Is Dark Matter Made up of Xons?

Journal of Modern Physics, 2013, 4, 121-125 http://dx.doi.org/10.4236/jmp.2013.48A011 Published Online August 2013 (http://www.scirp.org/journal/jmp)

Jean-Paul Auffray ex Courant Institute of Mathematical Sciences, New York University, New York, USA Email: [email protected]

Received June 5, 2013; revised July 8, 2013; accepted August 1, 2013

Copyright © 2013 Jean-Paul Auffray. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

ABSTRACT Three months before his untimely death in Paris in July 1912, Henri Poincaré formulated the conjecture that Planck’s action element could (should) be regarded as constituting a “véritable atome”, i.e. an “atom of motion”, whose integrity arises from the fact that the “points” it contains are equivalent to one another from the standpoint of probability. In this paper we investigate the possibility that this conjecture provides a clue to the origin and nature of dark matter.

Keywords: Smolin; Poincaré; Action; Space; Vacuum; Scale Relativity; Nottale; Dark Matter; Xons

1. Introduction 2. Premises Some years back, in “The Trouble with Physics”, Lee 2.1. Poincaré’s (Little Known) Action Conjecture Smolin wrote: “Many quantum-gravity theorists believe Leibniz’s invention of his Dynamica in 1689, at the heart there is a deeper level of reality, where space does not of which he placed the abstract concept of action (actio), exist (this is taking background independence to its logi- provided an alternate framework to Newton’s System of cal extreme)”. In Lee Smolin’s conception “background the World in which Absolute Space and Absolute Time independence” refers to the proposal that “the laws of enter as primary or “God-given” Principles. The subse- nature can be specified completely without making any quent discovery of the Principle of Least Action, fol- prior assumption about the geometry of space” and to the lowed by the invention of the elementary quantum of requirement that “space and time emerge from the laws action by Max Planck in 1900, placed action at the very rather than providing an arena in which things happen”. heart of modern physics. To achieve this objective, he offered this advice (his ital- As a physical entity worthy of consideration per se, ics): “Don’t start with space, or anything moving in the elementary quantum of action was investigated by space.” He proposed, instead: “Start with something that Henri Poincaré shortly before his untimely death in July is purely quantum-mechanical and has, instead of space, 1912 when he proposed that it constitutes a “véritable some kind of purely quantum structure [1].” atome”—an “atom of motion”—whose integrity arises It is noteworthy that, in this statement, Lee Smolin re- from the fact that the “points” it contains are equivalent fers to space—and not to time (nor to spacetime)—as the to one another from the standpoint of probability, a ingredient one should dispose of, suggesting to replace it highly significant remark [3].. by something having a “purely quantum structure”. There is a reason for this distinction: quantum gravity specialists 2.2. Origin of the Poincaré Action Conjecture have argued successfully that “the best strategy for un- derstanding quantum gravity is to build a picture of the Poincaré’s action conjecture has its roots in a remark physical world where the notion of time plays no role at originally made by Max Planck concerning his discovery all”. In his recently published review article “Forget of a connection between the quantum of action and the time”, Carlo Rovelli has shown convincingly how this Liouville Theorem of classical physics reformulated in can be done [2]. the framework of Gibbs’ statistical mechanical method of We attempt in this Note to provide a plausible answer representations in phase space. Poincaré apparently first to the request Lee Smolin has formulated concerning heard of the connection when he met Planck at the Con- space. Our scheme is founded in a (little known) action seil de Physique (“First Solvay Congress”) gathered in conjecture originally formulated by Henri Poincaré that Brussels in October-November 1911. Transcripts of the we shall now describe. Conseil meetings show that Poincaré was inquisitive of

