Elements And Processes Of Reality

Elements and Pro esses of Reality Antonio Le
on. May 2024. Abstra t.-This arti le re alls the on ept of element of reality and introdu es a new more general on ept: the pro ess of reality, whi h in ludes the rst one, the on ept of hange and the irreversibility of time. The problem of quantum non-lo ality is also dis ussed and a new dis rete s enario with a single universal lo ality is suggested. Keywords: quantum lo ality, realism, element of reality, pro ess of reality, arrow of time, ellular automata. 1 Introdu tion: innity and history Con erning the importan e of the history of the universe in the quantum debate on reality and lo ality, it seems to me onvenient to begin by remembering that things happen; and then immediately add: things happen and leave tra es that we an re ognize, interpret and onrm in experimental terms. This is what seems to have been happening in the observable universe for at least the last 13.8 billion years. And sin e it is only what it seems, we will need a guiding prin iple to onstru t our dis ussion of that observable reality, we will all it the Prin iple of Dire tional Evolution: In the literature related to the strangeness of quantum me hani s that I have had the opportunity to examine, I have always been stru k by the onstant absen e of two issues of apital importan e in the possible unraveling of this strangeness (one has the impression that strangeness adds value to ontemporary physi al theories). The rst of these issues is the hypothesis of the a tual innity, subsumed in the Axiom of Innity, whi h underlies the mathemati s with whi h physi ists onstru t their theories, in luding quantum me hani s. The se ond is The observable universe always evolves the very history of the universe and its obje ts, a history independently of its observers and in the same that has always o urred in the same entropi dire tion, dire tion of in reasing its global entropy. always without human observers and always leaving indisputable tra es of the pro esses that have o urred, where the term entropy ould be repla ed with the for example tra es in the ro ks of planets su h as the term isotropy [13℄. This prin iple is a generalization of Earth (quantum physi s should be interested, even if only the Se ond Law of thermodynami s. It is an indu tive minimally, in stratigraphy). prin iple for whi h there is maximum empiri al eviden e. Although the hypothesis of the a tual innity was From it some results are dedu ed almost immediately hotly debated for more than twenty-six enturies, it that are also worth onsidering as part of the formal suddenly eased to be dis ussed. It was in the early setting of the main dis ussion of this arti le (details and years of the last entury when the innitist works of proves an be found, for example, in [15℄): R. Dedekind, G. Frege, and espe ially G. Cantor, nally laid the foundations for modern set theories, all of whi h Theorem 1 (of the Consistent Universe) The uniaxiomati ally admit (Axiom of Innity) the existen e verse always evolves under the ontrol of the same and of the innite sets, whose innity is the a tual innity, unique set of invariant and onsistent physi al laws. not the potential innity [16℄. The elements of an a tual innite set exist as a omplete totality: any element that Theorem 2 (of Formal Dependen e) No on ept dean belong to the set is in the set. For example, the nes itself; no statement proves itself; no physi al obje t ordered list of natural numbers in their natural order is the ause of itself; and no ause is the ause of itself. of pre eden e ontains all the natural numbers, all of Theorem 3 (of the First Cause) The universe had them; even if there is no last natural number to omplete an origin whose ause is external to the universe itself the list. Or the real interval (0, 1) ontains all the real and s ienti ally unknowable in terms of knowledge numbers greater than zero and less than 1, all of them, extra ted from within the universe. even if there is neither a rst real number greater than 0, nor a last real number less than 1. This is the innity that physi ists assume without omplaint in all their theories. 