The Kochen - Specker Theorem in Quantum Mechanics by Vasil Penchev

Vasil Penchev

The Kochen – Specker theorem in quantum mechanics: A philosophical comment Highlights: Non-commuting quantities and hidden parameters – Wave-corpuscular dualism and hidden

parameters – Local or nonlocal hidden parameters – Phase space in quantum mechanics – Weyl, Wigner, and Moyal – Von Neumann’s theorem about the absence of hidden parameters in quantum mechanics and Hermann – Bell’s objection – Quantum-mechanical and mathematical incommeasurability – Kochen – Specker’s idea about their equivalence – The notion of partial algebra – Embeddability of a qubit into a bit – Quantum computer is not Turing machine – Is continuality universal? – Diffeomorphism and velocity – Einstein’s general principle of relativity – „Mach’s principle“ – The Skolemian relativity of the discrete and the continuous – The counterexample in § 6 – About the classical tautology which is untrue being replaced by the statements about commeasurable quantum-mechanical quantities – Logical hidden parameters – The undecidability of the hypothesis about hidden parameters – Wigner’s work and и Weyl’s previous one – Lie groups, representations, and -function – From a qualitative to a quantitative expression of relativity − -function, or the discrete by the random – Bartlett’s approach − -function as the characteristic function of random quantity – Discrete and/ or continual description – Quantity and its “digitalized projection“ – The idea of „velocity−probability“ – The notion of probability and the light speed postulate – Generalized probability and its physical interpretation – A quantum description of macro-world – The period of the associated de Broglie wave and the length of now – Causality equivalently replaced by chance – The philosophy of quantum information and religion – Einstein’s thesis about “the consubstantiality of inertia ant weight“ – Again about the interpretation of complex velocity – The speed of time – Newton’s law of inertia and Lagrange’s formulation of mechanics – Force and effect – The theory of tachyons and general relativity – Riesz’s representation theorem – The notion of covariant world line – Coding a world line by -function – Spacetime and qubit − -function by qubits – About the physical interpretation of both the complex axes of a qubit – The interpretation of the self-adjoint operators components – The world line of an arbitrary quantity – The invariance of the physical laws towards quantum object and apparatus – Hilbert space and that of Minkowski – The relationship between the coefficients of -function and the qubits – World line = -function + self-adjoint operator – Reality and description – Does „curved“ Hilbert space exist? – The axiom of choice, or when is possible a flattening of Hilbert space? – -function и -function – But why not to flatten also pseudo-Riemannian space? – The commutator of conjugate quantities – Relative mass – The strokes of self-movement and its philosophical interpretation – The self-perfection of the universe – The generalization of quantity in quantum physics – An analogy of the Feynman formalism – Feynman and many-world interpretation – The -function of various objects – Countable and uncountable basis – Generalized continuum and arithmetization – Field and entanglement – Function as coding – The idea of „curved“ Descartes product – The environment of a function – Another view to the notion of velocityprobability – Reality and description – Hilbert space as a model both of object and description – The notion of holistic logic – Physical quantity as the information about it – Cross-temporal correlations – The forecasting of future – Description in separable and inseparable Hilbert space – „Forces“ or „miracles“ – Velocity or time – The notion of non-finite set – Dasein or Dazeit – The trajectory of the whole – Ontological and onto-theological difference – An analogy of the Feynman and many-world interpretation − -function as physical quantity – Things in the world and instances in time – The generation of the physical by mathematical – The generalized notion of observer – Subjective or objective probability – Energy as the change of probability per the unite of time – The generalized principle of least action from a new viewpoint – The exception of two dimensions and Fermat’s last theorem Key words: Kochen – Specker theorem, generalized relativity, Hilbert space, Minkowski space, world line by -function, -function by qubits

At first glance, the work of Kochen and Specker reiterates well-known results:

The main aim of this paper is to give a proof of the nonexistence of hidden variables. This requires that we give at least a precise necessary condition for their existence (Kochen, Specker 1967: 59).

In fact, it was a revolutionary, new as a principle in regard to the proof and the foundation of the claim given initially by von Neumann. Before it, the non-existence of the hidden parameters in the quantum mechanics had been attributed to non-commuting operators and observables (e.g. in Dmitriev, 2005:435 summarizing the premises of von Neumann’s theorem). Kochen and Specker demonstrated the impossibility of hidden parameters even about commuting operators in quantum mechanics. Respectively, in the case of statements about commuting and therefore commensurable quantum-mechanical observables, classical logic is not always applicable, because its tautologies might turn out refutable and even identically false in quantum mechanics. Furthermore, after a more detailed look at their proof, we are going to underline the fact that, in their interpretation, the absence of hidden parameters is due to the necessity of common considering discrete and continual morphisms, i.e. to wavecorpuscular dualism in last analysis. Thereupon, they have tacitly comprehend hidden parameters as local ones since Lorentz invariance still remains in force restricting the generalization of the continuous functions as Borel ones, and this enables the precise translation of the commensurability of quantum-mechanical observables into mathematical language as a common measure in the rigorous mathematical meaning of the concept ‘measure’. So nonlocal hidden parameters − which are left outside the range of Kochen and Specker’s article − are completely and implicitly ignored by the justification that their Lorentz noninvariance implies their mathematical and physical incommensurability with the quantities to whose functions they should serve as arguments. On the other hand, Dirac delta function or Schwartz distributions (generalized functions) long ago involved in the apparatus of quantum mechanics do not require such mathematical commensurability of the areas of the argument and the values of the generalized function. Sometimes the local (Lorentz invariant) hidden parameters are unduly confused with hidden parameters in general (including the violation of Bell's inequalities opposite to Kochen and Specker’s results), but this confusion does not evolve neither explicitly, nor implicitly from their article. Kochen and Specker’s text – both rigorous and precise, also heuristic, and of radically new ideas and approach, not only gives rise to a great number of subsequent studies, but by now is away from depleting its intrinsic potential. In the beginning of their article the authors submit concisely their conception, which can be summed up as follows: if we look at the previous attempts to introduce hidden variables (e.g. the Bohm theory, 1952, or the description of the general model made by von Neumann - see Penchev 2009, ch.4), the paradigm of classical statistic mechanics shows up:

The proposals in the literature for a classical reinterpretation usually introduce a phase space of hidden pure states in a manner reminiscent of statistical mechanics. The attempt is then shown to succeed in the sense that the quantum mechanical average of an observable is equal to the phase space average (Kochen, Specker 1967: 59).

