We discuss the recently proposed model, where the spacetime in large scales is parametrized by the usual real line R, while at small (quantum mechanical) scales, the space is parametrized by the real numbers RM
from some formal model M of Zermelo–Fraenkel set theory. We argue that the set-theoretic forcing is an important ingredient of the shift from micro- to macroscale. The set RM, describing the space at the Planck era,
is merely a meager subset of R. It is Lebesgue non-measurable and all its measurable subsets have Lebesgue measure 0. According to this, the contributions to the cosmological constant from the zero-point energies of
quantum fields vanish. Moreover, the emerged irregularities in the real line can be considered as the source of the primordial quantum fluctuations

Bielas, Krzysztof

Klimasara, Paweł

Król, Jerzy

Acta Physica Polonica B

English

Repozytorium Uniwersytetu Śląskiego RE-BUŚ

             Title: The structure of the real line in quantum mechanics and cosmology  Author: Krzysztof Bielas, Paweł Klimasara, Jerzy Król  Citation style: Bielas Krzysztof, Klimasara Paweł, Król Jerzy. (2015). The structure of the real line in quantum mechanics and cosmology. "Acta Physica Polonica B" (Vol. 46, no. 11 (2015), s.2375-2379), doi 10.5506/APhysPolB.46.2375   Vol. 46 (2015) ACTA PHYSICA POLONICA B No 11THE STRUCTURE OF THE REAL LINEIN QUANTUM MECHANICS AND COSMOLOGY∗Krzysztof Bielas, Paweł Klimasara, Jerzy KrólInstitute of Physics, University of SilesiaUniwersytecka 4, 40-007 Katowice, Poland(Received October 19, 2015)We discuss the recently proposed model, where the spacetime in largescales is parametrized by the usual real line R, while at small (quantummechanical) scales, the space is parametrized by the real numbers RMfrom some formal model M of Zermelo–Fraenkel set theory. We arguethat the set-theoretic forcing is an important ingredient of the shift frommicro- to macroscale. The set RM , describing the space at the Planck era,is merely a meager subset of R. It is Lebesgue non-measurable and allits measurable subsets have Lebesgue measure 0. According to this, thecontributions to the cosmological constant from the zero-point energies ofquantum fields vanish. Moreover, the emerged irregularities in the real linecan be considered as the source of the primordial quantum fluctuations.DOI:10.5506/APhysPolB.46.2375PACS numbers: 02.10.–v, 03.65.–w, 98.80.–k1. IntroductionNowadays, the primordial inflation is a widely accepted scenario for thecosmological evolution of the early Universe (e.g. [1]). The scenario can besuccessfully described by a class of simple models based on general relativity(GR), with the addition of the single scalar inflaton field φ with the potentialV (φ). Even though these models are very effective, we still do not knowthe reason why the inflation emerged. The aim of this paper is to showthat turning to the fundamental mathematical structures, like the real line,improves our understanding of the origins of cosmic inflation and sheds somelight on the cosmological constant (CC) problem.The structure of the set of real numbers can be described algebraically asthe linearly ordered complete field or topologically (and smoothly) as a 1-di-mensional Euclidean manifold, which is the simplest manifold of dimension∗ Presented by P. Klimasara at the XXXIX International Conference of TheoreticalPhysics “Matter to the Deepest”, Ustroń, Poland, September 13–18, 2015.(2375)2376 K. Bielas, P. Klimasara, J. Królone. Higher-dimensional manifolds, like Rn, are directly modeled by R. It isa truism to say that physicists refer to differentiable manifolds, hence to realnumbers, when developing various models of reality. Besides, R is the basicmathematical object containing the formal representations of the results ofclassical and quantum measurements. Thus, it is the basic field to which(almost) all physical theories refer to. Let us just mention that space andtime coordinates are described by real numbers. The impact of the nontrivialstructure of the real field on physics was discussed mostly in the context ofquantum mechanics (QM) and quantum gravity [2, 3]. Here, we study themathematical model for the evolution of the Universe with varying set ofreals. The set of real numbers is a formal object in models of ZFC — theZermelo–Fraenkel set theory (with the axiom of choice). Since ZFC is thefirst order theory (its axioms are formulated in the first order language), itis not categorical, i.e. there exist infinitely many non-isomorphic models ofZFC. Every model M of ZFC