2 EPR and its elements of reality The other innity, the potential innity, is not In the rst page of the famous arti le known as EPR, its even onsidered in ontemporary mathemati s. For this authors already establish the ne essary requirement for potential innity, the ordered list of natural numbers a physi al theory to be onsidered omplete [8, p. 777℄: does not exist as a omplete list, but as an unbounded Every element of the physi al reality must have list in whi h it is always possible to add natural numbers a ounterpart in the physi al theory that are larger than any natural number in the list. Here, the in omplete annot be onsidered as omplete, And in the following paragraph they dene their famous as Aristotle would say [1, p. 291℄. It is ironi , on the other elements of reality [8, p. 777℄: hand, that it has been modern set theories that have If, without in any way disturbing a system, we ended up providing the formal instruments to prove the an predi t with ertainty (i.e. with probability in onsisten y of the a tual innity that underlies them. equal to unity) the value of a physi al quantity, For example, the ω -order of the natural numbers; the then there exists an element of physi al reality dense order of the real numbers and rational numbers; orresponding to this physi al quantity. transnite arithmeti ; supertask theory, et . More than 40 su h proofs an be examined in [16℄, and the shortest With them they build their argument on the and simplest I ould develop in the nal appendix to in ompleteness of quantum me hani s based on the this arti le: less than 300 words that take less than three in ompatibility between realism and lo ality that follows minutes to read. from the Copenhagen interpretation [8, p. 780℄: 1 and sin e both assumptions (lo ality and realism) were inalienable for them, they ended up de laring quantum me hani s in omplete. But the argumentative variations on EPR and their orresponding experimental veri ations gave reason to the opposite on lusion: reality is not lo al, for instan e, elementary parti les an be inuen ed instantaneously and at arbitrary distan es. Moreover, and a ording to some interpretations of quantum me hani s, it is not possible to deta h observers from the observed physi al fa ts. The question we pose here is what role do the elements and pro esses of reality play in the explanation of the physi al world? A ording to the formal s enario introdu ed above, the observable universe ould not have originated by itself and had to have a rst ause unknowable in terms of other known auses. It is a universe that, moreover, evolves always in the same dire tion and under the ontrol of the same set of formally onsistent laws. This is what it has been doing for 13.8 billion years, with no observers to inuen e the results. Consequently, it an be armed that all intera tions, mi ro and ma ros opi , ausing the dire tional evolution of the universe have always o urred in the same dire tion in whi h they would have to o ur, otherwise su h dire tional evolution would be impossible. An evolution arried out through trillions of reality pro esses, ae ting trillions of reality elements and leaving trillions of empiri ally veriable proofs (trillions is a way of speaking). We an, therefore, on lude this part of the dis ussion by suggesting an autonomous, observer-independent evolution of the observable universe. An evolution that has nely produ ed ons ious observers who wonder about the reality of the pro esses from whi h they themselves have emerged. Pro esses of reality that have left a multitude of observable eviden e onrming the autonomy of the dire tional evolution of the universe. It may be that ertain experiments, mental or real, require more exoti and observer-dependent explanations, but that does not seem to be the ase for the unobserved physi al obje t that the universe has been for most of its history. Interpretations of quantum me hani s that link a tive observations to observed fa ts should onsider the dire tional and autonomous evolution of the universe. Previously we proved that either (1) the quantum-me hani al des ription of reality given by the wave fun tion is not omplete or (2) when the operators orresponding to two physi al quantities do not ommute the two quantities annot have simultaneous reality. Starting then with the assumption that the wave fun tion does give a omplete des ription of the physi al reality, we arrived at the on lusion that two physi al quantities, with non- ommuting operators, an have simultaneous reality. Thus the negation of (1) leads to the negation of the only other alternative (2). We are thus for ed to on lude that the quantum-me hani al des ription of physi al reality given by wave fun tions is not omplete. And as is well known the story ontinued in theoreti al terms with Bell's theorem [5℄ and the GHZ states [10℄; and in experimental terms with the respe tive experiments arried out sin e 1982 [4, 3, 20, 11, 7℄, all of whi h ended up proving the Copenhagen interpretation was not wrong. 