Von Neumann used to underline quite explicitly that the half of the variables of the configuration space of micro-objects are “superfluous”, redundant and simultaneously fully adequate to describe again the same micro-system if the other half of the same variables, in number used in the first description are now left aside as redundant. The two descriptions are incompatible, complementary, or dual in the intention of Bohr, but they both give the same probabilistic description of the micro-system, which as Schrödinger (1935: 827) highlighted is the all possible knowledge of it. Because of that reason the phase space must be modified to be applicable in quantum mechanics: one modification is made by Wigner (1932) and Moyal (1949) on the base of the preceding fundamental work of Weyl (1927): e.g. the basic cell in the classical phase space is the product of quantities – position and momentum – which are non-commuting in quantum mechanics; therefore each cell is duplicated according to the order of multiplying the quantities. As this is independently valid for each of the cells in the phase space, the variants of the phase space that have to be referred to the same quantum system are found to be as a number instead of the only single one in classical consideration. Since the observables in the two sets are conjugated, each with the one to which it is relevant , and their operators do not commute (e.g. position and momentum for every particular micro-object according to the uncertainty relation), there may be propounded the hypothesis by analogy, illegal as a strict logical inference, that just the noncommutability of the operators (or the observables in quantum as contrasted to classical mechanics) is the premise, the precondition for the absence of hidden parameters. Hence it becomes obvious that if hidden parameters exist, the physical quantities would commute with each other in the same way as in classical mechanics. As the non-commutability does not allow a physically relevant interpretation of the product and even the sum of two such non-commuting quantities (demonstrated in Hermann (1935) – Bell (1966) argument), "the back door" of our ignorance, behind which the cherished "true" hidden variables could be found eventually, remains anyway. Notice that we speak of another (second!) heuristic hypothesis by analogy. Kochen and Specker show categorically and unambiguously (i.e. by a counterexample) that the non-commutability of the observable variables is not the premise for the absence of hidden parameters: Commutability is not an indispensable condition for hidden variables, and thus they clear their way for formulating a logically strict indispensable condition, instead of the “heuristic” and in fact wrong hypothesis based on a misleading analogy to classical statistic mechanics. Their interpretation on the commensurability of physical quantities in quantum mechanics by the mathematical concept of “commensurability” (and hereby, of “measure”) is a decisive step. The measure of a function does not require the latter to be continuous, but only almost continuous, i.e. the measure of the set of points where it is discontinuous to be zero. Two quantities of a common measure are commensurable and commutable.

The algebraic structure to be preserved is formalized … in the concept of a partial algebra. The set of quantum mechanical observables viewed as operators 3

on Hilbert space form a partial algebra if we restrict the operations of sum and product to be defined only when the operators commute (Kochen, Specker 1967: 59-60). Nevertheless, although being commensurable and commutable, they do not allow hidden parameters as Kochen and Specker showed, since the indispensable condition for their presence is not fulfilled: exactly the embeddability of “partial algebra” – according to two authors’ concept, by which they formalise commensurable quantities – of quantummechanical quantities in commutative algebra. Respectively the statements on such quantities – so-called partial Boolean algebra – is not embeddable in Boolean algebra, or in other words, more contemporarily said, one qubit is not embeddable in one bit, quantum computer is not Turing machine.

A necessary condition then for the existence of hidden variables is that this partial algebra be imbeddable in a commutative algebra (such as the algebra of all realvalued functions on a phase space) (Kochen, Specker 1967: 60). Then

it is shown that there exists a finite partial algebra of quantum mechanical observables for which no such imbedding exists. The physical description of this result may be understood in an intuitive fashion quite independently of the formal machinery introduced (Kochen, Specker 1967: 60). So it comes natural to ask how can be explained the different behaviour of physical quantities in classical and in quantum mechanics − the determinism of the former and the indeterminism of the latter − if the demarcation: commutability – noncommutability makes no already sense. Obviously the only difference left is the continuality of the quantities in classical physics and their discrete character as a rule in quantum mechanics, or in other words – the validity of its field of the principle of quantummechanical dualism. The real premise for the absence of hidden parameters could be formulated as invalidity of Einstein’s principle of relativity (Einstein 1918: 241) and the resulting from it suspension of Mach’s principle (ibid.): the concepts of speed or resp., of diffeomorphism are not universal in regard to mechanical as well as physical movement. Along with this, the requirement for Lorentz invariance may remain in force, whereas the discontinuities appear to be in space-time and it corresponds to the velocity confined to the same maximum, which is defined by the fundamental constant of the velocity of light in free space. That is exactly the implicit model, with which Kochen and Specker comply, suggesting the ordinary consideration for hidden variables as local ones. That is the reason why their statement for the absence of hidden variables concerns only local ones and does not affect neither the type of investigation made by Bell, nor the possibility of violation the inequalities introduced by him. Here we should raise again the question of ineradicable insolubility, which faces any profound philosophical discussion of quantum mechanics. Because of the Skolemian relativity of the discrete and continual, the absence of hidden parameters also seems to be 4