Copyright © 2013 SciRes. JMP 122 J.-P. AUFFRAY the true nature of Planck’s action element all through that number of possible distinct states; it jumps from one of week [4]. these states to another without passing through interme- The issue divided the Conseil into three “camps”: diate states [5].” those, a majority, who did not care one way or the other; Thus was born in 1912, under Poincaré’s pen, the con- those, a minority led by Arnold Sommerfeld, who concept of quantum jumps. sidered Planck’s action element to be the true fundamental entity one needed to take into account in the formula- 2.5. Transition tion of emerging quantum theories; and those, led by In what follows we examine the possibility that Planck’s Albert Einstein, who favored the “energy quantum” over action element, understood in the sense proposed by the action element. Poincaré, should indeed be regarded to be not simply a Obviously impressed by Planck’s connection of the “universal constant”—Planck’s constant” akin to other action element with the Liouville Theorem and by Som- “universal constants” such as π, e, c or k—but as the merfeld’s vigorous defense of the deep physical signifi- fundamental principle of quantum physics (we use here cance he attached to this connection, Poincaré formed his the word principle in the Aristotelian sense it originally conjecture shortly after the Conseil ended. To clarify the had: “that which comes first” [archē]). issue involved, we shall briefly outline how the connec- To implement this scheme, we shall engage in a line of tion with the Liouville Theorem arises. thought initiated by James Gray, Yang-Hui He, Vishnu Jejjala and Brent D. Nelson in their paper “Vacuum 2.3. Connection with the Liouville Theorem Geometry and the Search for New Physics” in which Consider two allowed states of a given (classical) physi- they explore the possibility of finding a “hidden struc- cal system. If one of the states is a necessary conse- ture” in vacuum space [6]. quence of the other, then the two states are equally prob- able. Let q represent a generalized coordinate of the sys- 3. The Concept of Space Revisited tem and p the corresponding conjugate momentum, then Ordinary space is thought (is said) to be composed of dpdq constitutes an infinitesimally small elementary do- dimensionless “points”—0-branes in M-theory—whose main of probability associated with the system. Planck’s (sole) characteristics is to “exist”. In Euclidian space, and quantum hypothesis consists in assuming that, rather than in relativistic spacetimes as well, these points are related being infinitesimally small, the elementary domains are to one another via notions of proximity and distance (in- all equal and given by the relation ddpq  h, where h terval). is Planck’s famous “second universal constant” (the first We postulate instead the presence in nature of a vac- is k, which has the dimension of entropy and intervenes uum space containing: to make kT, where T is the system temperature, an en- 1) An unstructured chaotic or “passive” substrate com- ergy). posed of dimensionless elements we shall call “i-points” pending further examination of their characteristics. i- 2.4. Poincaré Invents the Concept of Quantum points present in the substrate bear no geometrical rela- Jumps tionships with one another. Poincaré’s reasoning concerning the structure of Planck’s 2) A primitive or active principle that we shall call the finite domains is striking: “These domains are indivisible, xon and designate as h, since, as we shall show, it corre- he wrote. If we know the system to be in one of them, sponds to Planck’s 1900 invention of the action element. then everything is automatically determined. If it were In what follows, we shall call the interaction of xons not, if events that are to follow were not fully determined with i-points the “occurrence” of h in the chaotic sub- by that knowledge—in a word, if they were to differ de- strate and we shall say that the result of the occurrence is pending on the system being in such or such part of the the potential formation in the substrate of i-point clusters, domain—then, since the probability of some future designation which conveys no predetermined algebraic or geometric connotations. events would not be the same in its diverse parts, the domain considered would not be indivisible from the 4. The Crux of the Matter viewpoint of probability.” And he concluded: “This means that all the system states that correspond to a giv- If Nature’s underlying reality is indeed to be described in en domain are undistinguishable from one another and terms of a passive substrate which acquires geometric therefore constitute one and the same state.” A conclu- and/or dynamic properties through the occurrence of an sion that led him to assert unambiguously: “We are action element acting in its midst as a “first principle” or therefore led to formulate the following fundamental cause, then significant consequences must ensue. One in theorem: A physical system can exist only in a finite particular is that the symbol h, which represents the ac-