3 Pro esses of reality It is not possible to dene everything: we would fall into a potentially innite regress of denitions. For that reason we are obliged to use in all s ien es primitive on epts, on epts that are not dened in terms of other more basi on epts; if we did, those more basi on epts would be the new primitive on epts. Whi h justies B. Russell's ironi words that in mathemati s we never know what we are really talking about [19℄, although some do seem to know what they are talking about. The on epts of hange, event, modi ation, transformation and pro ess are intimately related and surely all of them are primitive on epts, of whi h it is not possible to give non- ir ular denitions. We have to onstru t our arguments with this inevitable restri tion, whi h, although it should not be forgotten, should not stop us in our eagerness to understand the physi al world. We will then say that if an element of reality, observed or not, is modied in su h a way that another element of reality always results and the modi ation always leaves the same observable tra es, then that modi ation is a pro ess of reality. Naturally, the pro esses of reality in lude the elements of reality and their modi ations or hanges. They also in lude the passage of time. Therefore, it is a ri her on ept than EPR's element of reality. The in lusion of the on ept of hange is really important. Indeed, the problem of hange has been posed and not solved for 25 enturies. What is worse, physi s, the s ien e of hange, has ompletely forgotten the old problem of hange, as if it were possible to explain the physi al world without having previously solved the problem of hange. On the other hand, it an be proved that the problem of hange has no solution in the spa etime ontinuum, but it does have a solution in a dis rete model of spa e and time. [16, p. 293-302℄ [14, p. 502-512℄. The EPR argument made its authors on lude that quantum me hani s was in ompatible with lo al realism, 4 The universe as a unique lo ality The elements and pro esses of reality serve to de ouple the dire tional evolution of the universe from observation. But they do not solve the problem of quantum non-lo ality. In this nal se tion we suggest a new way of dis ussion in whi h the spa etime ontinuum is repla ed by a dis rete spa e and time, with indivisible minimum units: qseats and qbeats respe tively. And the substitution is not arbitrary, among other reasons be ause: 1. The innite sets, su h as the ontinuum, are in onsistent (see one of the proofs in the nal appendix). 2. The vibrations of spa e are empiri ally dete table (gravitational waves) and therefore spa e must be a real physi al obje t: that whi h does not exist neither vibrates nor an modify the length of the arms of the interferometers that dete t gravitational waves. 2 Referen es 3 3. Spa e and time are involved in numerous mathemati al fun tions whose outputs must be dis rete values of energy, whi h is impossible if ontinuous variables are involved in these fun tions. Therefore, if energy is dis rete, spa e and time must also be dis rete. 4. The problem of hange an only be solved in a dis rete spa e and time [14, 12℄. And without solving the problem of hange it is impossible to explain the physi al world. 5. Dis rete spa e is mu h simpler than the spa e ontinuum: if a qseat had a Plan k volume, the observable universe (9.9 billion light years in diameter) would ontain 7.6 × 10184 qseats, while any arbitrary volume of the spa e ontinuum ontains the same non-numerable innite number of points: 2ℵo . 6. Dierent regions of dis rete spa e ontain dierent numbers of qseats, whereas a Plan k volume and the entire universe, whatever its a tual size, have the same number of points: 2ℵo points. 