Skolemian relative, including the manner of their exposition in Kochen and Specker’s article. After proving their famous theorem and its implications, they gave a counterexample introducing hidden parameters limiting their consideration to two-dimensional Hilbert space and a model of a single electron spin emphasizing that it is completely artificial and even invalid in the case of two electrons in a potential field according to their words. However, their intention was thus to show that von Neumann’s theorem requires in that case the absence of hidden parameters, while their own consideration would demonstrate the possibility of being introduced. In turn, we may easily show that this counter-example is isomorphic to a qubit, since it represents a sphere in three-dimensional Euclidean space, and because of the qubit additivity it can be transferred and consolidated for the whole Hilbert space. In other words, this is a counter-example also towards the very main theorem of theirs and is in direct contradiction to the immediate corollary. That is why the theorem should obtain the statute of one more claim unsolvable in quantum mechanics − the one side of a complementary, dual relation whose other side is precisely its negation. They together demonstrate the same suggesting that it is only one special case; on the basis of Skolemian type relativity we can talk about a special kind of quantum dualism: absence − presence of hidden parameters. But how then should we interpret hidden parameter? According to the illustration that Kochen and Specker have given, this is a random position on a disc, i.e. of a large circle of the sphere. In the general case of ordered sum of qubits representing Hilbert space, the hidden parameter will be the angle formed from the "axes" of Hilbert space, which represent an infinite number of embedded into one another decreasing oscillators. That angle may be interpreted as an initial moment in time: for example, if we have chosen a zero point of time for all oscillators, then Hilbert space as an ordered set of qubits will be displayed in a simple and determined manner by the hidden parameters as an infinite strip. That is also a respectively ordered set of zeros and ones according to the following (not only possible) rule: , if the hidden parameter determines a past moment in time corresponding to the chosen zero benchmark, and , if it determines present or future moment. Along with this, “curved” Hilbert space can be compared in a simple manner with pseudo-Riemannian space and thus be so interpreted. Therefore the parameter can be construed as gravity. These two interpretations of the hidden parameter − time and gravitational − again proved to be dual, which turns out to be a normal expectation in quantum mechanics. At the end of their article, in the last § 7 both authors suggest that their consideration may be logically demonstrated as the impossibility to be embedded (resp. weak embedding − homomorphism) the partial Boolean algebra of quantum-mechanical observables in Boolean algebra.

It is proved there that the imbedding problem we considered earlier is equivalent to the question of whether the logic of quantum mechanics is essentially the same as classical logic (Kochen, Specker 1967: 60).

Thence they deduce that there is a classical tautology , which is false even in meaningful substitution, i.e. the substitution with statements concerning commensurable quantum-mechanical variables:

Roughly speaking a propositional formula is valid in quantum mechanics if for every "meaningful" substitution of quantum mechanical propositions for the variables this formula is true, where a meaningful substitution is one such that the propositions , are only conjoined by the logical connectives in if they are simultaneously measurable. It then follows from our results that there is a formula which is a classical tautology but is false for some meaningful substitution of quantum mechanical propositions. In this sense the logic of quantum mechanics differs from classical logic (Kochen, Specker 1967: 60). And they immediately give a simple example of such a tautology. According to our principle position we will pay attention once again to the alleged relativity of this statement, i.e. from what kind of dual, complementary, but also quite legitimate position the opposite is true: the non-existence of such a classical tautology or no substitution of the quantum mechanical observables, which makes that classical tautology false. For this purpose the concept of hidden parameter should be transferred to properly logical consideration. That would be the presence of a hidden unsolvable statement, in other words, a hidden axiom. Thus whether it or its negation is accepted will determine whether the concerned statement on quantum mechanical observable is true or false. Embedability (weak embedment, homomorphism) is the necessary condition for the existence of such a logical hidden parameter. Respectively the absence of such embedability ensures its non-existence. Then our propositional formula which is a classical tautology will appear to be true also in substitution for commensurable quantum-mechanical obser-vables. Therefore, the very formula is from the desired type of unsolvable statement or a logically hidden parameter. In this case any propositional formula that is true in a classical sense and false in a quantum-mechanical substitution, as stated above, is such a logically hidden parameter, an unsolvable statement. There emerges a common and fundamental hidden parameter of such logical type, an unsolvable statement: whether randomly given formula will be considered as classical or as quantum-mechanical. That cannot surprise us at all, as it is built in the very foundation, in the structure and mathematical formalism of quantum mechanics as a theory about the system of a classical device and a quantum object. Accordingly, such insoluble, dually true propositions about the system can be solved when referred either only to the apparatus, or only to the micro-object. But the very second reference contains an element of insolubility and is hypothetical as a theory of the micro-object by itself does not seem possible. With a similar reservation reducing the mere statement about the existence of hidden parameters to insolubility, however, the opposition or the dualism between device and quantum object may be assumed and therefore interpreted in any special case as a universal hidden parameter in the logical sense.

Finally, the same situation can be demonstrated by the counter-example given by them, in which a sphere as a qubit will be compared to the propositional formula of quantum-mechanical observables and a usual bit, i.e. a binary unit, to the true value of the propositional formula classically interpreted. The conclusion of Kochen and Specker indicated the significance of their work to the overall development of thought in quantum mechanics, which we have already tried to sketch briefly:

This way of viewing the results of Sections 3 and 4, seems to us to display a new feature of quantum mechanics in its departure from classical mechanics. It is of course true that the Uncertainty Principle, say, already marks a departure from classical physics. However, the statement of the Uncertainty Principle involves two observables which are not commeasurable, and so may be refuted in the future with the addition of new states. This is the view of those who believe in hidden variables. Thus, the Uncertainly Principle as applied to the two-dimensional situation described in Section 6 becomes inapplicable once the system is imbedded in the classical one. The statement , we have constructed deals only in each of the steps of its construction with commeasurable observables, and so cannot be refuted at a later date (Kochen, Specker 1967: 86). Let us start our detailed discussion of the work of Kochen and Specker from the possibility, the difficulties and the ways to use the phase space of classical mechanics and thermodynamics since it acts as a bridge between the statistical interpretations of the latter by the former, and thus sets a successful example for the introduction of "hidden parameters". Therefore, any confirmation of such impossibility must clarify precisely what exactly is the difference between classical and quantum mechanics, which deters us from following this method. We also have the major works of Weyl (1927), Wigner (1932), Groenewold (1946) and Moyal (1949), which show mathematically rigorously the degree of correspondence between Hilbert and phase space. They demonstrate how and by what necessary generalizations of the classical phase space in the latter may be present and deployed by the standard formalism of quantum mechanics based on Hilbert space. Fundamental is the work of Wigner. As for the study of Weil, it is historically the earlier (1927) and is based on the theory of groups, ones of the most simple and fundamental objects of the abstract algebra equipped with a single binary operation, a reverse element to any and a single neutral element coinciding with its reverse element. Also interesting is the theory of representations1 â&#x2C6;&#x2019; Hermann Weil should be assumed as its founder2 â&#x2C6;&#x2019; and the study in quest, which clarifies the meaning of such an abstract mathematical theory to quantum mechanics at the same time.

