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Quantum modalities, interpretation of QuantumMechanics and Special Relativity, and an experimental

test for multiple worldsYannis Delmas-Rigoutsos

To cite this version:Yannis Delmas-Rigoutsos. Quantum modalities, interpretation of Quantum Mechanics and SpecialRelativity, and an experimental test for multiple worlds. 2019. hal-01983757

Quantum modalities, interpretation of Quantum Mechanics and Special

Relativity, and an experimental test for multiple worlds

Yannis Delmas-Rigoutsos∗

Laboratoire TECHNE (EA 6316) Universite de Poitiers

(Dated: January 1st, 2019, submitted to Stud.Phil.Phys.)

Abstract.- This paper investigates the physical, experimentable, reality of multipleworlds of Quantum Mechanics. On the basis of a reection on the fundamental prin-

ciples of physics (sec. I.A), rmly rejecting the incomprehension imperative often as-sociated with Quantum Mechanics (sec. I.B), and strongly relying on an experimentalmethodology (sec. I.C) and on the superposition principle, we propose a unied inter-pretation of quantum physics (sec. II), the potential world modal interpretation. Thisframework makes multiple-world and pilot-wave interpretations converge. It also clariesthe concepts of potential world and sub-quantical potential particle. Then, we turn toRelativity and sketch out an emergence of Special Relativity without rods and clocks(sec. III.A) providing a cinematic and phenomenal interpretation of SR. We suggest todistinguish between given and received gravitation and reason on it at a sub-quanticallevel (sec. III.B) and formulate the hypothesis that weight is active between potential

worlds (sec. III.C). If this eect follows the von Neumann chain, it should be observ-able with available techniques (sec. III.D). If correct, this hypothesis could contributeexplanations for cosmic ination, dark matter and the cosmological constant problem(sec. IV). Moreover, our cinematic interpretation of relativity in an expanding universemight lead to a justication of the MOND theory (sec. IV.B.3).Resume.- Cet article explore la question de la realite physique, experimentable, desmondes multiples de la Mecanique Quantique. En nous appuyant, d'une part, sur unereexion sur les principes fondamentaux de la physique (sec. I.A), rejetant resolumentl'imperatif d'incomprehension qui auble souvent la Mecanique Quantique (sec. I.B) ets'appuyant sur des principes methodologiques fondes sur l'experimentation (sec. I.C), et,d'autre part, sur le principe de superposition, nous proposons une interpretation unieede la physique quantique (sec. II), l'interpretation modale des mondes potentiels. Celle-cifait converger les interpretations de type mondes multiples et les interpretations de typeonde-pilote et donne une denition precise des concepts de monde potentiel et de par-ticule potentielle ou subquantique. Nous esquissons ensuite un mecanisme d'emergencede la Relativite Restreinte a partir de presupposes subquantiques simples (sec. III.A),sans regle ni horloge. Ceci fonde une interpretation cinematique et phenomenale dela Relativite Restreinte. En distinguant entre gravitation recue et donnee et en raison-nant a un niveau subquantique (sec. III.B) nous formulons l'hypothese que le poids agitentre monde potentiels (sec. III.C). Si cet eet remonte bien la chane de von Neu-

mann, nous proposons de l'observer a l'aide de technologies d'ores et deja disponibles(sec. III.D). Si notre hypothese s'avere correcte, elle pourrait contribuer a expliquerplusieurs phenomenes jusqu'alors isoles : l'ination cosmique, la matiere sombre et leprobleme de la constante cosmologique (sec. IV). Enn, notre interpretation cinematiquede la relativite dans un univers en expansion pourrait conduire a une justication de latheorie MOND (sec. IV.B.3).

Keywords: interpretation of Quantum Mechanics; multiple worlds; modal logic; Quantum Logic; Special

Relativity; gravitation; quantum gravity; dark matter; cosmic ination; cosmological constant problem; MOND

I. INTRODUCTION

A. Raising a question

For nearly 2 500 years, Plato's Cave allegory1 has in-spired many reections about reality. Drawing a line be-tween phenomena and theories, it emphasizes the dif-culty to tell the illusion of habit from the essence of

∗ yannis.delmas@univ-poitiers.fr1 Republic, 514a520a. Greek word for cave is σπήλαιον(speelaion): a natural cavity.

things. In some way, phenomena are the only tangiblereality; we can reach for the latter, we can bump onit. But phenomena are in the Cave, their expression iscomplex, sometimes hard to understand, and does notconvey any deep representation. Phenomena are, fun-damentally, a scattered reality: the Moon's orbit andthe fall of apples fuel no common experience. To gathera solid representation, we need to explore the Cave foran exit, and search for another form of reality: scien-tic theories. It is indeed some form of reality, a least amethodological one, since we can experience it. But thatreality fundamentally lacks objectivity. In Plato's nar-

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rative, Cave people do not want to believe the explorerand they end up killing him. To cope with subjectiv-ity, science has built up, over the centuries, a methodol-ogy; yet, as epistemology and the history of science haveshown, experiment under-determines theories: the sameexperimental corpus can lead to numerous dierent the-ories. Isaac Newton changed our theoretical vision ofgravitation and gathered in a single model the Moon'sorbit and the fall of apples, showing to mankind an un-derlying reality. This vision is more sound, more real ina way, but can pass with time: going further out of theCave, new experience can lead to other seminal theories.Some theories may also be concurrent for centuries, likewave and corpuscular theories for light. Some theoriesmay supersede others, like the Einsteinian v. Newtoniangravitation theories.Plato's Cave was somewhat small: as soon as he

withdrew his shackles, Plato's explorer managed to getout of it. Science requires more work; science explorersprogress slowly, to add, step by step, some light to an-cient puzzles. For us, scientists, theoretical explorationdoes not only consist of abstract reections, it is alsoa means to extend the phenomenal register: it does saywhere to look, and what to look for as shown by UrbainLe Verrier and Johann Galle for Neptune, or AlbertEinstein and Arthur Eddington for General relativ-ity. For that reason, Ockham's razor should not be tooquick to shave away interpretations, as long as they aimat some experimental validation. Thus, our rst guidingprinciple should not be too strict2.

Principle 1 (parsimony). We are to admit no morecauses of natural things than such as are both true andsucient to explain their appearances. Therefore, to asmany as possible natural eects (phenomena), one mustassign the same causes (explanations).

The rst modern relativity was the relativity of veloc-ity, formalized and argued by Galileo. This model doesnot rule out absolute speed, but does involve that an ab-solute speed is not phenomenal. On this basis, AlbertEinstein did theorize a relativity of space/time, and,later on, a relativity of inertia/gravitation. From sucha point of view, inertia, gravitation, space, time, veloc-ity and displacement are eects of perspective. One canread similarly Hugh Everett's interpretation (1957) asa relativity of quantum state3. Our rst main question, in

2 The rst sentence quotes Isaac Newton. The second is inspiredby his formulation. In many places he comes back to the idea ofa Nature [. . . ] most simple and consonant to herself.

3 On a rst level of analysis, Galileo and Einstein share acommon relativist approach. However, the parsimony principleleads to dierent interpretative conclusions for these authors.Galileo, in his Dialogue Concerning the Two Chief World Sys-

tems (Galilei, 1992) traces some sort of relativity principle back

this paper, is whether Everett's interpretation is purelytheoretical, i.e. a matter of metaphysics or pedagogy (asit is commonly understood without any signicant discus-sion), or whether the relative state theory could containexperimental reality. Plato's Cave explorer discoversnew landscapes and, thanks to them, explains shadowson the wall; Hugh Everett's work has not reached thatpoint yet. It oers a coherent vision for quantum phe-nomena, it does enforce principle 1 of parsimony, it iscompatible with all we know experimentally, but, still,it is lacking experimental support for at least part of it:multiple worlds. We introduce in this paper a new wayof considering this question, to formalize it as a rela-tivity, and to confront it to previous relativity theories.Pushing further the interpretation, we put forward a hy-pothesis on how gravitation could behave with respect toquantum state relativity. We show that this hypothesiscan have an experimental test with existing technologies.In a more speculative part of the paper, we argue thatour hypothesis could lead to a common explanation fordark matter, for cosmological ination, and for gravita-tion's extreme weakness, compared to other forces. Ifour hypothesis happens to be true, gravitation shall nolonger be seen, fundamentally, as an interaction in thesame sense as electroweak or strong forces. Incidentally,we sketch out some possible reconciliation between Rel-ativity and Quantum Mechanics.

B. Relying of an principle

Richard Feynman4 once said: I think I can safely saythat nobody understands quantum mechanics. This af-rmation circulates mostly in two popular versions: Ifpeople say they understand quantum mechanics, they'relying, or If someone thinks he understands quantummechanics, it's because he doesn't. These formulationsfollow suit with an old catchword: Those who are notshocked when they rst come across quantum mechan-ics cannot possibly have understood it (Niels Bohr

to archaic physics. Nicolaus Copernicus (1998, chap. I.8, p.63), for example, cite Aeneas: We set out from harbor, andlands and cities recede. [Virgil, Aeneid, III 72]. Galileo re-sorts to the example of the ship from Venice to Alep and herspeed like nothing for passengers (Balibar, 2002, pp. 12 sqq.).Galilean relativity is mostly a use of perspective rationale, itdoes not imply the nonexistence of an absolute celerity. On theother hand, Einstein's Special Relativity follows Ernst Mach

very strict interpretation of the parsimony principle and rulesout absolute velocities: only relative velocities do really exist. Inthe present paper we shall not restrict relativity to the Mach-Einstein meaning Everett's relativity seems, indeed, closerto Galileo's perspective relativity.

4 Probability and Uncertaintythe Quantum Mechanical View ofNature, 6th Messenger lecture (1964). Collected in The Char-

acter of Physical Law (1967), p. 129.

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(Heisenberg, 1958, cit.)). Many other thinkers of Quan-tum Mechanics propagate a similar conception, which wecould call the incomprehension imperative. In a way, ourcurrent scientic culture forbids, in a very strong man-ner, the understanding of Quantum Mechanics, leadingto some form of resignation: All of modern physics isgoverned by that magnicent and thoroughly confusingdiscipline called quantum mechanics... It has survived alltests and there is no reason to believe that there is anyaw in it... We all know how to use it and how to apply itto problems; and so we have learned to live with the factthat nobody can understand it (Murray Gell-Mann5).The incomprehension imperative may also be satis-

faction with abstruse, absurd or nonsensical explana-tions. For example, many courses or popular science,present particles in a double-slitYoung-type experimentas passing through both slits, or neither. Some au-thors, more cautious, present it as a question that cannotbe answered. Is it a failure of pedagogy? Not only! It isa failure of language and reasoning: Garrett Birkhoffand John von Neumann (1936) showed, a long timeago, that nature cannot be described by Classical Logicalone. To be general and sincere with quantical6 ex-periments, one must reason with Quantum Logic (QL)instead of Classical Logic (CL). The sentence that theelectron passes through the left slit and the right slitis false (zero measure). The statement that the electronpasses through the left slit orCL the right slit is also false(zero measure). What is true is that it passes throughthe left orQL the right slit with a quantum logical or,i.e. a superposition of states. To reason according towhat we experimentally know of our world, we need toadapt our language, and our way of thinking. Fortu-nately, we have shown that Quantum Logic is equivalentto Classical Logic when propositions commute7 (Delmas-Rigoutsos, 1997).

Principle 2 (logic). Expression should seek for clar-ity, comprehension and logical soundness (coherence).Therefore, we must reason about natural phenomena ac-cording to Quantum Logic.

Paul Dirac8 contended that The only object of the-oretical physics is to calculate results that can be com-pared with experiment [. . . ] it is quite unnecessary thatany satisfactory description of the whole course of thephenomena should be given. This statement has had

5 1977, quoted by Ierome Bernard Cohen, The Newtonian Revo-

lution, Cambridge University Press, 1983.6 To ease the formulation, in this paper, we prefer the adjectivequantical to mean quantum mechanical, instead of the nounquantum. Our intention is also vocabulary accuracy, since ref-erence to quanta are, in some cases, merely historical.

