testing of the physical elements described by the theory, we do not want the classical representation to infiltrate the theory. This has very often happened as a result of the influence of Bohr’s thought. Bohr introduced measurement into the heart of the theory itself (that is, into its formal-conceptual representation, and not just as a way of testing that representation), and took the experimental situation (no matter what we are measuring) as describable only by classical concepts. In this way, he managed to impose the need to understand quantum theory as a whole through classical concepts. Not only is this unnecessary, but it actually hurt the theory’s operationality, introducing a break with the experience that the formalism actually captured. There are two main Bohrian reasons behind this maneuver: on the one hand, he tells us that outside the concepts of classical mechanics, there is no possible communication of the experimental situation between scientists; and, on the other hand, Bohr would say that it is necessary to account within quantum theory for the macroscopic classical existence of the measuring devices because, at the moment of measurement, an uncontrollable interaction occurs that alters the physical element to be measured²⁸. With regard to the second point, as we have said previously, this does not in fact happen, since, if we refer to the intensive elements on which the invariance of the quantum formalism is based, there is no problem with the repeatability of experiments, there is no destructive influence of measurement. Regarding the first point, we believe that what will be said in the following pages will demonstrate the possibility of communicating the experimental situation without involving classical representation. Let us also add that it is not actually true that Bohr’s proposal manages to account for classical space and time within quantum theory. Within the quantum formalism, there is still no way to represent them, nor has it been possible to demonstrate the existence of a supposed quantum-to-classical limit. In other words, this impression of intelligibility is only the illusory effect of a homology, of using familiar language.

In order, then, to avoid classical concepts to infiltrate the theory, but also to deepen and specify the operational character of our proposal, and above all to show its potential to encompass an even wider range of phenomena, to expand the experimental information captured, it is necessary to describe specifically how to understand the experimental situation from this representation. To this end, we have developed⁵² —and will repeat here— how to interpret from it the notions of screen, detector, QLab, and Experimental Arrangement. The reason for this development is that, as Einstein and Heisenberg insisted, it is from the theory that we must make sense of what is observed; it is the theory that defines what has been observed. Furthermore, let us add that there are two important theorems which also allow to secure the invariant relation between bases and factorizations within the matrix formalism⁴². It is in this context that we will develop an extension of the mathematical formulation of QM beyond matrices⁵².

An essential point of the tensorial formulation we are now ready to present is its capability not only to take into account the phenomena captured by the matrix formulation but to extend even further the consideration of quantum phenomena in the lab. As we will see in the following, while the situation represented by a vector (or a rank 1 matrix) is limited to an experimental arrangement with only one screen and a matrix to an experimental arrangement with two screens, a tensor is capable to describe the general case of an experimental arrangement with n screens (and multiple detectors). Such a tensorial formulation could allow, for instance, among other things, for a simple and natural understanding of the phenomenon of quantum entanglement when considering multiple screens⁵⁴, something that has remained an open problem within the orthodox literature —referred to as ‘multi-partite entanglement”⁵⁶.

Let us begin with the definition of a (simple) graph as a pair $G = (V, E)$, where V is a set whose elements are called vertices (or nodes), and E is a set of unordered pairs $\{v,w\}$ of vertices, whose elements are called edges. While each vertex is related to the mathematical notion of projector operator and to the physical concept of power of action, each edge is linked to the mathematical concept of commutation and the experimental compatibility of powers within a particular measurement set up.

Definition 3

Graph of powers: Given a Hilbert space H, the graph of powers G(H) is defined such that the vertices are the projectors on H (called powers), and an edge exists between projectors $P_1$ and $P_2$ if they commute.

It is these powers, in their multiplicity and their relationships, that allow us to define an Intensive State of Affairs (ISA) —that contrasts to the Binary State of Affairs (BSA) that represents situations in classical physics and relativity theory. But first, we need to formalize the notion of intensity (or potentia). In general, the assignment of intensities is called Global Intensive Valuation (GIV).

Definition 4

Global Intensive Valuation: A Global Intensive Valuation is a map from G(H) to the interval [0, 1].