And in particular, Lie groups in the automorphisms of Hilbert space. A work (Peter, Weyl 1927) co-authored with his student of the same year should be mentioned in a properly mathematical aspect. Its main theorem essentially ensures that any group fulfilled certain broad conditions can be juxtaposed one or even one-one Hilbert space determined by its orthonormal basis if the group has a representation into it: In other words, representation is the condition (its boundaries of necessity or sufficiency could investigate in different cases) for identifying of a group with (a) Hilbert space. 2

The main idea of the theory of representations − the identification under certain conditions, namely the availability of representation in general, of the groups and of (the transformations of) Hilbert space will allow us to make a decisive step forward in studying the relativity of the continuous and discrete in a mathematical and in a physical, and in a philosophical sense as well. If the group is not only continuous but also smooth, i.e. differentiable, such as Lie groups are, we could equate it, at least mathematically, by its presentations, to Hilbert space of -functions, i.e. of quantum, therefore discrete states. If the very Lie group embodies Einstein’s principle of general covariance (relativity), we should clarify how exactly (or namely) -function presents a quantum, discrete state. It will help us to move forward from a merely qualitative relativity of continuity and discreteness to a quantitative (in a broad sense, by mathematical structures) description of their unity and the transition between them. -function presents the discrete by the random as follows. It is always a function of arguments consisting of exactly half the parameters in the configuration space than in the classical case and those parameters may be considered as continuous ones. The other half – according to Heisenberg’s uncertainty – proved to be completely vague, random, and discrete. Since there is a quantum leap, discreteness, that second half of the parameters appears to be a set of random variables, which may take one or another value with different probability. Then we will interpret -function, in the spirit of Bartlett (1945)'s approach, as the characteristic function of the discrete and therefore random coordinates in configuration space. The other half of coordinates in configuration space simply do not need a description by - function, since being continuous, they are not random. From this point of view "the problem of hidden parameters" appears to be a result of misunderstanding: -function does not summarize, but only complements the continual description of classical physics with its discrete "mate", where the discrete is represented by the accidental. The other "half", i.e. the continual description itself is given by the inevitable in quantum mechanics presence of classical device. Hence the importance of the theory of representations for the interpretation or creating the ontology of quantum mechanics: it provides the possibility, unity and quantitative equivalence of the discrete description of quantum phenomena in terms of micro-object and their continual description in terms of device. In such "translations" between both languages, we should pay special attention to the consubstantiality and the equivalent transformation of the speed from a smooth description (i.e. not only mathematically continual but also differentiable) in the probability from a discrete description. There comes the conclusion that Lorentz invariance (and respectively the postulate of no exceeding the speed of light in free space) is a direct result of a principle already involved in the previous sentence, which is valid for the imposed generalisation of Einstein’s relativity principle also for discrete motions: as gravity and inertia are treated equally in general relativity, as velocity and probability should coincide as to the sketched more general view. However, this would be possible only if there is a fundamental constant of maximal velocity, in relation to which any velocity is converted to a

dimensionless number that for all less or equal to the maximal velocity is respectively less than or equal to one and can therefore be interpreted as a standard probability. If however, using Bartlett’s approach, we have introduced negative probabilities (and hence those which are greater than one), then they should be discussed also speeds exceeding that of light according to a principle of equivalence of velocity and probability. Conversely, the emerging from the mathematical formalism of special relativity complex speed or other kinematic physical quantities getting complex values are immediately interpreted as the explained above complex probabilities or the physical quantities of entangled systems studied by quantum information: The tachyons theory developed in the second half of last century could be identified with quantum information or more exactly, with its translation into the diffeomorphism language of classical physics. So Wigner function (Wigner 1932: 750) is already in fact interpreted as the corresponding and earlier translation into the classical language of smooth transformation from the previously postulated discreteness of quantum mechanics. On the one hand, our world well-described by classical physics allows an equivalent quantum description towards a sufficiently massive mega-object losing its causality equivalently replaced by randomness. On the other hand, we could extrapolate the situation regarding micro-objects studied by quantum mechanics and information hypothetically introducing an analogous classical physical description for them (by diffeomorfisms, causal, using as a hidden parameter the moment of time within the almost eternity of their own present). A similar hidden parameter, of course, can no way be defined in terms (quantities) of the massive object. That is why we can genera-lize in the spirit of Skolemian relativity that both following statements are valid: there is and there is not ‘hidden parameter’ in quantum mechanics: The latter is from the viewpoint of the appliance, the former from that of the micro-object. In classical physics, in science or even in knowledge at all, the empirical and the objective never reach to such a direct impact to each other. On the other hand, however, a similar collision is typical for religious views, including and especially for the Christian ones. The objective is interpreted also as the hidden, non-empirical, also as the random, non-causally impacting on the practical world, also as the ideal, non-material, and also as the numinous, sacral, non-profane.3. If you return to a quantum reading of GR and take into account „the consubstantiality of inertia and gravity“4 (Einstein 1918: 241), you should mean that not only descriptions by gravitational and any potential field are equivalent to each other mathematically (or if we allow ourselves to express so, consubstantial), but from our viewpoint the quantum description of the same object to a mega-object is equivalent to 3

It is clear that the philosophy of quantum information looks at religion and religious experience with impossible for another scientific discipline or philosophy sympathy and interest in anticipation to be presumably synthesized the viewpoints of those two thorough enemies, which, being polar, but of the same doxography, outline or at least sketch the episteme of fundamental and historical moment preceding the articulation of fundamental history. (Пенчев 2010: 116124). 4 The exact quote is as follows: „b) Äquivalenzprinzip: Trägheit und Schwere sind wesensgleich. Hieraus und aus den Ergebnissen der speziellen Relativitättheorie folgt notwendig, daβ der symmetrische ,,Fundamentaltensor" ( ) die metrischen Eigenschaften des Raumes, das Tragheitsverhalten der Körper in ihm, sowie die Gravitationswirkungen bestimmt. Den durch den Fundamentaltensor beschriebenen Raumzustand wollen wir als " -Feld" bezeichnen ” (Einstein 1918: 241).