7 We recall this denition hereunder, see n. 29 and 25.8 Cf. The Principles of Quantum Mechanics, 1930, p. 7.

a great inuence: physicists put a great lot of creativeenergy on building phenomenological models and theo-ries, but largely neglected their interpretation. Surely,this is a respectable metaphysical creed, but, as surely,it is a sterilizing physical methodology. Representationsare psychologically inevitable, so we consider a better op-tion to discuss them, and to push them further to theirextreme consequences. To deal with interpretations is asupport for heuristics, a support for creativity and forinnovative options. We will, here, make a sound hypoth-esis, unifying various phenomena, but perhaps strangein its consequences. Would experimental tests be nega-tive, it would nevertheless provide valuable informationon how to interpret Quantum Mechanics.

C. Suggesting a way out

In astronomy before Johannes Kepler, one conceivedonly celestial movements with circles. For that reason,theoretical astronomers designed the epicycles to under-stand as circular movement combinations what were ac-tually ellipses. Today, the natural movement is con-ceived as being the inertial Galilean movement, so onetends to understand any movement in these terms orcontemporary equivalents: Newtonian gravitation is afall; Einsteinian gravitation is a pathway of a geodesic.Quintessence, dark matter, dark energy, etc. many ofthese notions are, quite certainly, contemporary epicy-cles. Indeed, these notions have a great phenomenal va-lidity, but they do put a heavy shroud of complexity overthe picture. Can we not just put together some puzzlepieces we know for sure?For us, the fundamental opposition between the quan-

tical and the relativist representations of phenomena isthat in Quantum Mechanics (QM) a coordinate is a mea-sure, and therefore it results from interactions, whereasin the three Relativities (Galilean, Special, General) acoordinate comes under what epistemology calls God'seye9: the possibility of an absolute objectivity. ButEinstein-Podolsky-Rosen (1935) (EPR) paradox sit-uations showed that such a concept as God's eye cannotexist, at least for particles. Any common vision notto mention theory pretending to encompass these tworepresentations of the phenomena must solve this om-nipresent contradiction. One must make a choice be-tween these two conceptions. In this paper, we will stickto experimental evidence10:

9 Following Hilary Putnam.10 The following principle is not neutral: since observations are

interactions, experimental evidence shows that one cannot reduceindenitely their eects. The reality can be independent fromthe observer, in a sense, but cannot be independent from theobservation. In other terms, the methodological realism cannotdeny the act of observation.

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Principle 3 (measure). All that we know about nature(the phenomena) is known through our senses. Coordi-nates, momentum and the other physical quantities re-sult from measure operations, and therefore from inter-actions.

We will suggest, in this paper, an interpretation of theenergy-momentum as a displacement of eervescence ofa sub-quantical scrum from which the particles emerge similar, in a way, to the virtual particles of Quan-tum eld theories (QFTs). From this eervescence,there emerges what could be interpreted, in relativity,as a coordinate density, that is a local increase of theremoteness11 measures any distance measure being in-teractions, and, as such, compatible in principle with aquantum interpretation. If we admit this vision, andthe observation that gravitation is proportional to theenergy-momentum norm, inertia-gravitation should op-erate at the level of the scrum, and not only at thelevel of the emerging particles. To separate clearly thesetwo levels of reality, we will rely on an interpretation ofQM we formulate in section II. Then, in section III, weformalize an interpretation of inertia-gravitation in thisframework and suggest an experimental test for our hy-pothesis. In section IV, we explore some consequences ofthis hypothesis, suggesting a new interpretation for darkmatter, cosmological ination and gravitation weakness.

II. A UNIFIED INTERPRETATION OF QUANTUMPHYSICS

A. The measurement problem and the interpretations

Several formalisms are available to cope with quantumphysics (Styer et al., 2002). Since these are generallyconsidered equivalent, we will mostly use wave functionformalism, but our considerations are meant for all ofthese formalisms.Historically, there is a fundamental methodological dis-

tinction between an observed system, quantical, an ob-server, considered classical, and a measuring process,that may be iterated, based on observables, which aremainly classical quantities. For that reason, epistemolog-ically, there are two clearly distinct processes12: quan-tical or unitary periods, during which evolution is de-scribed by a Schroedinger equation, and wave functioncollapses or reductions or projections, that serve asinitial conditions for the future and are linked to the pastby Born's rule, similar to an absolute form of probabil-ities.

11 See hereunder for the dierence between two types of distance:the remoteness and the interval.

12 Von Neumann's processes 2 and 1 (von Neumann, 1955).

This divide is a major epistemological diculty andhas raised a great deal of debate on the measurementproblem. In general, a science is the instrumented con-struction of an abstraction, called theory, of the realityit observes, called its object; this abstraction13 tends toencompass, eventually, the totality of the object. In thecase of physics, an object could be a physical system con-sidered to be isolated (suciently, or ideally), and poten-tially the whole universe, everything that surrounds us,and, in any case, experimenters. Experimental modelshave to be surpassed, thanks to theoretical thinking, inorder to encompass them. Scientic disciplines, to setup this approach, usually adopt a point of view calledmethodological realism, consisting essentially in assum-ing that there does exist an objective reality accessible byexperimentation. In a way, we have systematically pre-supposed methodological realism in section 1, especiallywhen we expressed our rst three principles.Some authors have objected that in the case of quan-

tum physics, objectivity does not exist, in the sense thatany experiment, which is subjective, disturbs the real-ity to be observed14. Indeed, this makes it more com-plex to integrate subjectivities into an objective whole,but there is no fundamental obstacle against it: this isvery common in the humanities and social sciences, forexample. However, one should be cautious. It is not,in general, possible to bestow a truth value to any vul-gar proposition, like the electron passed by the left slit;and, more generally, there is nothing like a propositionalGod's eye. But every experimental proposition doeshave a truth value (for example, the proposition L anelectron was measured to pass through the left slit hasa meaning and a truth value (if no experimental test hastaken place, then L is just false). Our principle 3 is atool to anchor rmly any discussion in methodologicalrealism.But there is another, deeper, diculty with realism,

i.e. superposition. In Erwin Schroedinger's approach,a fundamental fact, consistent with principle 2, is thatwhen |χ〉 and |ψ〉 are solutions of the Schroedinger equa-tion of a system, then any normalized linear combina-tion of them is also a solution. Three main classes ofinterpretations for the formalism aim at overcoming thisdiculty: the Copenhagen school interpretations (NielsBohr, Werner Heisenberg), the pilot-wave theories(Louis de Broglie, David Bohm, John Bell) and themany-world interpretations (Hugh Everett, Bryce S.DeWitt).

According to the Copenhagen school interpreta-

13 To outline, to abstract is to retain some qualities, some features,of the object, and discard others.

14 This is especially the case if, instead of an epistemological de-nition of objectivity, we consider a physical one: being invariantby change of observation conditions.

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tions (CSI), following Bohr's ideas, any measure-ment operation determines a state of the system,prepares it for future measurements, and denesits wave function for later evolution. Each mea-surement actualizes the possibilities of the system-observer (inseparable) couple.

In the pilot-wave interpretations (PWI), the sys-tem is described by a wave function (the pilot-wave) and by particles (Bohm), or singular waves(de Broglie). Pilot-wave and its evolution de-scribe all the possibilities for the system, and anobjective probability for each outcome of a mea-surement. In each measurement, particles revealwhich possible outcome happens to be real, whichpossibility comes true. Any of these particles mayhave some sort of a history, not necessarily classi-cal since there may be some non-local phenomena;but this history is only knowable at the time of themeasurement.

Third category interpretations, usually calledmany-world interpretations, following Everett'sideas, refrain from giving greater importance toone state over another. The rst approach is Ev-erett's relative state interpretation (RSI) (Ev-erett, 1955, 1957). Another is what Yoav Ben-

Dov (1990) calls popular many-world interpreta-tion (PMI) extremely popular indeed, and gen-erally mistaken for it15. In these interpretations,the dierent solution states, i.e. the dierent possi-bilities, the dierent possible evolutions of the sys-tem, are as real as one another. The measurementexperiment only provides the observer with partialinformation relative to his point of view. All ver-sions of the observer, who may be thought of asliving in parallel universes, have their own readingof the relative states of the observed system. Thesesinterpretations have an advantage and are a chal-lenge: the observer as well as the observed systemmust be taken together in the same possible worldmodels.

In these three interpretation classes there is some notionof possibility and some notion of reality. Clearly, the lat-ter is very dierent between the classes; moreover, thereis a debate about what reality is among pilot-wave inter-pretations and among many-world interpretations. Sucha discussion is not the purpose of this paper much

15 See Ben-Dov (1988) for an historical presentation. PMI emergedwith Bryce S. DeWitt reformulation, in the 1970's, putting RSIin the background. At the quantical level, DeWitt reformula-tion is not coherent with QM results, cf. Ben-Dov (1988) esp.p. 106. Nevertheless, due to decoherence, DeWitt reformulationis, in practice, convenient at the macroscopic level.

has already been written on that matter. We shall onlyfocus on common aspects to suggest a common model,especially the notion of possibilities, which seems moreunequivocal. All visions of Quantum Mechanics share acommon principle about these possibilities16:

Principle 4 (superposition). Possible outcomes of aninteraction, possible solutions of a system's equation, andsupposedly possible worlds are vector spaces. In otherwords, linear combinations of possible states are possiblestates.

We will now come closer to this notion of possibilityand dene carefully the concept of a possible world. Onthis basis, we shall build a common interpretation tobe used later on as a framework to deal with inertia-gravitation.Beforehand, we need to consider a usual diculty op-

posing principles 4 (superposition) and 1 (parsimony):the objectivity challenge.

B. The objectivity challenge

With principle 3 (measure), we have acknowledged thefact that observation processes are interactions and, thus,cannot be absolutely neutral. This raises up a dicultyon how to apply quantical models to the whole universe.The rst historical mainstream interpretations of QM,the Copenhagen school interpretations, focused on ob-served systems, S, and many of these, in one way oranother, denied the relevance of the application of QMto observers. Werner Heisenberg (1958), for example,makes the case that quantum systems integrate both theobserver and the observed: they cannot be separated17.For these interpretations, quantum theory is only phe-nomenal and cannot be generalized. Retrospectively, thissituation can be seen as a lack of a measure theory. Froman epistemological point of view, we then have two theo-retical frameworks: one for (quantical) observed systems,one for (classical) observers. As a rst step, this was con-venient, but it is now insucient to encompass phenom-ena like decoherence, which systematically involves theenvironment, E. There is also a need to get rid of theepistemological dependence of QM on classical physics.Physical works on this matter go back at least to John

von Neumann's Grundlagen (1932) showing that quan-tical probabilities reduce to classical ones in a measure-ment experiment, along what is now called the von Neu-mann chain. Nowadays, works on decoherence and clas-sical appearance of nature, follow this research path

16 We neglect here normalization considerations in formulation.17 So, to be exact, CSI denied application of QM to observer per

se. Since the system and the observation apparatus were con-ceived inseparable, QM precluded application to the macroscopicreality.