determines the object of real numbers RM ,which is the set of all internal in M subsets of NM ' N. The structure ofRM , the model-dependent real line is, in general, very rich and complicatedwith the properties relative to a model [4]. The general tool for exploringthe real line is the set-theoretic forcing, the procedure invented by Cohenin 1963 when proving the independence of the continuum hypothesis of theaxioms of ZFC (and ZF) [5].In this paper, we consider the model of the evolution of the Universe inwhich the change of a model of ZFC took place (during the Planck era).Thus, the set of real numbers is a varying and model-dependent objectrather than absolute. We apply the forcing on the measure algebra as atool to formally grasp the changes of the set of reals. From the point of viewof cosmology, forcing can be seen as a process that underlies the cosmicinflation and the inflation potential can be related with the change of theinherent density of reals. Moreover, the gravitational contributions of thezero-point energies of quantum fields described in some model M vanish inthe, extended by forcing, generic model M [G]. It is shown in Sec. 3. Webegin with some results about forcing and QM to justify the use of forcingin cosmology.2. Forcing in QMThis section is the summary of our previous article regarding the forcingemerging from QM. An interested reader is referred to [6] for a more detailedtreatment. A forcing can be seen as the deriving property of some Booleanalgebra1. By B, we denote some Boolean algebra of projections chosen fromthe lattice of projections L(H) on a Hilbert space H. The spectral theorem,1 The algebra should be complete and atomless to support a nontrivial forcing.The Structure of the Real Line in Quantum Mechanics and Cosmology 2377in general, gives the correspondence between the algebra of self-adjoint (s-a)operators which are in the Boolean algebra B [6, 7] and the measure algebradefined on the Borel subsets of X = R3. Actually, there exists the isomor-phism between the algebra B generating the family of s-a operators {Aα}and the measure algebra of the space2 (X,µ). Such measure algebra is com-plete and, in general, non-atomic. Thus, there exists a nontrivial measureforcing adding random reals to a model M and resulting in the extendedmodel M [G]. Roughly speaking, a forcing allows us to formally grasp thequantumness of the observables in the Boolean contexts [6].The direct consequence of Wesep’s analysis of the LHV program in [8] isthat the results of the continuous measurements of the position observable,in the semiclassical Boolean contexts, refer to the random forcing. Thus,as shown in [6], the (random) measure forcing has to be present when ap-proaching the classical geometric limit emerging from the quantum scales.Let us analyze some consequences of this fact.3. Forcing in cosmology: the CC problemSuppose that, indeed, the shift from the Planck era to the GR-based eraof the evolution of the Universe refers to different models of ZFC. Denotingthe model in the Planck era by M , we use M [G] for the shifted one. Thus,there corresponds the shift RM → RM [G] of the real numbers parameterizingthe physical content of the epochs. The description of nowadays large scalestructure is based on the full real line R, since the diffeomorphism invarianceof GR enforces it.Now, given a particle of massm, the zero-point contribution to the energydensity is formally calculated asEV=∫d3k(2pi)3√k2 +m22. (1)This expression is not Lorentz-invariant and is quartically divergent. Im-posing any ultraviolet cut-off ΛUV on the momentum integration, one canobtain the finite value. However, the cut-offs induce the loss of validity of atheory for very big momenta. Instead, working in our forcing-based model,let us evaluate this integral without using any cut-offs.To this end, let us observe that in the case of random forcing, the set RM(of real numbers in M) is a meager subset of the full real line: RM ⊂ R [4].Moreover, RM is Lebesgue non-measurable and has the lower measure zero2 The measure algebra of (X,µ) is the quotient of the Borel measurable subsets of Xalgebra modulo the ideal of null sets (subsets of X that have measure 0).2378 K. Bielas, P. Klimasara, J. Króland the full outer measure: µ∗(RM ) = 0, µ∗(RM ) = 1. Let µ(S) denotethe Lebesgue measure of a set S. Then, the following elementary propertyholds true:Lemma 1. Every Lebesgue measurable subset S of the non-measurable setA with the inner measure zero (µ∗(A) = 0) fulfills µ(S) = 0.Proof. Since S is measurable, there is