The Kochen - Specker theorem in quantum mechanics: A philosophical comment

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  • 1.Vasil Penchev The Kochen Specker theorem in quantum mechanics:A philosophical commentHighlights: Non-commuting quantities and hidden parameters Wave-corpuscular dualism and hiddenparameters Local or nonlocal hidden parameters Phase space in quantum mechanics Weyl, Wigner,and Moyal Von Neumanns theorem about the absence of hidden parameters in quantum mechanics andHermann Bells objection Quantum-mechanical and mathematical incommeasurability Kochen Speckers 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 Einsteins general principle of relativity Machs principle The Skolemian relativity of the discrete andthe continuous The counterexample in 6 About the classical tautology which is untrue being replacedby the statements about commeasurable quantum-mechanical quantities Logical hidden parameters The undecidability of the hypothesis about hidden parameters Wigners work and Weyls previous one Lie groups, representations, and -function From a qualitative to a quantitative expression of relativity -function, or the discrete by the random Bartletts approach -function as the characteristic func-tion of random quantity Discrete and/ or continual description Quantity and its digitalized projection The idea of velocityprobability The notion of probability and the light speed postulate Generalizedprobability and its physical interpretation A quantum description of macro-world The period of the as-sociated de Broglie wave and the length of now Causality equivalently replaced by chance The philoso-phy of quantum information and religion Einsteins thesis about the consubstantiality of inertia antweight Again about the interpretation of complex velocity The speed of time Newtons law of inertiaand Lagranges formulation of mechanics Force and effect The theory of tachyons and general relativity Rieszs 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 axesof a qubit The interpretation of the self-adjoint operators components The world line of an arbitraryquantity The invariance of the physical laws towards quantum object and apparatus Hilbert space andthat 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 -function But whynot 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 andmany-world interpretation The -function of various objects Countable and uncountable basis Ge-neralized continuum and arithmetization Field and entanglement Function as coding The idea ofcurved Descartes product The environment of a function Another view to the notion of velocity-probability Reality and description Hilbert space as a model both of object and description The no-tion 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 ormiracles 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 physi-cal by mathematical The generalized notion of observer Subjective or objective probability Energy asthe change of probability per the unite of time The generalized principle of least action from a new view-point The exception of two dimensions and Fermats last theoremKey words: Kochen Specker theorem, generalized relativity, Hilbert space, Minkowski space, world lineby -function, -function by qubitsAt 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).1

2. In fact, it was a revolutionary, new as a principle in regard to the proof and thefoundation of the claim given initially by von Neumann. Before it, the non-existence of thehidden parameters in the quantum mechanics had been attributed to non-commuting op-erators and observables (e.g. in Dmitriev, 2005:435 summarizing the premises of vonNeumanns theorem). Kochen and Specker demonstrated the impossibility of hidden pa-rameters even about commuting operators in quantum mechanics. Respectively, in thecase of statements about commuting and therefore commensurable quantum-mechanicalobservables, classical logic is not always applicable, because its tautologies might turn outrefutable and even identically false in quantum mechanics. Furthermore, after a more detailed look at their proof, we are going to under-line the fact that, in their interpretation, the absence of hidden parameters is due to thenecessity of common considering discrete and continual morphisms, i.e. to wave-corpuscular dualism in last analysis. Thereupon, they have tacitly comprehend hidden parameters as local onessince Lorentz invariance still remains in force restricting the generalization of thecontinuous functions as Borel ones, and this enables the precise translation of the com-mensurability of quantum-mechanical observables into mathematical language as a com-mon measure in the rigorous mathematical meaning of the concept measure. So non-local hidden parameters which are left outside the range of Kochen and Speckers article are completely and implicitly ignored by the justification that their Lorentz non-invariance implies their mathematical and physical incommensurability with the quantitiesto whose functions they should serve as arguments. On the other hand, Dirac delta function or Schwartz distributions (generalizedfunctions) long ago involved in the apparatus of quantum mechanics do not require suchmathematical commensurability of the areas of the argument and the values of the gene-ralized function. Sometimes the local (Lorentz invariant) hidden parameters are undulyconfused with hidden parameters in general (including the violation of Bells inequalitiesopposite to Kochen and Speckers results), but this confusion does not evolve neither ex-plicitly, nor implicitly from their article. Kochen and Speckers text both rigorous and precise, also heuristic, and ofradically new ideas and approach, not only gives rise to a great number of subsequentstudies, but by now is away from depleting its intrinsic potential. In the beginning of theirarticle 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 usuallyintroduce a phase space of hidden pure states in a manner reminiscent of statisticalmechanics. The attempt is then shown to succeed in the sense that the quantummechanical average of an observable is equal to the phase space average (Kochen,Specker 1967: 59).2 3. Von Neumann used to underline quite explicitly that the half of thevariablesof the configuration space of micro-objects are superfluous, redundant and simultane-ously fully adequate to describe again the same micro-system if the other half of the samevariables, 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 asSchrdinger (1935: 827) highlighted is the all possible knowledge of it.Because of that reason the phase space must be modified to be applicable inquantum mechanics: one modification is made by Wigner (1932) and Moyal (1949) onthe base of the preceding fundamental work of Weyl (1927): e.g. the basic cell inthe classical phase space is the product of quantities position and momentum whichare non-commuting in quantum mechanics; therefore each cell is duplicated according tothe order of multiplying the quantities. As this is independently valid for each of the cellsin the phase space, the variants of the phase space that have to be referred to the samequantum system are found to beas a number instead of the only single one in classicalconsideration.Since the observables in the two sets are conjugated, each with the one towhich it is relevant , and their operators do not commute (e.g. position and momentumfor every particular micro-object according to the uncertainty relation), there may be pro-pounded the hypothesis by analogy, illegal as a strict logical inference, that just the non-commutability of the operators (or the observables in quantum as contrasted to classicalmechanics) is the premise, the precondition for the absence of hidden parameters. Henceit becomes obvious that if hidden parameters exist, the physical quantities would commutewith each other in the same way as in classical mechanics. As the non-commutability doesnot allow a physically relevant interpretation of the product and even the sum of two suchnon-commuting quantities (demonstrated in Hermann (1935) Bell (1966) argument),"the back door" of our ignorance, behind which the cherished "true"