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(Zurek, 1991, 2003). We will suppose here that the de-coherence phenomena are well known, but let us empha-size that it does not solve per se the objectivity chal-lenge, nor does it constitute an interpretation of QM. Itseems correct to speak of decoherence as making wave-function evolve rapidly, resulting in its diagonalizationon a pointer basis for an experimental system (at leastfor pedagogy or heuristics) (Zurek, 1991). But it is non-sense to speak of the pointer basis, not to mention thepointer basis of the universe. Indeed, in some situationssuperposition continues on a macroscopic time scale: ifwe take microscopic aspects into account, pointer basesdo not commute18. Indeed, one cannot consider the uni-verse as being classical except for some periods/places.In many sciences, objectivity arises from crossing sub-

jectivities. Hugh Everett have been the rst to ad-dress this problem physically, and not only philosophi-cally. He notes (Everett, 1955) that interactions S •O1,i.e. a rst observer (O1) apparatus, are phenomena fora second observer O2 (19). According to Everett, onemust be able to analyze experimental observations as in-teractions between a system, S, an observer, O1, and,possibly, an environment, E. Moreover, a coherent mea-sure theory must account for intersubjectivity: if there isa second observer, O2, reciprocal observations must bereconcilable (O2 fromO1 andO1 fromO2). He discusses(E•)S •O1 •O2 as a fundamental problem and answersit with a relativist rationale, i.e. the relativity of states.It is important to emphasize that Everett resorts tothe usual wave function formalism of QM20, from theviewpoint of a second observer. On this basis, he denesthe quantical relative information and the relative statefunctions, and proves evolution rules from which he canderive usual measure observations in a coherent way.Technically, Everett relies only on QM to deduce his

relative states theorems, but this is not sucient to foundan interpretation (RSI) and to assert that all relativestates are equally real. For this, we need methodologicalprinciples. For some, principle 1 (parsimony) would beenough: a physical theory should describe not only theobserved system S, but also the environment E, and theobservers, i.e. the experimental apparatus O. But forothers, specially the Copenhagen interpretation follow-ers, there is a conceptual impossibility here. So we willset up an additional principle21.

18 We discuss an example at end of section II.C.19 The question of the consistency of the scheme arises if one con-

templates regarding the observer and his object-system as a sin-gle (composite) physical system. Indeed, the situation becomesquite paradoxical if we allow for the existence of more than oneobserver. (Everett, 1955, p. 4)

20 He himself insists that he uses only pure wave mechanics (Ev-erett, 1955, p. 97 e.g.).

21 We quote Everett (1955, p. 80).

Principle 5 (objectivity). Observers are observable sys-tems and observers who have separately observed thesame quantity will always agree with each other.

This epistemological principle, that allows a generalapplication of quantum theory, is also fundamental tofully endorse the hypothetico-deductive method22. It isalso a means to conform to the psycho-physical paral-lelism advocated by John von Neumann: it is a funda-mental requirement of the scientic viewpoint [. . . ] thatit must be possible so to describe the extra-physical pro-cess of the subjective perception as if it were in reality inthe physical world (1955, p. 418)(Everett, 1955, p. 7,cit.).To be more specic, Everett's fundamental point is

that in an interaction a system-observer pair, S • O1,will eventually evolve (Everett, 1955, p. 54) up to apoint where the relative system states become approx-imate eigenstates of the measurement (Everett, 1955,p. 59). If |L〉 and |R〉 are observer (i.e. apparatus)states, if |l〉 and |r〉 are the corresponding system eigen-states for the measure, and if S is before the interactionin the superposition state a |l〉+b |r〉, with |a|2 + |b|2 = 1,then S • O1 evolves eventually to the correlated statea |l〉⊗ |L〉+ b |r〉⊗ |R〉 (23). From |L〉 version of O1, (rel-ative) state of S is |l〉, whereas it is |r〉 in the |R〉 version ofO1. Probabilities are |a|2 and |b|2, respectively, in agree-ment with Born's rule. For an external observer O2,there is neither something like a separation or split ofS •O1 in two universes, nor a wave function collapse:S•O1 merely evolves unitarily, according to its (extraor-dinarily complex) Schroedinger equation. Moreover, thisevolution will generally become classical very quickly, dueto decoherence by the environment E.

C. Potential world modal interpretation

If we follow this example, we can view |left〉S•O1=

|l〉S ⊗ |L〉O1and |right〉S•O1

= |r〉S ⊗ |R〉O1as some sort

22 Roughly speaking, in traditional inductive methods, science pro-duces (induces) theories to sum up all observed experimentalfacts (and potentially observable ones, by an hypothesis of reg-ularity), whereas the hypothetico-deductive method starts withhypothetical theories, deduces experienceable facts and confrontsthem with experiments.

23 Some authors argued that not every situation will evolve this way.This is true; it is not necessary to suppose that such a S•O sys-tem will always evolve that way that is that S •O eigenstatesare tensor products of S and O states. Everett, indeed, onlyconsiders that it does when the interaction is a measurement.For our purpose here, it is sucient to consider cases where Sevolves with O, according to O's eigenstates, and accept theexperimental fact that when it does so, it does so according toBorn's rule. Why universe evolves that way and why classicaleigenstates are selected is another question we shall not discusshere, see Zurek (2003).

7

of possible worlds. The observer, in a way, exists intwo versions that cannot exchange information, in Ev-erett's framework. But what does the word possiblemean precisely? There is a lot of confusion about thisin the literature. In many readings of RSI, especially inPMI, possible means imaginable: there would exist apossible world for any and every imaginable situation,for example a possible world in which this paper wouldbegin with the word the. This is undue extrapolation:physics knows nothing about it. No experimental evi-dence suggests such a possibility. To discuss this moreconsistenly, we will need an unambiguous vocabulary forpossible worlds24.In logic, a formalism exists to deal with possibility (and

necessity): modal logic. Nevertheless, classical modallogics will be of no use: we need here a quantical logic,according to principle 2 (logic). In (classical) modal log-ics, one denes possibilities with (i) a set, the model, ele-ments of which are possible worlds we will say poten-tial worlds, (ii) a relation of satisfaction between worldsand propositions (or statements), and (iii) a relation ofaccessibility between worlds which describes what worldsare possible from the point of view of another world. Weshall similarly dene Quantum Modal Logic.One can dene a Quantum Logic system with alge-

bra as an orthomodular lattice25 (Birkho and von Neu-mann, 1936). One can also follow a syntactic approachusing deduction (Delmas-Rigoutsos, 1997; Gibbins, 1985,e.g.). In any case, one can conceive Quantum Logic asbeing the structure of sub-Hilbert spaces of a Hilbertspace in the same way as Boolean logic is the structureof subsets of a set26. If we want to stick to physics, asimple proposition is a statement that after a specic ex-periment a specic observable lies within a specic rangeof values; for example detector 1 has measured an elec-tron with a speed less than 10 m.s−1 or photon spin inthe direction −→n was measured as positive. Ultimately,simple propositions are always yes-no statements and al-ways about an observation at a specic moment. Tech-nically, one can model a proposition L =particle goes

24 Everett is very cautious about that and in (Everett, 1955), henever uses the phrase possible world, but only branch.

25 A lattice is a partially ordered (6) set in which every two ele-ments have a supremum (∨) and an inmum (∧). It is orthocom-plemented if there is a least element (0), a greatest element (1)and an involution reversing order, the orthocomplement (·⊥).It is orthomodular if a 6 b implies a ∨ (a⊥ ∧ b) = b (Birkhoand von Neumann, 1936). One can also dene orthomodularitysaying a 6 b implies that a and b commute or are compatible(a _^ b) (Delmas-Rigoutsos, 1997). In this formalism, 0 is false, 1is truth, order (6) is consequence, supremum (∨) is disjunction(or), inmum (∧) is conjunction (and) and orthocomplement(·⊥) is negation.

26 A theorem by Constantin Piron (1964) shows that any niteor denumerable Quantum Logic system can be represented in aprojective geometry, and thus as sub-spaces of a Hilbert space.

through the left slit either by a sub-Hilbert space, L, orby its corresponding projection operator, L. With theone-dimensional simplication frequently used in peda-gogic writing, L = |left〉 〈left| and L = C |left〉. Complexpropositions combine simple propositions by conjunction(and), disjunction (orQL) and negation. Conjunctionis the intersection of spaces: A ∧ B = A ∩ B. Quanticaldisjunction is the sum of spaces: A ∨ B = A + B, i.e.all linear combinations of elements in A ∪ B. Quanticalnegation, contrary to the classical one, is not only a falseconjunction, but corresponds to an orthogonal sub-space,L⊥; so there are many ways for a state to satisfy neithera proposition nor its negation. For example, in a double-slit experiment, the left and right propositions areexclusive, meaning L ⊥ R, so the L and R propositionsare negation of each other. In English, one can say thatthe particle goes either through the left slit, or throughthe right slit with an exclusive or, but a quanticalor!If we consider a whole timeline of the universe, ψ : t 7→|ψ(t)〉, the observed system may satisfy a lot of propo-sitions that can be contradictory; consider, for example,the following history: the photon has spin up in the di-rection −→n1, then it has spin up in the direction −→n2, thenit has spin down in the direction −→n1. So it is importantto index propositions according to time, whenever for-mulations can be ambiguous: the photon has spin up inthe direction −→n1 at time t1, then the photon has spinup in the direction −→n2 at time t2, then the photon hasspin down in the direction −→n1 at time t3

27.By denition, a wave function ψ satises a proposition

P concerning a time t, if P |ψ(t)〉 = |ψ(t)〉 or, equiva-lently, if |ψ(t)〉 ∈ P; one writes ψ P . Can one considerall states in H, the total reference Hilbert space, as beingpotential worlds? Probably yes, in very simple pedagogicexamples where we use only few dimensions and veryshort histories; probably not, in the general case: eachobservation at least doubles the necessary dimensions ofH, so any whole timeline may lead to dimensional com-plexities. Restrictions may need to be applied in the fu-ture, but, for a start, we dene a potential worldA to be anon-zero sub-Hilbert space of H. Starting from there, wewill restrict the use of the term potential to this precisemodal meaning, and leave possible to general or infor-mal use. We say that A satises proposition P or thatP is true inside A, and we write A P , if any |α〉 ∈ Asatises P , except perhaps for some zero-measure sub-space. As for Everett (1955), a potential world W willgenerally be expressed as a recorded history: observa-tion 1 measured value a for A, observation 2 measuredvalue b for B, etc.28. Let us imagine we add a new ob-

27 It is to be noted that in these propositions time is merely arelative index, not a parameter.

28 We use histories to express potential worlds, not more. This ap-

8

servation toW, for example X with two values + and −;this observation determines two sub-spaces: X+ ⊥ X−,with W = X+ ⊕ X−. In such a case, we say that X+

and X−, and more generally any other sub-Hilbert spaceof non-zero measure and non-zero orthocomplement Y isaccessible from W; we write W 3 Y. The accessibilityrelation can be thought of as the logical identity of pastobservable interactions; in (Everett, 1955) it correspondsto shared automatic apparatus memories. In PMI, it cor-responds to a world split (but this is not a technicallyprecise expression).As usually in modal logic, we say that P is necessary

in W, W |= P , if any world accessible from W satisesP : ∀X 2W, X |= P . By construction, if W |= P , thenW |= P , and reciprocally. We can, similarly, denethat P is possible in W, W |= ♦P , if at least one worldaccessible from W satises P : ∃X 2 W, X |= P . Inour example, W |= ♦”X = +”, W 6|= ”X = +”, andW |= ”A = a”.One always has A P ∨ P⊥, but, in general, a poten-

tial world may satisfy neither P nor P⊥: correspondingobservations may not be possible for experimentation. SoA 6|= P does not imply, in general, that A |= P⊥, evenwhen A is one-dimensional. In classical modal logics,necessity and possibility are dual: ♦P is equivalent to(P⊥

)⊥; this is not true, in general, in Quantum Modal

Logic. Nevertheless, in a case whereA and P are compat-ible (A _

^ P)29, A 6|= P is equivalent to A |= ♦P⊥ (30).Note that Quantum Modal Logic is only a formalism;

it does not rule, in general, what is and what is not apossibility. Neither does the Hilbertian wave functionformalism. Only the experimental register can give us aclue about what possibly happens and what cannot hap-pen when there is no observation. Nevertheless, one canuse these models to make a concept (not only a notion)of observer explicit. Following Everett (1955), one candene an observer as a record of observation memories.An observer O resides in a potential world, but not onlyone: when O resides in P, meaning P satises all his

proach, despite similarities, is fundamentally dierent from theconsistent histories of Robert Griffiths and Roland Omnes

or the decoherent histories of Murray Gell-Mann and JamesB. Hartle: both are essentially attached to Classical Logic. Wewill not go further on this matter here; it would need more de-velopment.