Let us remark that a GIV implies an account of quantum probability in terms of intensive values that do not supervene on binary values. Or in other words, this should not be confused with the mainstream interpretation of quantum probability which redirects the reference to binary events. Clearly, not all GIVs are compatible or consistent with the relations between powers. Thus, we will focus on those that define an ISA as follows:

Definition 5

Intensive State of Affairs: Let H be a Hilbert space of infinite dimension. An Intensive State of Affairs is a GIV $\Psi : G(H)\rightarrow [0,1]$ from the graph of powers G(H) such that $\Psi (I)=1$ and

$$\begin{aligned} \Psi (\sum _{i=1}^{\infty } P_i)=\sum _{i=1}^\infty \Psi (P_i) \end{aligned}$$

for any piecewise orthogonal operator $\{P_i\}_{i=1}^{\infty }$. The numbers $\Psi (P) \in [0,1]$ are called intensities or potentia and the vertices P are called powers of action. Taking into consideration the ISAs, it is then possible to advance towards a consistent GIV which can bypass the contextuality expressed by the Kochen-Specker Theorem^38,57.

Definition 6

Quantum Laboratory: We use the term quantum laboratory (or quantum lab or Q-Lab) as the operational concept of an ISA.

Within a Q-Lab, we have the concepts of screen, detector with which we can define more explicitly the observation of the power of action, its potentia and also specify what is an experimental arrangement:

Definition 7

Screen and Detector: A screen with n places for n detectors corresponds to the vector space ${\mathbb {C}}^n$. Choosing a basis, say $\{|1\rangle ,\dots ,|n\rangle \}$, is the same as choosing a specific set of n detectors. A factorization ${\mathbb {C}}^{i_1}\otimes \dots \otimes {\mathbb {C}}^{i_n}$ is the specific number n of screens, where the screen number k has $i_k$ places for detectors, $k=1,\dots ,n$. Choosing a basis in each factor corresponds to choosing the specific detectors; for instance $|\uparrow \rangle , |\downarrow \rangle$. After choosing a basis in each factor, we get a basis of the factorization ${\mathbb {C}}^{i_1}\otimes \dots \otimes {\mathbb {C}}^{i_n}$ that we denote as

$$\begin{aligned} \{ |k_1\dots k_n\rangle \}_{1\le k_j\le i_j}. \end{aligned}$$

Definition 8

Power of action: The basis element $|k_1\dots k_n\rangle$ determines the projector $|k_1\dots k_n\rangle \langle k_1\dots k_n|$ which is the formal-invariant counterpart of the objective physical concept called power of action (or simply power) that produces a global effect in the $k_1$ detector of the screen 1, in the $k_2$ detector of the screen 2 and so on until the $k_n$ detector of the screen n. Let us stress the fact that this effectuation does not allow an explanation in terms of particles within classical space and time. Instead, this is explained as a characteristic feature of powers. In general, any given power will produce an intensive multi-screen non-local effect.

Definition 9

Experimental Arrangement: Given an ISA, $\Psi$, a factorization ${\mathbb {C}}^{i_1}\otimes \dots \otimes {\mathbb {C}}^{i_n}$ and a basis $B=\{|k_1\dots k_n\rangle \}$ of cardinality $N=i_1\dots i_n$, we define an experimental arrangement denoted ${{\,\textrm{EA}\,}}_{\Psi ,B}^{N,i_1\dots i_n}$, as a specific choice of screens with detectors together with the potentia of each power, that is,

$$\begin{aligned} {{\,\textrm{EA}\,}}_{\Psi ,B}^{N,i_1\dots i_n}= \sum _{k_1,k_1'=1}^{i_1}\dots \sum _{k_n,k_n'=1}^{i_n} \alpha _{k_1,\dots ,k_n}^{k_1',\dots ,k_n'}|k_1\dots k_n\rangle \langle k_1'\dots k_n'|. \end{aligned}$$

Where the number N is the cardinal of B and is called the degree of complexity (or simply degree) of the experimental arrangement.