7. In a dis rete spa e-time it is possible to dene models in whi h all qseats evolve in unison, the whole system being the same deterministi lo ality. on e ompared with all elements of Q01 , the urrent value of x were not the least rational in Q01 , there would exist at least one element f (n) in Q01 su h that f (n) < x. But this is impossible a ording to (1). Therefore, it was ompared with f (n) and redened as f (n). So, it is impossible that f (n) < x. But it is also immediate to prove that: On e ompared with all elements of Q01 , the urrent value of x is not the smallest rational in that set. In ee t, on e ompared with all elements of Q01 , and whatsoever be the urrent value of x, ea h element of the innite set {x/2, x/3, x/4 . . . } is an element of Q01 less than x. This ontradi tion proves the Axiom of Innity legitimizing the existen e of Q01 as an a tual (not potential) innite totality is in onsistent.  (1) This is formally proved by indu tion in [16℄, and an also be proved by Modus Tollens and by supertask theory. Bibliographi al Referen es [1℄ Arist
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es de Bell par mesure de orrelation de polarization de photons. PhD thesis, Universidad de Paris-Orsay, Paris-Orsay, 1983. [4℄ A. Aspe t, J. Dalibard, and G. Roger. Experimental test of bell's inequalities using time-varying analyzers. Physi s Review Letters, 49:18041807, De 1982. [5℄ J. S. Bell. On the Einstein Podolsky Rosen paradox. Physi s, 1(3):195200, 1964. [6℄ D. Bleh, T. Calar o, and S. Montangero. Quantum Game of Life. ArXiv e-prints, pages 1  5, O tober 2010. [7℄ G. Carva ho, F. Gratti, V. D'Ambrosio, B. C. Hiesmayr, and F. S iarrino. Experimental investigation on the geometry of ghz states. S ienti Reports, 7(1):13265, 2017. [8℄ A. Einstein, B. Podolsky, and N. Rosen. Can quantum me hani al des ription of physi al reality be onsidered omplete? Physi al Review, 47:777 780, May 1935. [9℄ A.P. Flitney and D. Abbott. A semi-quantum version of the game of Life. eprint arXiv:quantph/0208149, pages 1  6, August 2002. [10℄ D. Greenberger, M. Horne, A. Shimony, and A. Zeilinger. Bell's theorem without inequalities. Ameri an Journal of Physi s, 58(12):11311143, 1990. [11℄ Paul G. Kwiat, Salvador Barraza-Lopez, Andr
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on. Towards a dis rete osmology. There is a model ompatible with all the above requirements in whi h, and a ording to the last one, the quantum lo ality problem ould be solved: ellular automata (of whi h quantum versions are also available [9, 18, 2, 6℄). This may seem an extravagant model, although I nd the alternative of multiple universes mu h more extravagant. A tually, I am not proposing that the universe is a ellular automaton, but suggesting a hange of perspe tive in the interpretation of quantum me hani s, a new perspe tive based on the nite and dis rete nature of spa e and time in a universe in whi h all its dis rete elements are a tualized in unison, as in ellular automata. Appendix: The in onsisten y of the a tual innity (The next theorem is a very abbreviated version of the argument [16, p. 59-63℄. A slightly more omplete version than the following an be found [17, here℄) Theorem 4 The Axiom of Innity is in onsistent. Proof.-The open interval of rational numbers (0, 1) is denumerable and densely ordered. So, it an be put in one-to-one orresponden e f with the set N of natural numbers in their natural order of pre eden e; and the interval (0, 1) an be rewritten as the set Q01 = {f (1), f (2), f (3), . . . }. Let now x be a rational variable initially dened as f (1); and let (the urrent value of) x be ompared with the su essive elements f (1), f (2), f (3). . . so that x is redened as f (i) if, and only if, f (i) is less than the urrent value of x. Sin e, a ording to the Axiom of Innity, all elements f (1), f (2), f (3), . . . of Q01 are rational numbers whi h exist as a omplete totality, x an be su essively ompared with all of them: ∀n ∈ N : x is ompared with f (n), and redened as f (n) i f (n) < x (1) On e ompared with all1 elements of Q01 , the urrent value of x is the smallest rational in that set. Indeed, if 3 Automata. ArXiv e-prints, pages 199  204, June Indpendently published in KDP. Free pdf, 2023. 2009. [16℄ Antonio Le
on. Innity put to the test. Self edition in KDP. Printed at amazon. om. Free pdf, 2023 [19℄ Bertrand Russell. Misti ismo y l
ogi a y otros ensayos. Aguilar, Madrid, 1973. (2021). [17℄ Antonio Le
on. The Axiom of Innity Is In onsistent. [20℄ W. Tittel, J. Brendel, B. Gisin, T. Herzog, H. Zbinden, and N. Gisin. Experimental The General S ien e Journal, 2024. demonstration of quantum orrelations over more [18℄ N. Ollinger. Intrinsi ally Universal Cellular than 10 km. Phys. Rev. A, 57:32293232, May 1998. 4