29 With (Delmas-Rigoutsos, 1997), let us dene EF = E∩(E⊥+F),the projection of F on E, and dene E and F to be compatible,E _^ F , by:

E F = F E ⇐⇒ E F = E ∩ F ⇐⇒ E F ⊆ F def⇐⇒ E _^ F .

30 Proof.- If A is a potential world, if P is a proposition and P thecorresponding subspace, and ifA _

^ P, thenA = AP+AP⊥ =A ∧ P +A ∧ P⊥. If we suppose that A 6|= P , then A 6|= P , soA ∧ P⊥ is a non-zero subspace. Since this is a world accessiblefrom A, we infer A |= ♦P⊥.

memories-propositions, then O resides identically in anypotential world B from which P is accessible (B 3 P).There are also dierent versions of O in each potentialworld A accessible from P (P 3A). These B-worlds arethe only logical past of O (before) and the A-worlds arethe various potential futures of O (after). In each A, Osees himself as the only existing version of O, even if hecan conceive other imaginable versions of himself notethat an imaginable version is not necessarily a potentialversion, in the strict sense of our formalism. Note alsothat, for O, P and all its B-worlds are mutually indistin-guishable: no physical means can dierentiate them.

In our Quantum Modal Logic formulation, poten-tial worlds correspond to Everett's branches (Everett,1955), i.e. correspond to sets of parallel universes, andnot to individual universes of PMI31. Note that, in anycase, this parallel universes (PMI) formulation can beuseful at a macroscopic level, but it is incorrect at aquantical level, as noted by Yoav Ben Dov (1988) andBernard d'Espagnat (1994). For example, consider anexperiment measuring an electron spin in the direction−→n1: this leads to worlds S+1 and S−1 . Neither of theseworlds can be conceived without superposition: in eachone, one can add a measure of spin in a new direction, −→n2,not parallel to −→n1; S+1 is a superposition of the two acces-sible worlds S+1

+2 and S+1

−2 . So RSI and Quantum Modal

Logic are coherent with a parallel worlds popular for-mulation, but only if one is cautious enough not to gurethese worlds as being in Boolean disjunction. In particu-lar, one cannot enumerate the accessible potential worldsin general. As stated by Jean-Marc Levy-Leblond:from the point of view of the universe, there is only oneworld32. One cannot count potential worlds, but, still,if we reason relatively to some global wave-function, webenet from conditional probabilities for relative states(Everett, 1955); this provides us with quantical probabil-ities, similar to classical (Kolmogorov) ones but notto be confused with them33. These probabilities providea measure, i.e. Born's coecients, which we will uselater in this paper.

With our now formal vocabulary, let us come back toour initial question: from inside a potential world, canwe get to know something about alternative worlds?

31 Closer to David Deutch approach, see (Ben-Dov, 1988), p. 113.32 The many worlds idea [PMI] again is a left-over of classical

conceptions. The coexisting branches here [. . . ] can only berelated to worlds described by classical physics [. . . ] To me, thedeep meaning of Everett's idea is not the coexistence of manyworlds, but, on the contrary, the existence of a single quantumone. (Ben-Dov, 1988, cit. p. 79)

33 Karl Popper (1959) also noted this similarity and founded onit his propensional interpretation of quantical probabilities. Healso argued that satistical interpretation is not suitable for it.Contemporary reader would probably be more confortable, onthis matter, with Bell's inequalities violation arguments.

9

D. Qubals and sub-quantical framework

Let us be more precise. Imagine we are in a world Wand we set up an experiment leading to two potential val-ues, for example yes and no. Note that we say herepotential values and not imaginable values: there are twopotential worlds A+ and A−, both accessible from W.We aim here some Schroedinger's cat-like experiment,but without its complexity: in Erwin Schroedinger'straditional mind experiment (Schrodinger, 1935; Wheelerand Zurek, 1983, cit.), the observer may contemplate theeect of a disintegrating substance, after a certain periodof time, so this mind experiment integrates the eect ofmany dierent independent atoms we are here in a sit-uation of mixture rather than superposition. Even if weare content with a unique atom, there may be complexinteractions and if the observation is a yes-no experimentat the macroscopic level, we are not always sure it is alsothe case at the quantical level. The physical descriptionis complex, and the discussions, physical or philosoph-ical, are often very confused, sometimes even nonsen-sical34. We need more unequivocal situations. To bereasonably sure the imaginable alternatives correspondeectively to potential outcomes, we will focus here onvery simple experimental protocols, similar to the onesused to experiment upon the EPR paradox, for examplethe spin of correlated photon pairs. We can also imagine,for example, a measure of spin of a photon in a direction,following a measure in an orthogonal direction. Let uscall such a situation a quantical binary simple alternative a QuBAl. It is important here that uncertainty shouldbe of quantical origin and not of statistical origin (thereshould be no mixture), also that it should not be dis-turbed by an environmental decoherence in a sense,qubals are anomalies since in most experimental situa-tions the decoherence makes such alternatives disappearvery quickly along the von Neumann chain.Typical qubits, quantical binary digits, seem to be

good candidates to provide qubals, quantical binary sim-ple alternatives. The dierence between the two is thatwe will have here no use of the qubit phase, nor, as amatter of fact, of its value. There is also a practical dif-ference: the qubal is a situation, an experimental frame-work, whereas one now uses the word qubit mainly fortechnical devices.In a situation where we really have a quantical binary

alternative (a qubal), we have two accessible worlds, X+

and X−, each one having a non-zero probability, accord-ing to Born's rule. In that experimental context, ourquestion is now: if we are in X+, can we know anythingof X− (if we follow RSI) or, simply, does X− exist or not

34 True or false, a legend reports that Stephen Hawking said:When I hear of Schroedinger's cat, I reach for my gun. Thisis a disconcerting vision, but, a sure way to end the argument.

(in PWI, it does not)? If we strictly follow Everett'sframework, the answer is straightforward: we cannot. Noinformation can go from one world to an inaccessible one.Would this be nal, we could not settle between RSI andPWI, and the choice would be essentially a matter ofpersonal convenience one's metaphysical preference35.But Everett relied only on pure Quantum Mechanics,and QM is not nal as a description of nature; notably,we still seek for a framework integrating what we know ofthe quantical phenomena and what we know of inertia-gravitation.What could be a ner description of reality than QM

a sub-quantical theory? Many options already exist,in many variations, of a dierent status and of a dier-ent extension, especially quantum eld theories (QFTs),but, at the time writing, none is universally adopted, norcompletely satisfying. A convenient future theory willprobably have to alter some aspects of QM or perhapsadd some others.Let us be conjectural now, and combine principle 4

with many physicists' insight of quantical particles as thesuperposition of virtual particles. Quite generally, thesevirtual particles follow Feynman's paths conceptions(1948; 1949), as a means of computation or as a heuris-tic representation for quantum elds theories36: thesevirtual particles are meant to encompass all possibili-ties within a mathematical model. In our attempt ata theory, let us consider only potential particles: parti-cles existing in potential worlds accessible from quanticalparticle worlds. These potential particles, again, are nei-ther imaginable particles nor virtual particles of a modelor another; these are conceived as physical, and hope-fully experimentable. Note, however, that their poten-tial worlds may not be quantical potential worlds, butonly sub-quantical potential worlds, that might not beaccessible to direct experimentation (we will come backto that matter hereunder). Potential particles, surely,do have properties very dierent from those of quanticalparticles in the same way as complex ions in solutionmay have very dierent properties from bare ions. Forexample, if mass is an emerging property, one can imag-ine that, having no mass, all potential particles propa-gate at a displacement pace c, whereas quantical par-ticles can be measured at various velocity magnitudes.Physics for sub-quantical level has to be theorized; it maybe extremely dierent from quantical and macroscopicones. At the sub-quantical level of description, there

35 Many authors discuss this preference based on a version or an-other of the parsimony principle, but their conclusions are veryformulation dependent and did not achieve to convince a largeaudience. For this reason, we chose a methodological formulationof our principle 1, and aim for no metaphysical use of it.

36 Virtual particles can also be seen as mathematical artice torepresent elds, similarly to Huygens-Fresnel principle, seee.g. (Feynman, 1948, esp. sec. 7).

10

may be nothing like a space-time we are used to, but,perhaps, only contiguity and consecution relations. TheSchroedinger's equation at the quantical level mightjust be the result, by symmetry, of the superposition ofnumerous (perhaps innite) potential interactions of po-tential particles. (37)For more than a century and a half, elds have been a

very fruitful concept of physics. General Relativity (GR)and quantum eld theories (QFTs) made it a prime con-cept of today's physics, pivotal to make wave and cor-puscular behaviors converge elds overtaking classicalwaves, and corpuscles being quanta of these elds. Thisapproach has led to new understandings and new ways toconsider particles, especially with quasi-particles (Cooperpairs, phonons, holes, etc.). But, what are these elds ac-tually? Are they physical objects or merely theoreticalreications of physical relations38? Fields represent in-nite numbers of degrees of freedom even a continuousinnite number (i1 = 2ℵ0), for example in the form of aFock space. What could this mean physically? We willsuppose here that all elds emerge from potential parti-cles39. The reader can consider this as real (as a hypoth-esis), or act as if and consider this as mere heuristics.Since pure Quantum Mechanics phenomena cannot

distinguish between PWI and RSI and does not allowinformation to go from one potential world to anotherinaccessible world, we will now consider another option,within our conjectural sub-quantical framework, relyingon gravitation.

III. INTERPRETATION ELEMENTS FOR GRAVITATION

We will not unify General Relativity (GR) and Quan-tum Mechanics (QM) here, but our framework can helpsketch out some aspects of a merged theory. It couldcertainly not be something like GR⊕QM (metaphor-ically speaking), since many phenomena are on the fringeof both application domains, even in our near phenome-nal environment. So, many authors are rather searchingfor something like GR ⊗QM: this line of research isappealing and somehow general, but it does assume that

37 Other authors proposed similar hypothesis, following hydrody-namical interpretation of QM (Madelung uid), notably Bohm(1957, p. 33), presenting quantum particle as a condensation ofa sub-quantical uid, see (Ben-Dov, 1988, p. 190).

38 By reication of a relation we mean constituting as an objectwhat is really relation between objects: for example the Newto-nian gravitational potential of a body M is a reication of thegravitational force between M and test masses (real or virtual).

39 Newton, relied on forces for its mechanics, but prevented reex-ion about their nature: hypotheses non ngo. In 1936 Einstein(Cohen-Tanoudji and Spiro, 2013, cit., p. 471) argued we shouldfound physics theory on elds. Here we are, now, with a very ef-cient tools, but again not forging hypotheses about it. Perhapsare we in a time to build other tools.

fundamental phenomena of one theory are not alreadyconsequences of the other. We shall discuss this point,relying rmly on Schroedinger's linearity (principle 4)and some experimental facts.In the search for such an integration, we immediately

encounter a conceptual diculty, well summed up byDavid Bohm: In relativity, movement is continuous,causally determinate and well dened; while in quantummechanics it is discontinuous, not causally determinateand not well dened40. One could add that QM dealswith particles of no dimension, while GR diverges onpunctual bodies, dealing rather with densities. Clearly,a mere rapprochement would lead nowhere. We need tohave a look at core concepts.