Definition 10

Potentia: The number that accompanies the power $|k_1\dots k_n\rangle \langle k_1\dots k_n|$ is its potentia (or intensity) and the basis B determines the powers defined by the specific choice of screens and detectors.

Now, assume that in a Q-Lab we want to change or modify an experimental setup. We also have two theorems that allow us to predict the possible outcomes that will be obtained in a new experimental arrangement. If the number of powers (i.e., the complexity) remains the same after the rearrangement, then the Basis Invariance Theorem tell us that the new experimental arrangement is equivalent to the previous one. However, if the complexity of the new experimental arrangement drops, then the Factorization Invariance Theorem tell us that all the knowledge in the new experimental arrangement was already contained in the previous one (see⁵²).

Theorem 3

(Basis Invariance Theorem) Given a specific QLab $\Psi$, all experimental arrangements of the same complexity, are equivalent independently of the basis.

Theorem 4

(Factorization Invariance Theorem) The experiments performed within an ${{\,\textrm{EA}\,}}_{\Psi }^N$ can also be performed with an experimental arrangement of higher complexity N+M, ${{\,\textrm{EA}\,}}_{\Psi }^{N+M}$, that can be produced within the same QLab $\Psi$.

Now, in order to treat this situation formally, we must work with tensors. Specifically, assume that we have two bases B and $B'$ obtained from two experimental arrangements in the same Q-Lab $\Psi$,

$$\begin{aligned} B = \{|k_1\dots k_n\rangle \}_{1\le k_j\le i_j}, \quad B' = \{|\kappa _1\dots \kappa _{m}\rangle \}_{1\le \kappa _j\le \iota _j}, \quad \end{aligned}$$

From B we infer that the first experimental arrangement has n screens, where the first screen has $i_1$ detectors, the second $i_2$ detectors and so on. The second experimental arrangement has m screens, where the first screen has $\iota _1$ detectors, the second $\iota _2$ and so on. Assume that the two bases are related by the following transformation,

$$\begin{aligned} |k_1\dots k_n\rangle = \sum _{\kappa _1,\dots ,\kappa _m=1} ^{\iota _1,\dots ,\iota _m} \lambda _{\kappa _1,\dots ,\kappa _m}^{k_1\dots k_n} |\kappa _1\dots \kappa _{m}\rangle ,\quad 1\le k_1\le i_1,\dots ,1\le k_n\le i_n. \end{aligned}$$

Then, we can convert the first experimental arrangement into the second one through the algebraic properties of the tensors. If

$$\begin{aligned} {{\,\textrm{EA}\,}}_{\Psi ,B} = \sum _{k_1,k_1'=1}^{i_1}\dots \sum _{k_n,k_n'=1}^{i_n} \alpha _{k_1,\dots ,k_n}^{k_1',\dots ,k_n'}|k_1\dots k_n\rangle \langle k_1'\dots k_n'| \end{aligned}$$

then,

$$\begin{aligned} {{\,\textrm{EA}\,}}_{\Psi ,B'} = \sum _{\kappa _1,\dots ,\kappa _m=1} ^{\iota _1,\dots ,\iota _m} \sum _{\kappa _1',\dots ,\kappa _m'=1} ^{\iota _1,\dots ,\iota _m} \left( \sum _{k_1,k_1'=1}^{i_1}\dots \sum _{k_n,k_n'=1}^{i_n} \alpha _{k_1,\dots ,k_n}^{k_1',\dots ,k_n'} \lambda _{\kappa _1,\dots ,\kappa _m}^{k_1\dots k_n} \overline{\lambda _{\kappa _1',\dots ,\kappa _m'}^{k_1'\dots k_n'}} \right) |\kappa _1\dots \kappa _{m}\rangle \langle \kappa _1'\dots \kappa _{m}'|. \end{aligned}$$

Notice that in standard multi-index notation (a notation better suited for these type of algebraic expressions) the last equation can be written more compactly as

$$\begin{aligned} {{\,\textrm{EA}\,}}_{\Psi ,B'} = \sum _{\kappa ,\kappa '} \left( \sum _{k,k'} \alpha _{k}^{k'} \lambda _{\kappa }^{k} \overline{\lambda _{\kappa '}^{k'}} \right) |\kappa \rangle \langle \kappa '|. \end{aligned}$$

It now becomes clear that this new formulation produces a bridge between the mathematical formalism and the concepts required in order to refer in a meaningful manner to the experience produced within the lab (Table 1).