them, too. Clearly, the theory of representations, offering a mathematical language to identify and explain the isomorphism of potential (gravitational) and quantum randomness, along with that, guides us to their eventual consubstantiality. If quantum mechanics by -function shares an equivalent discrete description of any potential field, and general relativity has already enacted the indistinguishability of arbitrary potential and gravitational field, we have no more choice except concluding that -function should be an equivalent, but discrete description of gravitational field: For example, then the descriptions of the universe by its -function or by metrical tensor in any point of space-time are at least isomorphic, and following an Einsteinian kind of Pythagorean ontology (Пенчев 2010: 300), one in essence, besides. In our physical world a model can be represented by dissipative system from chaos theory. By it we could read as the alleged hidden parameter of quantum mechanics any “gently swinging the wings of a butterfly” in the appliance that leads to the measured result about the investigated quantum object in a dissipative, but causal way, anyway. Such an approach very reminisces Bohm (1952, I: 171)’s interpretation. However according to Kochen – Specker’s theorem and its corollaries a coherent system is impossible to be reduced to any de-coherent state. Consequently, dissipation can be accomplished only by the system environment or in other words, by the system non-standard, entangled, external parts, i.e. by the device as which all the rest can be considered. If we think of the universe as a whole without parts or environment, then all other possible states of its force it to leave the coherent state in favor of the single real state. Who accomplishes a choice? According to the axiom of choice a system can make the choice by itself. According to the Kochen – Specker theorem it is impossible to do that by itself, and the choice is only forced by its environment or by its possible states. Therefore we should again speak of a Skolemian type of the relativity of freedom and necessity, this time. Our central interest is the pending identification of the alleged superluminal area of SR (the tachyons theory) with GR on the base of the already proposed identification of any potential field (GR), but also with one-one discrete morphism, and hence with a certain superluminal speed transferring us into the real domain of Minkowski space (SR). The idea of interpreting GR as the theory of tachyons, i.e. as the superluminal GR generalization, is not only quite new and unexpected, but suggests even having to be particularly highlighted. The reasons are several: Even the very name and the intention of general relativity suggest that it to be interpreted as a generalization of SR as to non-inertial (or arbitrary) reference frame. Along with that the quoted already many times principle of Einstein’s relativity (1918: 241) constrains all physical movement to diffeomorphisms, so that non-inertial systems which are obtained from one another by a discrete (quantum) leap are not considered. Just because of that the idea of a possible generalization of the principle or theory of relativity as well regarding such a kind of reference frame is able to be suggested. Moreover, it seems that the notion of relative speed cannot be defined in a nontrivial way, since it is always infinite, and besides, violates the principle of no exceeding the light sped in free space, correspondingly, Lorentz invariance.

It is useful to emphasize a relationship with the representation theorem (Riesz 1907) to try to clarify how the already sketched identification of GR with the superluminal extension of SR (the theory of tachyons), first, is easy to be transferred as isomorphism between pseudo-Riemannian and Minkowski space, and secondly, gains some redundancy of the thought of “curved” Hilbert space, since that isomorphism can be conducted, namely by the Riesz theorem, also between usual standard and alleged “curved” Hilbert space. Suffice it to introduce the concept of covariant world line matching the usual contravariant one if and only if the curvature of pseudo-Riemannian space in any point of the world line is zero. Obviously, the condition is met for each world line in Minkowski space. If you introduce a more natural and reasonable, and one can say, and traditional principle of invariance of physical laws in the transition from quantum micro-object to measuring instrument (in particular, it implies the mega-interpretation of quantum mechanics), which establishes the equivalence of any usual continual and a new, discrete description of the surrounding macro-physical reality in relation to mega, i.e. universe objects. The resulting common micro and mega-interpretation is called relative interpretation (Пенчев 2009: 127). Its relativity affects the equivalence as of the relation of micro and macro and of macro and mega as of the relation of discrete and continuous models. This approach is similar to that of Schrödinger: -function to be interpreted as a „list of expectation“ (Schrödinger 1935: 827-828), i.e. as a „description“. He also discussed it as “reality”, therefore in an epistemological sense, which is opposite. There is more information in the world line than in the -function and it should regard as reality from this perspective. From the other hand, we cannot recover the real world line from all the set of experiments, as we have not any empirical or experimental access the alleged hidden parameter, which we could call the moment of projection and which appears to be due to an uncontrollably random superposition of a large number of contributing factors associated with macro-measuring: if we did so, we would be near to Bohm (1952, I: 171)’s position. There is always a morphism, whose physical interpretation is the time reversion, swapping places of calculation and meta-calculation, or respectively, of the co- and contravariant world line as well as, of the set of all subset of a given one and it itself, which is rather unexpectedly. That will be allowed only if any set is always a set of subsets of another (Пенчев 2009: 235). The conclusion is: continual representation by force or (gravitational) field be-

tween parts, interacting by means of it, of a system is equivalent to entanglement between them if representation is discrete. Gravity (force field) and entanglement are two different, correspondingly continual and discrete, images of a single common essence. Here we encounter a Skolemian type of relativity between discrete and continual models, between a system as an indivisible whole and as an ensemble or even sum of its components, between entanglement and (incl. gravitational) forces. On the poles of philosophical reflection, continual and discrete, countable and countless, “curved” and “flat”, physical and logico-mathematical, material and algorithmic, holistic and calculate models proved to be really united, but splitting up into these polar images, which should 11

be different only and seemingly rather only by tradition. It is Hilbert space that manages to display that unity. Getting once again on a properly philosophical position, we have to raise the issue of mapping between description and reality, or on the poles of inevitable speculation, between subject and object, in our case: a quantum physical object described by -function. For classical philosophy, the description is always different from reality, in most cases it is not more than an extremely rude and imperfect copy. This viеw, prevalent among scientists and philosophers of science, is opposed to the position got a complete expression in classical German philosophy, in turn reflecting Christian faith, about an absolute entity, God, an abstract philosophical essence as the “absolute subject”, or human being creating reality and thus, conversely, the reality is a slight and very imperfect copy of a subjective intent. It is hardly the 20th century philosophy, namely phenomenology, the doctrines of Husserl or Heidegger, that put the problem about such reality that is synonymous with its description, so-called ‘phenomenon’, and by its postulation to legitimize philosophy as science, the science of “phenomena”, i.e. “philosophy as a rigorous science” to be just ‘phenomenology’. Is there a mathematical structure to model both an object and its description and which can display the relationship of identity between them, figuratively speaking, a “phenomenon” mathematical structure? Hilbert space matches such a type. We could look at world line as an object by itself, and on -function as its description. On the one hand, we could add to -function the corresponding hypermaximal operator or a set of such ones so as to get the object just as the movement of a material point or as the change of the quantities characteristic of it. On the other hand, the very -function choses exactly a single world line as the movement of a material point and in a sense it is that description which coincides with the object, it is a “phenomenon”. So, in this second case, we deal with the phenomenon of the thing in relation to the latter. The phenomenon can play the role of logical subject in spite of being variable in time and having many physical quantities, which are not phenomena as the -function has to be complemented by the corresponding self-adjoint operator characterizing a concrete quantity. However, any quantity can be considered as the substance of the thing, and any else as its predication.This allows us to make identification, by which to move ourselves to the position of classical ontology and logic, by taking the entire class of specific quantities as the very thing, since each one of them can be considered as that substance – a phenomenon, to which the rest are predications. The prospect of an also Skolemian relativity of phenomenon and object, of individual and class is now outlined in front of us. @@@@The made just discussion should be added to in the following sense. Not only the mechanical movement can be seen in the logical and discursive perspective of description, but conversely, any logic of something or a concrete discourse is isomorphic of some mechanical movement and consequently, we can direct to them also from this position, again by means of -function, but already discussed as an iterating and iterative set of projective operators in the spirit of von Neumann’s classical foundation of quantum logic.