A. Emergence of Special Relativity

1. Relativity without rods and clocks

Howard Robertson (1949) showed that Special Rela-tivity (SR), and thus grounds for GR, can be deducedfrom some general postulates and three experimentalfacts. We will follow his reasoning but reduce its hy-potheses, to enforce our previous principles.His rst postulate is that there exists a reference frame

Einstein's `rest-system' in which light is propa-gated rectilinearly and isotropically in free space withconstant speed c. The notion of reference frame, withabstract rods and clocks is, in general, incompatiblewith our principle 3: no measure exists in abstracto, butone can adapt Robertson's postulate without impairinghis reasoning41:

Principle 6 (reference observer). An observer O exists,for which, in free space, light propagates rectilinearly andisotropically at a constant velocity c.

Robertson (1949), then, postulates the existence ofa reference frame [. . . ] Einstein's `moving system' which is moving with any given constant velocity [. . . ]with respect to [rst reference frame], also supplied withabstract rods and clocks. We will disregard this postu-late and only suppose that we can observe, in moving sys-tems S, relative remoteness and relative duration, as anapplication of our principle 3. Both are evaluated frominteractions internal to S, but outcomes of these mea-sures can be traced from the outside, O. Here, note thatthe remoteness and duration are space and time distances

40 David Bohm, Wholeness and the Implicate Order, 1980, xv.41 Isotropy and constancy of light speed is one of the best veried

facts, up to 17 orders of magnitude (Eisele et al., 2009), still,we state it as a principle, not merely as a fact, since we needto postulate, at least methodologically, a reference observer (notnecessarily unique).

11

at the macroscopic and quantical levels for S; one canspeak also of coordinates. Their measure emerges fromsub-quantical mechanisms, and therefore can have struc-tures very dierent from those of sub-quantical contigu-ity and consecution intervals. For speed, we will use theterms velocity and rapidity, as usual, for the macroscopicand quantical phenomena; at the sub-quantical level, wewill speak of displacement pace (table I). Epistemologi-cally, in strict application of principle 3, this means thatwe do not necessarily view matter as being in space-time,but merely as having coordinates, as descriptive qualities(emerging qualities, as we shall see).

macroscopic / quantical sub-quantical

position in space-time placement in sub-spaceand time

remoteness, duration,coordinates

interval (of contiguity andconsecution)

velocity, rapidity displacement pace

Table I vocabulary for usual quantities

We know nothing, yet, about sub-quantical intervalsand displacement paces. As far as we know, the sub-quantical space (the sub-space) may be continuous, evenEuclidean, or may be a grid, a network, or merely a rela-tion, with the interval relation having emerged by somesort of geometrization mechanism a la Perelman42. Wewill not suppose much more on that matter now; but,with the two previous postulates, we shall suppose thephenomena regular enough so that coordinates are nearEuclidian and local transformation can be linearly ap-proximated.So, with Robertson (1949), let us suppose S is mov-

ing at velocity c−→β and use clever axis representation

(movement along x axis, especially); the transformationhas (locally) the following form:

Λβ =

a0 βa1/c 0 0βa0c a1 0 0

0 0 a2 00 0 0 a2

. (1)

The rst fact to determine Λβ is the Michelson-Morley type experiments, in S:

Fact 1 (Michelson-Morley). The total time requiredfor light to traverse, in free space, a distance l and toreturn is independent of its direction.

42 Grigori Perelman uses Ricci-Hamilton ow to homogenize themetric on manifolds, and proves that any 3-dimensionnal mani-fold can be geometrized. One can imagine a similar mechanismto make the local sub-space interval relation build up from aninitial simple unstructured contiguity relation.

If we suppose this fact to be general, one can con-sider a light beam making any angle from the x axis,and then one nds the Lorentz-Fitzgerald contractionas a consequence: a2 = a1/γ with Lorentz factorγ = 1/

√1−β2. Then, Robertson (1949) adds Kennedy

and Thorndike experimental result as: The total timerequired for light to traverse a closed path in [a refer-ence frame] is independent of the velocity [of this frame]relative to [rst reference frame]. The important thingis that we do not need to suppose c constant in S, butsimply deduce it from being constant in O. We will statethis experimental result as:

Fact 2 (Kennedy-Thorndike). The total time re-quired for light to traverse a closed path, as experiencedfrom a system S, is independent of the velocity of Srelative to O.

Following Robertson, again, one nds that a0 = a1 =γgβ and a2 = gβ . Eventually, one may conclude that Λβis a Lorentz transformation (gβ = 1), if we add a thirdfact i.e. the Ives and Stilwell experiments:

Fact 3 (time dilation). The frequency of a movingatomic source is altered by the factor 1/γ relative to thevelocity of the source with respect to the observer.

This fact makes use of two relatively moving observers,but this can be interpreted coherently with our princi-ple 3.Robertson wanted to show that one can replace the

greater part of Einstein's postulates with ndings drawninductively from the observations, but our small refor-mulation, here, shows a more general fact. In any theoryobeying our postulates and coherent with these three ex-perimental facts, local coordinates, measured from sys-tems moving at constant velocity for a reference observer,should be transformed according to a Lorentz transfor-mation, at least for linear approximation.As a consequence, any sub-quantical theory from which

one could derive these facts would be coherent with SRand would have locally a Minkowskian metrics (with lin-ear approximation). Let us now specify our attempt ata theory. Next section will be essentially speculative.

2. Special relativity emerging from potential particles

Some clues, mainly from contemporary theories, es-pecially due to renormalisability questions, suggestthat sub-quantical potential particles should be thoughtof without inertial mass. The latter would be ac-quired through a coupling mechanism. Present researchfavors the Brout-Englert-Higgs-Hagen-Guralnik-Kibble (BEH) mechanism, i.e. interaction with BEHbosons, with some recent experimental support (ATLAScollaboration, 2012). We will not rely on the BEH the-ory but, in coherence with it, let us only suppose all

12

sub-quantical potential particles are massless. Accord-ingly, all of them have a displacement pace of the samemagnitude (in some sort of sub-space-time). Betweentwo interactions, the displacement pace of a potentialparticle is uniform and has a specic direction43. Po-tential particles move like ideal non-interacting photons,whereas quantical particles do engage in interactions indierent potential worlds, and so correspond to super-positions of potential particles. The less a photon inter-acts, the closer the mean displacement pace of its su-perposition will be, in magnitude, to the common pace.So, if we suppose that velocity accounts for some sortof mean displacement pace, light with minimum inter-action will always have the same velocity magnitude, c,which we can identify with the only displacement pacemagnitude of potential particles (to a factor44). If thesub-space-time is supposed isotropic, this would also ac-count for principle 6 and fact 1. Note that we presupposeno sub-quantical notion of inertia or relative motion: onthe contrary, all displacements occur at same pace, andall interactions emerge from combination and emission ofsub-quantical potential particles in diverse directions45.This accounts for fact 2.For fact 3, we need to be more precise in our hypothe-

ses. Let us imagine that the velocity of a quantical par-ticle is some sort of mean displacement pace of potentialparticles. For a (set of) potential particle(s) p, let us noteνp its quantity and −→np its displacement pace direction46,both operators on quantum states. For potential parti-cles participating in a quantical system S, let us dene amean pace:

−→βS =

∑p∈S

νp−→np∑

p∈Sνp

. (2)

43 We suppose it to be rectilinear, for now, but geometry of sub-space-time may appear to be more complex.

44 This is only for supporting understanding, since we have no ac-cess to contiguity intervals independently of consecution inter-vals. We could as well consider displacement pace magnitude tobe 1 and state the contiguity interval to be exactly the consecu-tion interval between two events directly connected by a potentialparticle. For that reason, we use the same term interval, forboth space and time.

45 For potential particles, there can probably be no notion of atrajectory: since quantical particles of the same type are indis-tinguishable, i.e. logically identical, so should be underlying po-tential particles. A potential particle, in a potential world, onlyexists between two interactions.

46 Quantity and direction are time dependent. The quantity termshould be specied in a more formal theory. For now we canthink of it either as homologous to frequency for photon or as anumber of inseparable potential particles or as some sort of in-trinsic Born's rule coecient (since we do not take into accountpotential worlds in following formula). It is also a convenientmeans to keep non existing particles (not yet existing or no moreexisting) in same formula; in that case νp = 0 and direction isundened.

In this formula, one can recognize velocity: c−→βS , energy:

ES = h∑p∈S

νp, and momentum: ES

c

−→βS = h

c

∑p∈S

νp−→np (47).

If these potential particles have a balance between thepotential directions (isotropy in pace), emerging velocityis zero. The velocity magnitude would rise with a com-mon orientation in some direction of displacement pace.In the limit, if a vast proportion of displacement paces ofpotential particles heads in one direction, the emergingvelocity magnitude will be close to c. Now, let us imag-ine two mono-directional streams of potential particles;if they have the same direction, they have no potentialfor interaction, since they are displaced at the same pace the greyhound cannot catch the rabbit. At angle π,on the opposite, the probability of interaction by unitof time is at a maximum. At angle θ, we can imaginea relative interaction progressing with 1 − cos θ. In thistheory, accelerating a quantical particle in a directionis just adding potential particles in the same direction:each interaction then tends to replace more frequently apotential particle in the opposite direction by a potentialparticle with another direction, and so raises the level ofdirectionality.Let us suppose that for two quantical particles poten-

tially interacting, there is a factor (1 − cos θ) on theirwave function describing their propension to interact;this factor reduces the potential presence of particles andthus increases the time before interaction occurs. At thesub-quantical level, this (1− cos θ) factor corresponds to(1−−→np1 · −→np2). It is linear, and so the factor for two po-tential particle sets, for example two quantical particles,

becomes (1−−→βS1 ·−→βS2). If these two sets participate in an

object i.e. a greater set with velocity c−→β , then let

us decompose−→βSa

=−→β +−→ba . If we suppose this bigger set

to be homogeneous enough, then ba β (48). Therefore,the dynamics factor becomes

1−−→βS1·−→βS2

= 1− β2 −−→β · (−→b1 +

−→b2)−

−→b1 ·−→b2

≈ 1− β2.(3)

If we share this factor equally between the two sets, bysymmetry, we get the following factor:

√1− β2 = 1/γ.

With these hypotheses, γ appears as a directionality fac-tor, and 1/γ as a time dilation factor for moving objects.This accounts for fact 3: proper time emerges as a col-lective property (without need to consider mass).The theory we have sketched out could make SR

emerge in a QM context, without any need to natural-ize a Minkowskian space-time. Contrary to Einstein'sSR, in our theory, length contraction and time dilation

47 If βS < 1, we can set γS = 1/√

1−β2Sand dene usual relativist

homologous of these quantities.48 The closer β will be to 1, the more homogeneous the object will

be, relative to the displacement pace.

13

would appear as phenomena, not perspective eects49.Note also that it makes relative four-velocities emergefrom displacement paces, which may thus be absolute.Scholium 1. Our theory, which technically provides

an alternative to Einstein's SR, supports an interpreta-tion of SR which we will name cinematic interpretation,as opposed to the usual Einsteinian ontic interpretation.These two interpretations dier metaphysically, since thecinematic interpretation allows an absolute space and hasgot absolute displacement paces, whereas the ontic inter-pretation acknowledges only relative space-time and ve-locities. But, at a physical experimental level, bothinterpretations expose only relative velocities, and bothacknowledge only Galilean frames as privileged frames.Scholium 2. With our attempt at a theory, the phe-

nomena (i.e. causality) propagate at a maximum veloc-ity c, but, still, some correlations, as in EPR-like experi-ments in qubals, are instantaneous, since they dependon potential worlds.