Table 1 Relations between physical and mathematical concepts.

Let us add that, given that classical, Euclidean spatiality plays no role in quantum theory, it in turn plays no role in the interpretation of the experimental situation. Of course, no physical theory can account for all the aspects involved in physical reality; each theory is capable of accounting for a specific field of phenomena. And quantum mechanics is not capable of representing the spatial existence of measuring devices, but it can represent a series of experimental phenomena that classical mechanics is unable to explain. But furthermore, not only does quantum mechanics not speak of the spatiality of macroscopic objects, but that spatiality has no influence, no effect on the observations that quantum theory predicts. As has already been shown in another context⁵³, in QM there is no “spatial distance” that can be represented by the mathematical formalism; there is no meaningful way to define the spatial distance between two detectors or screens. Detectors in different screens are represented by vectors in different mathematical spaces, thus, it is impossible to compute any distance between them. The “distance” that can be actually computed, via Pythagoras’ theorem, is that between two powers (or between detectors in the same screen) and it always gives the same result, $\sqrt{2}$ —and is thus meaningless. Hence, the Euclidean distance that could be measured within the lab between two detectors simply cannot be represented within the mathematical formalism of the theory. Furthermore let us notice, for example, that the Euclidean distance between two detectors of an experimental set-up constructed within a lab has no effect whatsoever on the quantum phenomena in question. In fact, placing the two detectors in two screens at a distance of 1 meter will produce exactly the same quantum phenomena as placing them at a distance of 100 kilometers [Let us add, however, that an undoubtedly interesting question, and one that would be worth reflecting on —but which has nothing to do with classical space-time representation of experimental devices— could be whether we can conceive of concepts that account for a specifically quantum space-time, that is, what space-time might be appropriate to quantum formalism and correspond to the powers of action. We have already made some references to this question —albeit still insufficient— in another context⁵³.

It is important to stress the fact that this is not a simple re-naming of mathematical and conceptual elements. There are many important differences between the notions in SQM and TQM. For example, while projectors in SQM are related to the properties of quantum systems which are non-invariant, in TQM projectors represent powers of action which are invariant; while the numbers $\alpha _{k_1,\dots ,k_n}^{k_1,\dots ,k_n}$ are understood in SQM as the probability of finding a particle, in TQM they express the intensive quantification of powers themselves. Furthermore, while in the case of SQM the factorization of a Hilbert space is interpreted as the separation of quantum systems into sub-systems, in TQM it represents the number of screens in an experimental set up; and while bases in SQM represent an experimental arrangement which is incompatible with other ones, in TQM bases characterize the detectors of screens. This means that what is obtained is not just a re-assignment of words, or a different picture, but a radically new account of quantum phenomena, their experimental preconditions as well as their formal interrelations which, in turn, lead to experimental consequences that could be tested in the lab. While within TQM we have a Basis Invariance Theorem and a Factorization Invariance Theorem^42,52, this is not the case in SQM. Furthermore, the analysis of quantum entanglement leads to possible solutions of the open problem of multi-partite entanglement in terms of multi-screen entanglement^53,54. These developments open of course the door to the future experimental testing of TQM in the lab.