The notion of time as in physical as in philosophical sense allows us also to correlate infinite and finite choice: infinite choice mathematically guaranteed by the axiom of choice and embodied in the totality of time usually named eternity, and finite choice which is empirically given to any human being in present. If we now apply the Skolemian type of relativity, but no longer to the continual and discrete, and to the infinite and finite, then we head to an existential reading of eternity like that, witnessed extensively in the work of numerous authors from which we choose to mention Berdyaev, Shestov, Heidegger and Assen Ignatov. One could formulate the following principle of a rather philosophical nature: description is not more complex than described (reality), the cardinal of the former is less than or equal to that of the latter. Description can be treated as encoding the described, and the physical quantity of information as the product of those two factors. Along with the above, ‘forces in reality’ can adequately and equivalently be represented as the entanglement of descriptions. Jumps, „miracles“ from the discrete description have a parallel continual physical description only within which they are able to be thought as physical forces or fields according to the setting or prejudices of modern science. A-causal description by means of feedback from the future is systematically overlapped by causal one, which has as if the crucial advantage of successful scientific predictions, which do not influence, do not “shift” the future by their very nature. The notions of subjective or objective probability allow for us to distinguish correspondingly subjective or objective time being reconstructed on the base of them (e.g. as the expectation of a subject or the frequency a given event to occurs). In both cases, time is assumed discrete as it jumps from the present to the fixed future moment when the event is going to occur. In the case of objective probability interpreted e.g. as frequency, a statistics is available as to the realization of the event in many cases: the numbering of the latter may be regarded as a parameter or as a name of many parameters. The case of subjective probability means a unique event, which rejects the possibility of any hidden parameter because of its indivisibility. An iterative procedure can be imagined such that it converges on a hypothetical absolute subject gradually covering more and more real ones who form a common (to call it “expert”) opinion on the probability for the target event to take place in reality. We can express the hypothesis that the two limits each tending to infinity correspondingly of individual events or actors will coincide, besides suggesting its fundamental, axiomatic character. Another option is its negation, namely that there is in general a mismatch between the value of the objective and subjective probability, possibly varying from one to another event: We will call it ontological difference. Let us emphasize that we define a strictly quantitative expression for ‘ontological difference’: the difference or the ratio between the limits (in infinity) of subjective and objective probability, to which a difference or a ratio of the restored to their base discrete (subjective and objective) times will correspond.

However, if you stand on the position of a match, therefore continuity between subjective and objective time, then you could weaken the hypothesis of smooth transition between them: It is only the infinite values that have to coincide, but not both first derivatives correspondingly “on the right” and “on the left”, which both must exist. The limits of converging by subjective and by objective probability do not coincide. Such an option split the concept of ontological difference by the hypothesis of the identity/ non-identity of infinity: we will denote it as “theological difference”, and being different from “ontological difference”, it suggests a topic about “onto-theological difference”, of course, in a partly quantitative and emphasizing its limited aspect. But which is the case realized in quantum mechanics if we look at it as a check on that: which of the above suggestions as theoretical possibility has taken place in reality? There is a statistics (frequency) as in the case of objective probability, but it precludes any hidden parameter, as in that of subjective probability. One (and perhaps the only possible) solution is matching the subjective and objective probability, i.e. zero ontological and therefore onto-theological difference. We are going to denote that case as “quantum probability” and understand exactly matching the subjective and objective probability. We can add that the concept of probability seen as the ratio of infinites in a finite limit allows to clarify how and why its involvement in quantum mechanics gains a definite quantitative expression for any discretization as regards the infinite speeds obtained as a result because of the zero time to perform any quantum leap. While infinity is not empirically and therefore experimentally attainable, it is already subjective probability (of an absolute subject), postulated as coinciding with the objective one and by means of it, that is. While and if we have assumed that present is before the infinity of past, and that of future is after it, we could use the term of “non-finite” for a similar status after arithmetization. Non-finite sets are built as follows: that set whose set of subsets is countable is non-finite. (Пенчев 2005: 60-62). Present is located in non-finiteness: after any finiteness, but before any infinity. It is the relation between non-finiteness and infinity that is represented as probability and can describe discreteness. In the context of the above, another approach is possible to the comparison of subjective and objective probability, respectively, of subjective and objective time. We would like to recall the idea of a non-standard whole, which includes its externality as its unusual part. In this sense, one part is considered as the very whole: the other parts towards the standard whole are accepted as non-standard, or external, ones of that part being simultaneously the whole. We could use the descriptive term “here-and-nowintegrity” if we have understood “here and now” as that privileged part, on whose position we have stood. Appropriate is the analogy to the established and fundamental term of Heidegger’s philosophy “Dasein” as well as to the neologism “Dazeit”, which is possible to be introduced on its base (Пенчев 2009Х: 48). Further we could assign the subjective probability to the privileged part, while the objective one to its “negative image”: i.e. by means of the non-standard, external 14