B. Inertia and energy induction of weight

The usual General Relativity interpretation viewsgravitation as a reciprocal eect between particles anda special object called space-time. This does not com-pletely rule out mechanisms where gravitation wouldemerge from eects of the curvature of coordinates onquantum phenomena. For example, Andrei Sakharov(1967) suggests that gravitation could emerge fromquantum uctuations of the vacuum if space is curved.So, should gravitation be quantied, or should it be dif-ferently dealt with? Let us start with an analysis of someelements of knowledge about motion and gravity.John Philoponus rst introduced a notion of impulse

(impetus) or of a power to move to account for thecontinuation of movement50, a notion later expandedby Ibn Sna, Jean Buridan and Galileo, leading tothe laws of conservation of momentum and (what wenow call) kinetic energy by Christiaan Huygens (Loc-queneux, 2009). Isaac Newton's Principia (1687) de-ne inertia as a vis insita or innate force of matter, asa vis inertiae or force of inactivity, or as a power ofresisting by which every body as much as in it lies en-deavors to persevere in its present state whether it beof rest or of moving uniformly forward in a right line;Newton adds it is essentially mass. A few lines above,Newton opens the Principia with a paragraph stating

49 Bernard d'Espagnat (1994) relates that John Bell emphati-cally advocated an interpretation of relativity as phenomenal.

50 Before him, movement was not something to model mathemati-cally; for example, for Aristotle, weight was a clinamen, a naltendency to be in a low place, and motion needed an eectivemotor to continue its course.

that weight is proportional to mass as an experimentalfact. Inertia, mass and weight are here proportional orequivalent.

Einstein's equivalence, Newton's equivalence andalsoGalileo stating that all weighting objects fall at thesame velocity, are nowadays considered to be variationsof an equivalence principle. This principle is often ill-dened as a physical concept since it is generally deeplyinterwoven with mechanics. For example, in Newtonianmechanics, Newton's and Galileo's equivalences arelogically equivalent. We will not completely specify thisnotion, and we will consider the weak equivalence princi-ple (WEP) to be the equivalence of inertia and gravita-tion: inertial mass and weighting mass for Newtonian me-chanics, inertial energy and ponderous energy nowadays.The experimental register for this equivalence is consid-erable; it goes back, at least, to Simon Stevin's Princi-ples of statics (1586) and have nowadays a precision of10−14 with the MICROSCOPE satellite experimentation(Touboul et al., 2017); so no theory can dispense withit. Some form of equivalence is also deeply rooted in SR.As early as 1905, Einstein notes, in the E = M.c2paper (1905), that, in SR, energy content (whatever thismeans) decomposes into kinetic energy (K) plus a con-stant, the energy content at rest. On that basis, if a bodysymmetrically emits light for energy L, Einstein notesthat Kbefore −Kafter = (γ − 1)L. Since K = 1

2M.v2, atrst approximation, he concludes: If a body gives o theenergy L in the form of radiation, its mass diminishes byL/c2 [. . . ] If the theory corresponds to the facts, radia-tion conveys inertia between the emitting and absorbingbodies. Here, Einstein assimilates inertia (Tragheit),moving mass, and energy content (Energieinhalt) lateron, this will be a cornerstone of GR. This reasoning isvery general in its form, and thus valid also in our sketchframework with emerging kinetic energy: emitting lightdiminishes the rest energy content.

So, let us acknowledge this equivalence and considerinertia and (ponderous) energy as equivalent. What linkcan we make between this energy and quantical or sub-quantical energy, especially in our sketch framework?

Let us turn to gravitation. Newton's gravitation law(1687) shows a reciprocal force between two bodies C(context) and T (test) being G.MC .MT /d

2. Newton'swork also expresses acceleration for a test body, basedon WEP: G.MC/d

2. This expression is independent fromthe test mass, which helps virtualize it and reify potentialgravitation as a eld for a test mass small enough notto disturb the context body. Experiments (now numer-ous) show that gravitation concerns every object, withor without mass. Einstein generalizes this considera-tion and postulates we can decompose the gravitationphenomenon into two parts which we call here receivedand given gravitation. Received gravitation is theorizedby the fact that inertial movement, that is movementreceiving no other inuence than gravitation, follows a

14

geodesic of a special object, the space-time. Receivedgravitation concerns inertia, and experimental evidencesuggests that it is compatible with equivalence: it is thesame for all forms of energy content. Given gravitationis theorized by the fact that bodies with a energy den-sity eld T = (Tµν) create a curvature of the space-timeobject such that(

1

2R− Λ

).gµν −Rµν = −8πG

c4Tµν , (4)

where (gµν) is the metric of space-time, (Rµν) is this met-ric's Ricci curvature tensor, modeling its propagation, GisNewton's constant, and Λ is a cosmological constantwhich we will come back to below51. We only know directexperimentation of given gravitation for mass energy, buton the one hand cosmological evidence suggests that itcan account only for a small part of the observed grav-itation phenomenon, and, on the other hand symmetrywith received gravitation suggests that tensor T shouldenroll any form of energy content. So, in GR, inertia(inertial energy) is now better described by tensor T , ageneralization of energy-momentum in the form of energycontent density and ux52. Nonetheless, the question re-mains partially open for QFT vacua: since they allowno energy transfer, and since they are omnipresent andisotropic, whether they should be counted in this energycontent. This question also makes sense for CMB andCνB53, that may have no cinematic reality with respectto GR.More generally, according to Emmy Noether's the-

orem, in QM, energy-momentum is conceived as a sym-metry of the model (i.e. of the Lagrangian). So it doescorrespond to the physical phenomena, but only to a con-stant. It may also incorporate model artifacts, i.e. somesymmetries of the model with no phenomenal (experi-enceable) counterpart. Certainly, these artifacts can becompensated for by a categorial quotient or gauge theo-ries, but at the cost of ease and heuristics. So the quan-tical energy-momentum may not correspond exactly tothe GR energy content.In our sketch framework, any potential particle, mass-

less, is supposedly associated with a quantity ν corre-sponding to an energy in h.s−1 or a momentum in h

c .m−1.

As a heuristic, one may thus view energy-momentumas some sort of eervescence of potential particles (eachone counted with its quantity ν), or some sort of sub-

51 There is experimental support for G being constant, under a10−12 per year (Will, 2014, p. 50). According to usual conven-tions, Greek letters indices range for space-time coordinates.

52 So, it is still a model of matter presence in space-time, but notonly massive matter. Matter is to be thought of as anything withenergy content.

53 Cosmic microwave / neutrino background radiations. Seesec. IV.C.1.

quantical scrum. Emerging as a superposition of sub-quantical potential particles, energy is this scrum's timefrequency, and momentum its space frequency. More po-tential particles make more inertia emerge. We insistthat this is a heuristic view: since potential particles arecontinuously being absorbed and emitted, from the quan-tical or above level viewpoint, we essentially count thesuperposition; much in the same way as the Huygens-Fresnel principle represents wave propagation as a con-tinuous re-emission of punctual sources on wave surfaces,one can think of a trajectory as a superposition of nu-merous Feynman diagrams of potential particles, cor-responding to minimum dimensional sub-spaces of po-tential worlds54. In this picture, from a sub-quanticallevel viewpoint, quantical particles merely exist as corre-lated propagations of interaction. If our view is correct,the traditional view of gravitation as an interaction be-tween two inertias, or two energy contents, holds no more.Two collectives cannot indeed interact per se: interactionshould also mean something at the sub-quantical level.We will now formulate an experimentable proposal onthat matter.

C. An hypothesis about weight

Cliord Will, in his review of experimental tests ofGR (2014), considers that it is possible to argue con-vincingly that if EEP [Einstein's equivalence principle]55

is valid, then gravitation must be a `curved space-time'phenomenon, in other words, the eects of gravity mustbe equivalent to the eects of living in a curved space-time. As a consequence of this argument, the only the-ories of gravity that can fully embody EEP are thosethat satisfy the postulates of `metric theories of gravity'[. . . ]. Note that living in a curved space-time, in ourframework, following principle 3, is a matter of measuredcoordinates and may be an emerging property of space-time, or even of sub-space.So, pushing our investigations further, the question is

now: on which level of physical modeling does gravitationplay? We can see three main possibilities.1) We can imagine gravitation as being essentially a

quantical level phenomenon.In that case, GR curved metrics may emerge from sub-

quantical phenomena in some way similar to what we

54 It may be innite, but we only consider innite here as a limitcase, according to principles 1 and 3. In nite case, 1-D sub-spaces are potential worlds, because of non-zero measure.

55 Following Robert Dicke, Einstein's equivalence principle isWEP plus local Lorentz invariance: the outcome of any localnon-gravitational experiment is independent of the velocity of thefreely-falling reference frame in which it is performed, plus localposition invariance: the outcome of any local non-gravitationalexperiment is independent of where and when in the universe itis performed (Will, 2014, p. 9).

15

suggested for SR. Or, one can also imagine that gravita-tion plays essentially at the sub-quantical level, with twooptions:

2) Sub-space, i.e. intervals, is curved by energy con-tent, not unlike the case of QFTs in curved space (B. Un-ruh, S. Hawking, R. Penrose, R. Wald).

3) Some sort of force is exerted between potentialparticles.

There has been a lot of research on quantical gravita-tion, leading, for the time being, to no generally acceptedtheory. We will therefore leave the rst hypothesis aside.The second hypothesis is at the very least untimely: itwould be too speculative for our roughly sketched theory,since we have no idea, yet, of the mathematical structureof intervals. We will explore here the third hypothesis.

Is gravitation exerted at the quantical particle level?Does it use some sort of vector potential particles, i.e.gravitons, the favored path for present research? Or doesgravitation dier from the other interactions? There isanother way of addressing these questions: is weight anin-world phenomenon, i.e. a phenomenon limited to eachpotential world, like other interactions, or is it exertedacross worlds, that is also between worlds otherwise in-accessible?

In our hypothetical theory, inertia and gravitation ap-pear to be proportional to the energy of all potential par-ticles in a given location. More precisely, it correspondsto linearity with respect to mass for given and receivedNewtonian gravitation, to linearity of given Einsteiniangravitation of the stress-momentum-energy tensor, andto linearity of the geodesic movement of received Ein-steinian gravitation. If this theory is correct, this sug-gests that gravitation given by a body, resulting fromits energy content would apply to all potential particlesindierently, and thus its given gravitation would applyto all potential worlds. Considering this phenomenonthe other way round, test bodies would receive a gravita-tion that would be the superposition of gravitation fromall possible worlds. Pushing the hypothesis further, if

|ψ〉 = a |ψ1〉⊥+ b |ψ2〉, with |a|2 + |b|2 = 1, we could imag-

ine that received gravitation should be something likeg = |a|2 g1 + |b|2 g2. Let this be our hypothesis, and letus look at possible experimentable consequences.

Hypothesis (trans-world gravitation). Gravitationacts at the sub-quantical level, and produces its eects be-tween worlds in coherence with the coecients of Born'srule.

A lot of conjectures and suggestions have been made toexplain some experimental diculties with gravitation.Our assumption, on the contrary, is an experimental hy-pothesis, as we shall see in the following section. Wouldit be in accordance with the experiment, it would alsoinstitute Born's coecients, or propensity probability,

as a directly observable quantity56.Before considering experimentation, we would like to

underline some aspects of our hypothesis:Scholium 3. Newton's third law, the action-

reaction or interaction law, results classically from thesymmetry of physical laws. It does exist at the macro-scopic level and, in a certain way, at the quantical level,even if the eld formalism does not make it obvious. Withour hypothesis, gravitation might not respect this law in-side each potential world (in-world action), but only be-tween potential worlds (trans-world action). Thus, grav-itation would not be an interaction in the same way aselectroweak and strong forces.Scholium 4. Without gravitation, potential worlds

do not mingle because quantical interference maintainsthem phenomenologically separate. On the other hand,gravitation, as it is now understood, is not linear with re-spect to wave-function, thus inaccessible potential worldsshould gravitationally interact.Scholium 5. If our hypothesis is correct, for trans-

world reasoning, the most adequate representation of al-ternative worlds is RSI except that when gravitationjoins the game the phenomenological world isolation ofRSI vanishes. On the other hand, for in-world reason-ing, the best heuristic support is PWI again, whengravitation is out of the game.