µ(S) = µ∗(S). Let us assume thatµ(S) > 0. Then, we have µ∗(S) > 0. Since A = S ∪ (A \ S) and the innermeasure is additive, there is µ(A) > 0. But A is non-measurable — a con-tradiction. Hence µ(S) = 0. Note also that the Lebesgue integral of an integrable function f over theset of measure zero vanishes even if the set is uncountably infinite (which isthe case here)3.In the considered cosmological scenario, the Planck era is described bythe tools of some model M of ZFC, hence the zero-modes of quantum fieldscontributing to the vacuum energy too. In particular, physical quantitiesare described with respect to the real numbers from RM . Recall that thechange of the model M forcing−→ M [G] refers to the random forcing, accordingto Sec. 2. Then, RM has lower measure 0 as a meager subset of RM [G] (andhence of R) and the Lemma 1 can be applied here.The description of spacetime at the present epoch requires, accordingto GR, the differentiable manifold built on the full real line R. This iswhy we evaluate the integral (1) from the nowadays perspective of the GR-scale-based observer. The zero-point energies lie in the model M , so theintegration is taken over the non-measurable subset4 R3M ⊂ R3EV=∫R3Md3k(2pi)3√k2 +m22. (2)Although such Lebesgue integral does not exist in general, we can evalu-ate the contributions given by the integration over all measurable subsets3 The Lebesgue integral of a simple function s =∑nj=1 αjχAj , where χAj is the char-acteristic function of Aj , reads∫Esdµ =n∑j=1αjµ (E ∩Aj) .If µ(E) = 0, then µ(E ∩ Aj) = 0 for all j, hence∫Esdµ = 0. The Lebesgue integralof any non-negative function f is the supremum of integrals of all simple functions ssuch that 0 ≤ s ≤ f . Since all these integrals are 0, the supremum is 0 too.4 In the model M , the real numbers RM parametrize both x and k coordinates.The Structure of the Real Line in Quantum Mechanics and Cosmology 2379of R3M . But, by Lemma 1, all such contributions vanish and the value ofthe integral (2) calculated this way is zero. Thus, the model of the evo-lution of the Universe based on set theory has mechanism which allows usto neglect the contributions from the zero-modes of quantum fields. Themeasure-theoretic argumentation is universal and represents the solution ofthis part of the CC problem.Moreover, one can assign the forcing between the models of ZFC to theinflationary epoch so that the real line RM becomes rarefied in RM [G] and R.However, such quantitative approach to the cosmic inflation requires the useof the non-standard (exotic) geometries on S3 × R or R4 [9].4. DiscussionWe proposed the model in which, during the evolution of the early Uni-verse, the change of a model of set theory took place. Such change is encodedby the forcing on the measure algebra of R3 and leads to the change of thereal line. It is shown that the vanishing of the contributions of the zero-modes of quantum fields to the cosmological constant is the consequence ofsuch forcing-based evolution. This is only a partial success of the proposedmodel. Namely, the following question remains to be answered (which isthe part of the CC problem): why the value of density ρΛ is non-zero andso small? As shown in [9], the answer can be based on the 4-dimensionalnontrivial curved geometries emerging during the shift M → M [G] at theprimordial era. Thus, the curvature is the source for the correct energydensity and the shape of the inflation potential. The forcing origins of suchgeometries will be discussed elsewhere.REFERENCES[1] A.D. Linde, Phys. Lett. B 108, 389 (1982).[2] A. Döring, C.J. Isham, J. Math. Phys. 49, 053515 (2008).[3] C. Heunen, N.P. Landsman, B. Spitters, Bohrification in: Deep Beauty:Understanding the Quantum World Through Mathematical Innovation(Ed. H. Halvorson), Cambridge University Press, New York 2011,pp. 271–313.[4] T. Bartoszyński, H. Judah, Set Theory: On the Structure of the Real Line,A.K. Peters, Wellesley, Massachusetts, USA, 1995.[5] P.J. Cohen, Proc. Nat. Acad. Sci. USA 50, 1143 (1963).[6] P. Klimasara, J. Król, Acta Phys. Pol. B 46, 1309 (2015).[7] G. Takeuti, Two Applications of Logic to Mathematics, Kano MemorialLecture 3, Math. Soc. Jpn., Princeton, USA, 1978.[8] R.A. Van Wesep, Ann. Phys. 321, 2453 (2006).[9] T. Asselmeyer-Maluga, J. Król, Adv. High Energy Phys. 2014, 867460(2014).

The structure of the real line in quantum mechanics and cosmology

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The structure of the real line in quantum mechanics and cosmology

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