For completeness let us explain why the Basis Invariance Theorem works in TQM and does not works in SQM. Let H be a Hilbert space, ${\mathcal {G}}$ its graph of powers and let $\Psi :{\mathcal {G}}\rightarrow [0,1]$ be an ISA. Let ${\mathcal {C}}_1,{\mathcal {C}}_2\subseteq {\mathcal {G}}$ be two contexts. Let $\rho$ be the density matrix associated to $\Psi$ in the basis ${\mathcal {C}}_1$. Then, the intensity of a particular power P can be computed with the Born rule $Tr(\rho P)$. The formalism of SQM says that the probability of a particle of being, say spin up, is given by the number $Tr(\rho P)$. Now, if we change the context to ${\mathcal {C}}_2$ and $\rho '$ is the density matrix associated to $\Psi$ in the basis ${\mathcal {C}}_2$, then there exists a unitary matrix U such that $\rho '=U\rho U^\dag$. Hence, P is transformed to $P'=U^\dag P U$ and the intensities of P and $P'$ are the same without contradicting any postulate of TQM,

$$\begin{aligned} Tr(\rho P) = Tr(U \rho ' U^\dag P) = Tr(\rho ' U^\dag P U) = Tr(\rho ' P'). \end{aligned}$$

However, in the SQM formalism, due to KS theorem, there is no such transformation of the definite binary values from one basis to another basis. Thus, as it well known, the change of basis implies the change of the binary values of projector operators and consequently contextuality needs to be accepted and the Basis Invariance Theorem is precluded. Indeed, let $\{u_1,u_2,u_3,u_4\}$ be an orthonormal basis and consider the projection operators $P_1, P_2, P_3, P_4$ on these vectors which are all mutually commuting (and, hence, correspond to compatible observables, allowing a simultaneous attribution of values 0 or 1). Given that $P_1+P_2+P_3+P_4 = I$, it follows that

$$\begin{aligned} v(P_1)+v(P_2)+v(P_3)+v(P_4)&= v(P_1+P_2+P_3+P_4)\\&= v(I)\\&= 1. \end{aligned}$$

It follows that out of the four values, one must be 1 while the other three must be 0. In²⁴, the authors considered 9 orthogonal bases, each basis corresponding to a column of the following table, in which the basis vectors are explicitly displayed. The bases are chosen in such a way that each projector appears in exactly two contexts, thus establishing functional relations between contexts.

It is proven in²⁴ that the following is impossible: to place a value, either a 1 or a 0, into each compartment of the table above in such a way that:

• 1. the value 1 appears exactly once per column, the other entries in the column being 0;

• 2. equally colored compartments contain the same value – either both contain 1 or both contain 0.

With this proof, it is shown that the value of a projector $P_{k}$ in a context is incompatible with the value of that same projector $P_{k}$ in a different context. Hence, the Basis Invariance Theorem does not apply within SQM.

Final remarks

As we have shown, the replacement of matrix mechanics by a vectorial formulation implied the loss of an enormous amount of physically meaningful information about experimental situations and phenomena. We presented this impoverishment of the theory as an argument for a return to the matrix formalism. But, as we have shown in the last section, if we advance even further into a tensorial formulation, we can not only recover the phenomena captured by matrix mechanics, but also extend its reach and consider even more experimental situations that can be observed in the lab. Notice that while the case of a single screen corresponds to the orthodox vectorial approach, the case of two screens is linked to the orthodox extension to density matrices. It is important to remark that in our tensorial formulation these are particular cases of the more general situation where we have n screens. We are thus able, within Tensorial QM, to consider as many screens as we want without any complications. This leads, for example, to the possibility to naturally consider a multi-screen analysis of entanglement which is capable to evade the many problems found when attempting to measure quantum entanglement⁵³ and also within the multi-partite orthodox account of entanglement⁵⁴.

The originality of TQM and its program, which takes distance from SQM, can be now summarized in three main points:

• I. Recognize the operational-invariant theory of Heisenberg grounded on the empirical reference to intensities.

• II. Provide arguments against the unnecessary reference to binary outcomes and the existence of “collapses” —both of which destroy the operational-invariance of the theory— due to the dogmatic restoration of the classical picture.

• III. Taking as a standpoint Heisenberg’s original matrix formulation, produce not only a conceptual representation anchored in the invariance of that formalism (and maintaining its operationality), but also a formal extension (trough tensors) that promises to encompass a wider spectrum of experimental phenomena.