D. Crucial experiment

We presented the main tool of our proposed exper-iment in section II.D: the qubal, or quantical simplebinary alternative. Its general ambition is similar toSchroedinger's cat or Wigner's friend but with aclearly formulated and experimentable quantical proto-col. In this experiment, we would have a qubal withresults 1 and 2 determining the position of a heavymassive body M . In branch 1 of the alternative, that isif result 1 appears, body M will be put in place r1 atsome date t (previously dened). In branch 2, that is ifresult 2 appears, mass M will be put in place r2 at thesame date t. If we except gravitation, any observer inbranch 1 can note then that M is in position r1, by viewor contact, for example; if our hypothesis is valid, it willnot be the case with gravity.Let us suppose this qubal has Born coecients a1 and

a2. Classically, measurable gravity strength (accelera-tion) given by M would be g1 in case 1, and g2 in case

56 There have been many attempts to prove Born's rule, but theyare still controversial, even in RSI context. An experimentalconrmation of our hypothesis would change the purport of theseattempts. It would also bring some support to Lev Vaidman'sterminology for Born's rule coecients: measure of existence,for a world in a many-worlds interpretation (1998).

16

2. The experimental context body may be, for example,a 1 t quarry stone moved from a distance to a point 1 maway from a gravimeter in case 1, and remaining distantin case 2 57. With a rough evaluation g2 g1, and g1 =G.M/d21 ' 7.10−11 × 103/12 = 7.10−8 m.s−2 = 7µGal.This value comes in the range of today's microgravime-try apparatuses. If our hypothesis is correct, and if eectfollows the von Neumann chain, received gravity strengthwould be g ' |a|2 g1. If the qubal is symmetric, withequal probabilities of the two branches, then g ' 1

2g1.

What can we learn from this experiment? Beforethe turn of the 20th century, physicists thought thatmolecules would never be observed. For that reason,some authors, like Ernst Mach or Wilhelm Ostwald,rejected atomism (Cohen-Tanoudji and Spiro, 2013, p.51). The situation has some similarities with that ofpotential worlds. For RSI, all potential worlds are phys-ically real; for PWI one is real and the others merelyvirtual. Would such a protocol be conrmed by exper-iments, our potential worlds modal interpretation, con-vergence of RSI and PWI, would be well established asan interpretation framework. Still, this result is highlycounter-intuitive, perhaps for good reasons, and thus maywell fail. Would this happen, it would be an indica-tion that inaccessible worlds (in sec. II.C sense) are phe-nomenologically impervious, and so that distinctions be-tween RSI and PWI are essentially metaphysical. Phys-ical discussion could nevertheless remain on the soundgrounds of quantum modal logic and potential worlds.

Concerning gravitation, an experimental success wouldurge us to revise large scale gravitational reasoning andtake into account the gravitation received from alternatepotential worlds. We will shortly explore some possibleconsequences of such an eect in the following sections.These hypothetical consequences are incentives to exper-iment our hypothesis, since it would then link severalcurrent diculties. On the contrary, a failure of this ex-periment could suggest that gravity may be ultimatelylinear in some wave-function representation, and be aninvitation to elaborate quantical theories of gravitation.

IV. EXPLORATION OF CONSEQUENCES

We will now explore some possible consequences of ourhypothesis of trans-world gravitation.

57 One can also mention the extremely heavy bodies of the buildingindustry: bridge elements, building parts, or even entire build-ings (on May 22nd, 2012, a 6 200 t historical building was moved60 m away in Zurich, Switzerland).

A. Potential worlds and cosmic ination

The idea of cosmic ination was developed in the early1980's by Alan Guth (1981). It is generally thought tobe a phase transition of the early universe, and is still inwant of an explanation. It could be linked to a term simi-lar to the cosmological constant in the energy-momentumstress tensor, and/or be an early eect of a scalar eld.Cosmic ination is modeled as a metric expansion of theuniverse. It is considered to be exponential and to corre-spond to 26 orders of magnitude, at least considerablymore in some models. Its duration is extremely hypo-thetical; in some theories, ination is even perpetual insome parts of the universe. The main justication for thisidea is that it solves the horizon and atness problems58.The main observational evidence in favor of ination isthe CMB, which is very homogeneous, and the variationsof which are interpreted as inhomogeneities correspond-ing to quantical uctuations scaled-up by the inationprocess.How could we address this question within our frame-

work? Some models of perpetual cosmic ination predictthat it produces causally independent universes; this sit-uation is called cosmic multiverse. Some independentpublication, notably (Nomura, 2011), suggest that themultiverse may be linked, or even corresponds to, themultiple worlds of QM. This proposal relies heavily onthe idea that anything that is possible (in the model, butsometimes in one's imagination) will happen59. We willtake another path.Cosmology suggests that there are two main eects

of gravitation or space-time curvature, whatever thecause: the rst one is attractive and directed, the otheris (metrically) expansive and isotropic. In a symmetricsituation, the balance of forces nullies the rst eect.In an early stage of the universe, before the separationof relative states (or potential worlds), such a situationoccurs, leaving unbalanced the expansive component ofgravitation only. Cosmic ination is a measure of theuniverse's gravitational symmetry. Without our hypoth-esis, this symmetry is very soon destroyed and ination isoften supposed to last something like 10−33 s or 10−32 s.If our hypothesis corresponds to the experiment, the uni-verse perceived by gravitation is much more symmetri-cal and homogeneous than the same universe seen byelectromagnetic radiations: quantical uctuations pro-duce no gravitational asymmetry and so ination may

58 The horizon problem is that of the relative large scale homo-geneity and isotropy of the universe. The atness problem isthat of the almost critical density of a at space-time in GR.See, e.g. (Cohen-Tanoudji and Spiro, 2013). Observations, es-pecially from the Planck satellite, conrm that the space-timecurvature is under 0.005 (Planck Collaboration, 2016, p. 3738).

59 See, e.g., the beginning of (Nomura, 2011). Sure, there is noexperimental support for this and cannot be any, by denition.

17

last longer. In such a situation, ination would cease,or become negligible, when trans-world received gravita-tion (i.e. gravitation given by alternate worlds) becomesrelatively small, compared to in-world inhomogeneities.Quantitatively, we only know that this situation hap-

pens before the decoupling epoch (the CMB wall), atredshift z ≈ 1000. To investigate further, one has toevaluate the quantitative incidence of the two eects.First, the cosmological horizon, similarly to a black-holehorizon, causally separates potential worlds, diminishinggravitational inuence on each other, thus causing inho-mogeneities to be unbalanced. Secondly, worlds dier-entiate by an eect of decoherence, causing some sort ofdistance between worlds, which we will call modal prox-imity. Let us elaborate a little further on that matter inthe next section.

B. Modal proximity and mass discrepancy

1. The discrepancy between the visible and dynamical mass

As early as 1933, Fritz Zwicky noticed a large dis-crepancy between matter visible due to its electromag-netical emissions and dynamical mass according to ob-servable gravitational eects60. This is the origin of thegeneric term dark matter. [By 1980] Astronomers ingeneral thought in terms of rather conventionnal darkmatter cold gas, very low-mass stars, failed stars (or su-per planets), stellar remnants such as cold white dwarfs,neutron stars, or low-mass black holes. This conven-tionnal dark matter is generally called baryonic darkmatter, since most of its energy content is made ofbaryons. Part of this dark matter, mainly X-ray emit-ting gas in galaxy clusters, has now become visible; partis still invisible, but its proportion in the universe's en-ergy content is quite precisely estimated. The WMAPsatellite observations of the CMB uctuation spectrumare in excellent agreement with a at universe as it istoday (Ωtotal = 1.099 ± 0.1) and with the concordancemodel of the Universe: ∼ 5% baryonic matter, 25% colddark matter, 70% dark energy. Observations and theo-retical models of the early universe have set a constraintfor baryonic matter quantity: Ωb = 4% · · · 5%, so onlyone-tenth of the baryons are actually shining61. Theremainder of the mass-energy content of the Universe isthought to consist partly of dark matter that is uniden-tied, and primarily of dark energy of even more un-certain nature. The dark matter lls the Universe, pro-

60 In this section, quotations and historical facts about dark matterand MOND without explicit reference are from (Sanders, 2010).

61 Luminous part: Ωv ≈ 3h, X-ray emitting gas of galaxy clusters:Ωg ≈ 2.5h, so shining baryonic matter: Ωv + Ωg ≈ 5h. Othercomponents can also be estimated; CMB: ΩCMB = 5 × 10−5,neutrinos: Ων ≈ 3h · · · 10%. See (Sanders, 2010) for references.

motes structure formation and accounts for the discrep-ancy between the visible and dynamical mass of boundastronomical systems such as galaxies and clusters; it isthe major constituent of such systems. In 2009, andstill today, the candidate dark matter particles have notbeen detected independently of their presumed gravita-tional eects. Robert Sanders (2010) concludes thatthe existence of dark matter remains hypothetical andis dependent upon the assumed law of gravity or inertiaon astronomical scales. So it is not at all outrageous toconsider the possibility that our understanding of gravityis incomplete.

2. Dark matter and modal proximity

Now, consider some potential world W, branching inpotential worldsW0 andW1 after a qubal. Whatever theobserver's world, received gravitation is the sum of grav-itation of matter in these two worlds, each one countedwith its Born coecient. Next, let us imagine a sequel ofn qubals, leading to 2n potential worlds, Wi1i2···in . Letus suppose, by symmetry, that the observer's world isW00···0. The longer the initial 0-sequence, the closer isa potential world to the observer's. To simplify formula-tion, if qubals are symmetrical, each world has coecient2−n, but, as seen from observer's world, it's eectively2−(n−`), with ` the initial common history length. Let usdene the modal proximity of a world to be the productof Born coecients after branching from the observer'sworld. Modally close worlds, that is worlds with a re-cent shared history (i.e. a small `), are generally moresimilar and have a higher modal proximity. Modally dis-tant worlds have a lower modal proximity and may havea dierent distribution of matter.

Note that for general reasoning we have dened worldson a very wide basis, encompassing all matter, but, inpractice, we can use potential world model on subsys-tems of the universe, in as much as we can suppose themisolated enough. This allows us to dene a modal prox-imity for individual objects. With Newtonian approxi-mation, a body M in a potential world gives gravitationin proportion to its modal proximity and inverse squareddistance.

If our trans-world gravitation hypothesis is correct, wereceive gravitation from all worlds, from matter presentin our world and from matter existing in alternate poten-tial worlds. In a way, gravitation is the contribution ofour world and of some sort of gravitational shadow of al-ternate worlds. In other words, this gravitational shadowacts as if it were matter (i.e. energy content) interactingwith other matter by no other means than gravitation.Thus, it does contribute to dark matter, even if modalproximity is quickly decreasing with qubals.

Now, let us imagine the early uctuations that initi-ated the build-up of a galaxy. In modally close alternate

18

worlds, counterparts of this galaxy may have vastly dier-ent orientations and positions. In fact, if universe struc-tures are distant descendants of quantum uctuations,the alternate worlds should reect that any interactionmay correspond to multiple potential outcomes in multi-ple directions and that any subquantum trajectory linemay correspond to multiple moments of interaction. Atthe galaxy scale, one can imagine that gravitation givenby alternate worlds' counterparts produce something likea dark matter halo, interacting purely by means of grav-itation, and with no instability problem. This may ac-count for galactic stability and, perhaps, for the observedat terminal rotation velocity curves of spiral galaxies62.Being more speculative, this might explain why large-

scale structures look so much like potential interactiondiagrams: alternate world galaxies populate a galaxy la-ment, in the modal vicinity of an initial potential particlediagram.