parts of its. In a generalized sense, the trajectory of a part can be determined by an analogous to the principle of least action, according to which the trajectory is accomplished through the smallest, in particular zero ontological difference. The pathway of least or no ontological difference may be indicated, using the terms of Chinese philosophy, which have been entrants after Bohr in that of quantum mechanics, such as Tao, and subjective and objective probability, or subjective and objective time, correlate the couple of Yang and Yin5. When we generalize the concept of “trajectory” from three-dimensional to Hilbert space, we replace point (an element of a function) with function. The concept of “continuous trajectory” must therefore be reviewed. In our examination by means of qubits, continuous trajectory refers to all of them and thus is limited to the continuous path of points. Furthermore, the change of probability per time unit is directly interpreted by the quantum of action, i.e. for the values smaller than it. We have to pay special attention to the text in italics, because, by the assumed and interpreted relationship between energy and probability, we can transform sub-quantum into super-quantum reality including the macro-world. Roughly speaking, the entire universe can look like the inside of one quantum; inversely, the super-quantum area can be considered as the outside of the universe: It is very interesting that the vicinity of any its internal point is able to be discussed as its outside. Indeed, Hilbert space interpreted as a space of -functions equivalent to world lines has precisely this property, which can be called fractal probably not only metaphorically. Each line describes an eternity of the universe as the continuity of past, present and future: note that the description originates from one reference frame. The impossible and self-contradictory6 external movement of the universe is described as the internal one: It is enough to “change the Gestalt” and look at each -function like a description of the state of the universe and only secondarily, and as a result of the former like a description of a physical object inside. The universe consists (internally) of its (external) states. The transition from stochastically determined chaos to probabilistic quantum states converts its parts into its states. Therefore, the Skolemian relativity of continuity and discreteness is easily to be transformed into the relativity of externality and internality permitting a single common description. Then energy corresponds to the values of change within a quant or a discrete jump. On the other hand, we have used the probabilities for infinity: the probability is a bridge, through which we are able to think uniformly as infinitely great as infinitely small quantities as infinitesimal, but in terms of discreteness embedded by quantum. Since in our approach, action corresponds to the change of probability, the principle of least action is converted into a principle of least probability change, which is 5

Such a transformation clears that Tao is just another, but more accurate way to be described onto-theological difference insofar as the latter is, as a kind of speaking, “a function of here and now”, or using an aphorism: Tao is the destiny of Dasein. In other words, the Tao presents the presence of God in this world, or translating by the terms of classical philosophy and therefore being bound by an oxymoron, as “transcendental transcendence” (Пенчев 2007: 10). Obviously, such an approach is much closer to the kind of ecumenism, to “God as a human ability”, or conversely, to detecting the possibility of cooperation between the sacred and profane. 6 Since it unites the non-self-identity of external identity and the definitive reflexivity of the universe as a single whole. Therefore the outcome which “is found” by the universe we can apply to solve similar difficulties lush in many paradoxes and in “the crisis in the foundation of mathematics” as to the set theory. For this purpose, a minimum base structure should be drawn from Hilbert space which would allow us to transform infinity into a point of that infinity.

intuitively perfectly acceptable: the transition from one to another -function must take place through the least change of probability. In particular, in the case of the transformation corresponding to a physical quantity (i.e. by a hypermaximal operator), the amendment of probability is zero, and we have already noted that all transformation of that class may be related to the present of the object possessing that -function. Time can be described as the density of probability for transiting from one to another -function, in which the former is interpreted as the present of the quantum object in question, while the latter as future or past states. The probability density function will be maximal in the present, to the maximum of which the past values will grow monotonically by different slope, and those of the future in turn will decrease, forming a typical bell shaped probability density curve. Negative probability will be then naturally inter-preted as a probability for converting in past states, i.e. back in time, while the complex one will be referred to the time axis distortion. The fundamental constants – the Plank and the speed of light – are a natural units set allowing the parts of them to be interpreted as probabilities, and continuity to be transformed into the internality of a discrete (quantum) jump. If we mean to type a summary of the principle of least action, it appears that may cover a wider area, namely Hilbert space, and as long as the amendment of the probability in time is interpreted as the frequency of de Broglie wave. Such an interpretation is natural because probability, thought even elementarily, is defined as the ratio of the part of the alternatives accepted as favorable (whether in anticipation or as a statistical frequency) of a choice towards all of them. Then de Broglie wave corresponds to the amendment of the positive alternatives if the set of all the alternatives remains the same (in other words, the change of the favorable ones is for the account of the negative ones). Such an approach, besides suggesting an unusual perspective on energy, respectively matter, assumes that the law of energy conservation roughly approximates the linear area of increasing (decreasing) the probability of the bell curve of probability density distribution, or of the linear change of the positive alternatives. In areas, in which such an approximation is significantly different, forces, fields, and their energies should appear to compensate for the inaccuracy of approximation. We tend to interpret the somewhat organized abundance of elementary particles and the four main forces (fields) of the standard model as “Ptolemaic cycles”, figuratively speaking. It is the probabilistic approach of “classical” quantum mechanics that favors the almost linear section as it prevails decisively. The principle of least action as a principle of least changing the probability actually includes also the case of its amendment in time corresponding to the above introduced concept of theological difference. In terms of the alternatives accepted to be favorable of a choice, ontological difference is the difference between the amount of favorable and unfavorable alternatives and all the alternatives7, while theological difference is that 7

The said does not contradict the interpretation of subjective probability as a jump of an expectation, which is realized as a smooth change of the objective probability by an also gradual shift of the favorable alternatives in a real movement. Zero ontological difference postulates their coincidence. We can likewise use a metaphor for the subjective probability as a negative imprint and "expectation" determined by anything else in relation to a given thing so that its amendment to modify the rest in the slightest possible degree.