3. MOND and cinematic interpretation of relativity

In 1983, Mordehai Milgrom published an alternativeproposal to non-baryonic dark matter: a modicationof Newton's second law of motion or of Newton'slaw of gravitation itself63. In this phenomenal theory,Newtonian gravitational acceleration, a = GM

R2 , becomesa2

a0= GM

R2 in the case of tiny accelerations: a a0. As oftoday this Modied Newtonian dynamics (MOND) hasno theoretical foundation, but it is greatly resilient toexperimental evidence. Furthermore, it explains muchmore phenomena than the cold dark matter model. Theterminal velocity of spiral galaxies is straightforward64:v4/R2a0 = GM/R2, so v = 4

√GMa0. Supposing a uni-

form mass-to-light ratio in spiral galaxies, one can derivethe TullyFisher relation (1977), now known to be avery good correlation between the luminosity and theterminal rotation velocity, L ∝ V 4, and settle an experi-mental value for reference acceleration a0 ≈ 10−10 m.s−2.MOND also explains why surface-brightness does notexceed Freeman's limit. Last but not least, MONDaccounts for the FaberJackson relation (1976) con-cerning all near-isothermal pressure-supported systems,M ∝ σ4, where σ is velocity dispersion, measured byspectral line width.

62 Velocity measures of gas galactic fringe in spiral galaxies showthat this velocity tends to be constant, and not to decrease ina Keplerian manner (the equilibrium between centripetal andgravitational accelerations gives v2/R = GM/R2, therefore v ∝1/√R where M(R) is constant). Cf. e.g. (Sanders, 2010).

63 The facts in this paragraph sum up Sanders (Sanders, 2010),chap. 10.

64 Moreover, It even appears that details in the rotation curvesare matched by the predicted MOND rotation curves (Sanders,2010, p. 141).

Being again very speculative, let us put forward anidea which could lead to some theoretical justication ofMOND. Classical PoundRebkaSnider experimentsconrmed that radial gravitational redshift is quantita-tively equivalent to a frontal DopplerFizeau eect.Now, let us turn to our cinematic interpretation of spe-cial relativity and represent any quantical particle as asuperposition of potential particles all displaced at thesame pace magnitude (corresponding to velocity c). Eachmomentum is something like ~p = h

c ν~n. If it under-goes the same gravitational frequency shift as photons,√

1−RSr√

1−RSr′

≈ 1+ RS

2r′ −RS

2r , one can imagine that centerward

potential particles are blueshifted and outward ones areredshifted. The eect would be in the order of δrr2 .

GM2c .

hc ν,

and would induce a gravitational strength in the orderof GM

r2 for higher level massive particles. One can alsoimagine a similar reasoning with (non-directional) red-shift due to the universe expansion, H0.d

c . If we considerradial potential particles, both eects could compensatefor H0.δr

c ≈ δrr2 .

GM2c2 , i.e. H0c ≈ GM

2r2 . Sure, not all poten-tial particles are radial: one would have to build a coher-ent theory out of these ideas. Heuristically, if this eect isconrmed, one can expect that something will happen tothe gravitation when gravitational strength goes down tothe order of magnitude of H0.c ≈ 7×10−10 m.s−2 thatis the MOND order of magnitude. We could foresee acombination of these two phenomena, gravitation beingdominant for high accelerations and becoming something

like√

GMr2 .H0c for small ones. This is still very specu-

lative, but might account for the MOND phenomenaltheory which we should probably call more accuratelythe Milgrom eect.

On the galactic and sub-galactic scales, MOND ex-plains well the observed mass discrepancy. On a largerscale, it reduces it from ∼ 6 · · · 7 to ∼ 2 · · · 3, but doesnot eliminate it (Sanders, 2010). So, even if the Milgromeect is conrmed, this does not rule out the need forsome sort of non-baryonic dark matter.

Finally, the observation of the Bullet galaxy cluster(1E 0657558, z = 0.296) showed that [a]ny nonstan-dard gravitational force that scales with baryonic massis insucient to account for the decoupling between thevisible mass (galaxies and X-ray emitting plasma) andthe map of gravitational strength (projected along theline-of-sight) by weak gravitational lensing methodology(Clowe et al., 2006)65. More precisely, this map sug-gests that unobserved matter, whatever it is, behaves

65 Douglas Clowe et al. (2006) also mention that other mergingclusters, MS 105403 (Jee et al., 2005) and A520 (in prepara-tion), exhibit similar osets between the peaks of the lensing andbaryonic mass, although based on lensing reconstructions withlower spatial resolution and less clear-cut cluster geometry.

19

like the stars and not like the hot diuse gas it is dis-sipationless (Sanders, 2010, p. 144). This is preciselywhat we could expect if this matter is the result of theshadow gravitation given by galaxies of alternate poten-tial worlds.

C. The cosmological constant and trans-world gravitation

1. The cosmological constant problem

For each quantum eld theory, even a space totally de-void of particles (its idealized zero point state) still con-tains virtual particles: this is called its vacuum. QEDvacuum, for example, is the current preferred interpreta-tion cause for the Casimir eect. The question is: dothese vacua contribute to weight? It is generally be-lieved they do, in the form of the cosmological constantΛ (eq. 4), but this belief does not come without trouble.In their historical survey, Svend Erik Rugh and HenrikZinkernagel (2002) distinguish at least three dier-ent meanings to the notion of a cosmological constantproblem: 1. A `physics' problem: QFT vacuum ↔ Λ[. . . ] 2. An `expected scale' problem for Λ [. . . ] 3. An`astronomical' problem of observing Λ. We will focushere on second point, named the vacuum catastropheby Ronald J. Adler et al. (1995), noticing that numer-ous papers have been written about it.Zero point energy density is, theoretically, evaluated

by counting QFT modes and applying a cuto to eludedivergence. This (strong) divergence is not per se anextraordinary diculty and it is physically somewhatunderstood; the vacuum catastrophe, named after thehistorical ultraviolet catastrophe, is the fact that evenwith a reasonable cuto, the zero point energy densityis evaluated to be way bigger than astronomical observa-tions and this fact is not at all understood. If we limitourselves to the electroweak theory and set a 100 GeVcuto, ρEWvac ∼ 1046 erg.cm−3 = 1045 J.m−3. This is al-ready a huge amount of vacuum energy attributed tothe QED ground state which exceeds the observationalbound on the total vacuum energy density in QFT by∼ 55 orders of magnitude (Rugh and Zinkernagel, 2002,p. 676)66. If we set the cuto at the Planck energy level,the same evaluation rises up to ∼ 120 orders of magni-tude. This is probably the worst theoretical predictionin the history of physics!67 Moreover, this is only forQED; accounts of other quantum eld theories are ex-tremely complicated: introducing the BEH mechanism

66 Gravitational observations (in the context of a model) constrainρ > 10−9 J/m3 see e.g. (Adler et al., 1995; Rugh and Zinker-nagel, 2002).

67 M. P. Hobson, G. P. Efstathiou, A. N. Lasenby, General Rel-ativity: An introduction for physicists, Cambridge UniversityPress, 2006, p. 187.

requires a massive choice, so to say, and the overall pro-cedure is very model dependent68. Finally, we shouldadd that a huge value for the cosmological constant iscoherent with the inationary model69.The theoretical connection of zero point energy den-

sity and gravitation is, for the time being, highly specu-lative as, on the one hand, energy in QM, as in classicalphysics, is dened only to a constant, and, on the otherhand, gravitation in GR is given by any form of energy,with an absolute value. In the future, any expected con-nection between QFTs and GR will have to deal withthis gap70. At present, we know no experimental resultlinking gravitation and any QFT, so it is only specula-tive whether or not we can assimilate energy in the senseof QFTs to energy in the sense of GR. We can broadenthe question, taking into account one of the best pre-dictions of physics: gravitation is so faint, compared toother forces that one can compute the magnetic mo-ment of the electron up to eleven exact digits withouttaking gravitation into account at all.

2. Trans-world analysis of vacuum

Can our framework shed some light on this situation?It can certainly raise some questions. Does QFT vacuaexert their eects in each potential world or are thesevacua trans-world phenomena? Is vacuum gravitationdiluted with time or is it essentially time-independent?If we consider given and received gravitation, it wouldcertainly be a surprise if QFT vacua would not give grav-itation, but what could be the meaning of receiving grav-itation for a fundamental state? Even more fundamen-tally, are the QFT vacua to be thought of as objects oras mere reication of relations between objects?Let us rst come back to the physical meaning of con-

cepts. Reection about motion started to be questionedrationally, in the Ancient times, by conceiving the emptyspace. Later, Galileo used this notion for his relativityprinciple, and Newton considered it a necessary basisfor establishing his laws. Some notion of free spaceis also fundamental for relativity: we supposed it, im-plicitly, when expressing facts 1 and 2, above. BlaisePascal, a philosopher and experimenter, made a cleardistinction between empty space (vide, emptied space,space devoid of matter) and nothingness (neant): noth-ingness has no quality, contrary to empty space. The

68 Even in the case of QED, far better understood than QCD, theprocedure is not sound: the energy density estimation dependson the cuto which is based on a belief and on the regu-larization of a delta-function by a volume the symmetries ofwhich have a major incidence on the computation.

69 The original idea of Alan Guth (1981) was that a huge vacuumenergy drove the ination.

70 In his historical survey (1989), Steven Weinberg introduces itas a veritable crisis.

20

interpretation of his experimental setups would be dier-ent today, but we can still keep the fundamental dier-ence between a space physically devoid of matter and anideal space-time with no content. Sticking to experimen-tal facts, we know now that intergalactic space contains∼ 10−12 m−3 molecules. Even the extreme empty spacebetween galactic clusters contains at least the cosmic mi-crowave background. Closer to us, extreme laboratoryempty space would certainly also contain the cosmic neu-trino background71. So, everywhere, there is matter or,to be more exact, a probability of the presence of matter.Vacua do not exist as physical states of a place.In our framework, any quantical particle is a swarm of

potential particles, the list of which is mostly the choiceof a representation. In a way, a vacuum contributes toall quantical particles. Vacua are often presented as uc-tuations in time of void space. This representation isnot actually coherent: the vacuum state corresponds tothe lowest energy eigenspace of the free Hamiltonian ofa QFT, so it is not much a uctuation in time (it doesnot change), but a variation between potential worlds.Thus, vacua should be best viewed as trans-world phe-nomena, and one can therefore expect that they dilutethemselves with the increasing modal proximity betweenall worlds. If this is the case, 120 orders of magnitudewould indeed not be that big, corresponding, roughly,to log210120 ≈ 400 symmetrical qubals. Since they arenot really objects, QFT vacua do not gravitate, strictosensu; the participating particles give and receive gravi-tation, but not the vacua themselves.

V. CONCLUSION

This paper has sketched out an assembly of puzzlepieces between Quantum Mechanics, Special Relativity,gravitation and cosmology, but a lot is still missing. Torecall only some:

The rst thing to do would be to experiment andtest our hypothesis of trans-world gravitation tobetter understand possible articulations of gravi-tation and quantical physics.

If our hypothesis is conrmed, our framework an-ticipates that gravitation received from inaccessibleworlds would manifest itself as shadow gravitationin our world and explain at least part of the ina-tion and dark matter phenomena. To go further,it is necessary to build cosmological scenarios andsee how much of these phenomena the shadow grav-itation could explain. On the contrary, would our

71 Estimated ∼ 5.10−5 m−3 at 1.9 K on a theoretical basis (Cohen-Tanoudji and Spiro, 2013, p. 412).

hypothesis be invalidated, one would have to ex-plain how gravitation could be conned to poten-tial worlds, i.e. respect Everett's relative stateseparation or, at least, behaves as if it did.

On the theoretical side, the cinematic justicationof Special Relativity should be explored further. Weinvestigated only a displacement pace factor on thewave function. An interval factor might be likewiseinvestigated.

Finally, one has to design hypotheses about the ge-ometry of the sub-space and its links with gravita-tion. A mechanism a la Perelman would be aninteresting scenario of emergence of the sub-space;it could also give some clues to sub-quantical grav-itation. This line of research line can be tracedback to Richard S. Hamilton's research which in-troduced the Ricci ow (1982) and drew a parallelwith the heat equation. This parallel may have amore profound physical meaning.

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