between all the alternatives (e.g. at different points in time). Theological difference, however, should be understood sufficiently generally: since all alternatives form a totality, it means that we can assume for the difference as preservation as change in an equal degree. Therefore, theological difference seeks to quantify the relationship of the totality to itself. Finally, the context of the work can be used to address a few − seemingly disparate and unrelated − exceptions for the dimensionality of : in Gleason’s (Gleason 1957: 132), in Kochen – Specker’s theorem (Kochen, Specker 1967: 70; § 6; Specker 1975: 139-140), and at last … Fermat’s last theorem proven at the end of last century by Andrew Wiles. Kochen and Specker directly indicated that the displayed result can be obtained from the theorem of Gleason. (Kochen, Specker 1967: 70). However our work has grounded on the isomorphism of qubit and three-dimensional sphere (which in fact is similar to Kochen and Specker’s counter-example of the § 6 of their article), on the representation of Hilbert space by qubits and hence, by Minkowski space. Further a surprise occurs: as three-dimensional sphere is obtained from the complex Hilbert space of dimension , and Minkowski space is an additive combination of qubits, at that representable as Hilbert space, then there is a direct pathway the exception about dimensionality two to be transferred to infinite dimensions. It should however again be noted that this is not a way to carry over to any finite dimensions. Further the axiom of choice carries the outcome to any transfinite power provided it to be valid for that power. We have statements − all of those that appear by the exception of dimensionality , − which are not true for any finite integer, but they are true for an infinite number. The only way out of the situation, if we are to preserve the principle of induction, moreover it is included in the Peano axiomatic of integers, is to accept that there is a number that we cannot point out8, for which that type of statements are not valid. Argued could immediately referred to Fermat’s last theorem if we have taken a sufficiently powerful axiom of choice as long as higher dimensions are obtained multiplicatively. Is the axiom of choice or weaker version of it (the theorem of prime Boolean ideals9) used in Wiles’ proof (McLarty 2010) to be displayed that the field of rational numbers has an algebraic closure or all fields have algebraic closures Wiles (1995; Taylor, Wiles 1995)? On the base of just made considering as well as the whole context of the article we tend to insist on a negative answer: rather Fermat’s last theorem as in Wiles’ proof as at all10 is equivalent to the negation of the axiom of choice or even of a stronger version of it. It could display by its eventual deduction from the Kochen – Specker or Gleason theorem. Instead, in the spirit of the philosophical nature of this discussion, we may propose the following problem: whether the propositions valid for any instant of time are valid for the eternity, i.e. whether they are tautologies. In statements on eternity, we find ourselves in an analogue of well-known difficulties on the set of all sets; in the case: the 8

Pure, i.e. nonconstructive in principle existence proof. It states that all ideals in a Boolean algebra can be extended to prime ideals. A variation of this statement for filters on sets is known as the ultrafilter lemma. 10 We would like to pay special attention to the notion of the admissible representation of a group (incl. Lie group) in Hilbert space (Cornel, Silverman, Stevens 1997: 165). 9

alleged validity / invalidity of the claim on all valid claims. Obviously, the proposed isomorphism between Minkowski space and Hilbert space and their physical interpretation because of the exception of alleged impossibility usual Boolean logic to be embedded (Kochen, Specker 1967: 70; Specker 1975: 139-140) as to dimension or transfers its statements which are valid, i.e. tautological, for any moment of time into “valid at all”, i.e. valid as to eternity, however even if they are not valid as to any future moment (but not present one!)11. It turns out that the usual binary logic has an unusual privilege with regard to eternity: in a sense it makes or brings an equivalence characteristic to existentialism, between present and eternity, interpreting eternity by generality quantifier as “for all presents”. Finally we will apply the used argument also to the theorem that a field (i.e. multiplication is commutative) of dimensionality higher than two does not exist: the principle of induction requires such to exists anyway: Of course, it comes to a clean and unconstructive proof of its existence. But later, according to the conventional interpretation of the theorem known as the “paradox Banach − Traski“12 (Banach, Tarski 1924: 244), a qubit is equivalent to two ones, hence ultimately to Hilbert space as a whole, as a model of the Universum. According to the theorem of Kochen and Specker (more precisely, as its direct consequence) it cannot be represented as a bit. However being valid the axiom of choice, it should be able, as well as according the very example (as a counter-example given by them in § 6 of their article). In last analysis, the starting point for their consideration may be reduced to a kind of a “bit”: wave-corpuscular dualism, or according to the discussion made in this study, the Skolemain or Einsteinian type of relativity between discreteness and continuity. Choice within a relativity is guaranteed, but immaterial. Therefore, the primal philosophical choice we have made is between the importance of the very choice and the relativity of its alternatives, and hence their immateriality, ultimately, of the very choice. In gnomic words, the being of the world is reduced to a single choice (even of the simplest kind, between two equal possibilities, i.e. to a single bit). On the other hand, this is trivial, because the question might be: is there a world? But that problem turns out most surprisingly (of course, not to the successors of the anti-metaphysic trend à la Wittgenstein in the con-

The statements about invalidity towards an arbitrary future, but not present instant of time turn out to be selfcontradictory on the base of the proposed argument. So „the problem of unobservables“ (Feyerabend 1975: 110) after interpreting quantum mechanics by means of three-values logic (Reichenbach 1975) reveals its self-contradiction. They are statements about unobservables which turn out to be contradictory as such ones about future moments in our approach. 12 Particularly should be emphasized that in this case − and in conjunction with the previous ones, − we encounter again a remarkable exception for dimension or : "In the Euclidean space of dimensions any two limited sets containing their internal points are equivalent in a finite decomposition. There is an analogous theorem for sets located on a sphere, but the corresponding theorem on Euclidean space with dimensions or is incorrect "(Banach, Tarski 1924: 244). The proof of the theorem uses the axiom of choice (ibid.: 244-245). Actually, received two spheres from one is widespread interpretation (or even replacement) of the theorem, which does not appear in the original work. Whether a set consisting of two spheres is “a limited set, which contains its internal points”? One area is a compact set, but two ones are not in general. If we take two hemispheres of the initial one and by the theorem, generate those two areas from them, whether that one of the two hemispheres will not contain dividing them circle would violate the conditions of the theorem? Whether another arbitrary set including its entire contour can also build by a set in the Euclidean space of three or more dimensions, deprived of its contour partly or entirely? Maybe we are faced with a topological equivalent of the relativity of one or two quantum systems.

temporary philosophy) to be immaterial: in Skolemian way, even being and non-being are relative. If we pass the route in reverse order, we can create the universe, including gravity, the ensemble of all the possible states of it: each of them is simultaneously the actual state of some part of it. In other words, we can create the universe as “consisting only of itself”, but not limited to acting as one, i.e. a whole, and also of all of its own parts equivalent to the states of the whole. Summarizing the whole current statement, we can highlight several major problems: 1. The fundamental importance of the axiom of choice in the discussion of issues around the theorem of Kochen and Specker. 2. The status of the theorem of Kochen and Specker: Is not it an axiom? 3. The relationship of the axiom of choice and the theorem of Kochen and Specker: whether and how the latter can be seen as a direct negation of the former?

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