Copyright © 2013 SciRes. JMP J.-P. AUFFRAY 123 tion element, should no longer be considered to designate structures. We shall assume instead that they relate at simply a “universal constant of nature” akin to other best to fractal structures. universal constants such as π, e or c, but as represent- To explore this idea we shall borrow extensively from ing a physical entity possessing characteristics in its own Laurent Nottale’s Theory of Scale Relativity [8]. rights. Let D be the fractal dimension and let δ designate the To implement this requirement, we shall make use of resolution dimension defined as the underlying fractal two fundamental axioms concerning action proposed by dimension minus the topological (observable) dimension: Leibniz in his legendary 1688 treatise, Dynamica de Po- δ = D – DT. Nottale has called the resolution dimension δ tentia: thus defined the system “fifth dimension” or “djinn”. Axiom 1. “One can understand what action in a body The use of the word resolution in this context refers to is only in an indirect way”; the following. The relations lp = h and Ed = h allow any Axiom 2. “Motion’s formal actions are the results of value of l or d to occur. When particular values prevail, the composition of diffusions and intensions [7]”. we will say, following Nottale, that they specify the Implementing these two axioms, we seek representa- cluster state of resolution. This assigns to the cluster a tions of h in terms of the composition of a diffusion or new kind of coordinate which is neither space-like nor extension, l, and an intension or intensity, p, yielding the time-like but reflects instead a fundamental dependence fundamental equation of the cluster geometry on resolution. To specify more closely the significance of this con- lp h . cept, we shall reason concurrently within two frame- If we assume the “extension” l thus introduced to rep- works: 1) the framework that arises from our 2-principle resent a “length” of some sort, then this equation recov- underlying scheme; and 2) the framework of ordinary ers Louis de Broglie’s 1923 relation p = h/ which asso- four-dimensional spacetime physics in which distances ciates a wavelength  to an electron moving with mo- between events (world-points) are the relativistic inter- mentum p. In our scheme, however, the extension thus vals that connect them. To simplify the presentation introduced assumes an entirely different significance: it however, we shall use the word “length” rather than the represents a “geometric structure” of some kind induced word “interval” to designate distances between events in the i-points substrate by the active principle, h. (points) in spacetime. The De Broglie relation, which we wrote as lp = h, may also be written in the alternative form 5.1. Measuring Distances in Spacetime Ed h Distances can be measured in spacetime only if one dis- where E measures an energy and d designates a “time- poses of “yardsticks” to make the measurements. like” extension or “duration”. This recovers the photon re- Let the extension l associated with an h-induced i- lation E = hν. points cluster be l(ε) when the yardstick length is ε in Any physical theory must ultimately be related to what spacetime. In the framework of his Scale Relativity The- one might want to call “reality as classical physicists see ory (SR), Laurent Nottale has shown that l(ε) tends to it” which constitutes a particular representation or inter- infinity when ε tends to zero. The yardstick length ε is pretation arising from the underlying reality. For the only defined relatively to the length ε’ of another yard- purpose of the present demonstration, we take the view stick, however. In the simplest formulation of the theory, that “reality as classical physicists see it” depends pri- the resolution dependence of l assumes the simple form  marily on the assumed validity of the Euclidian geometry ll   . When δ = 1, which arises when the fractal axiom according to which there exist in nature straight dimension is 2 and the topological dimension is 1, the  line segments whose lengths can be measured by means resolution dependence becomes ll   . Inasmuch as  of fixed-lengths rods taken as units—a method, inciden- it corresponds to the supposition tt 1 which charac- terizes classical mechanics, the supposition δ = 1 may be tally, Albert Einstein used, and later rejected, in his first said to impart to this formulation a Galilean character. In 1905 paper on Special Relativity. a more sophisticated formulation of the theory a transition scale ε occurs such that for    the system is 5. From Chaos to Fractal Geometry 0 0 fundamentally resolution-dependent, while for   0 , On this basis, we seek to establish a connection between it becomes essentially resolution-independent. The tran- the assumed underlying chaotic “reality” in which action sition scale thus defined establishes a connection be- is assumed to be the active ingredient and ordinary tween quantum and classical behavior, a highly desirable spacetime physics. feature for a scheme that seeks to define a new kind of There is indeed no compelling reason to assume that background for M-Theory brane propagation and inter- h-induced extensions—l or d—relate to “Euclidian-type” actions. To help visualize what is involved, we shall out-

line a representation of the process even though it is not the case when ln  ln 0  . Nottale introduces here strictly applicable to the proposed scheme since it in- in his theory the postulate of the uniformity of scale volves geometrical concepts not explicitly present in the space in general and of the gauge function scheme. It is based on original work by Polish mathema-  ln ln  tician Waclaw Sierpinski. In constructing this illustrative 0 example we selected on purpose the group-theoretical in particular. When this is done, the gauge function is concept of “substitution” to designate the self-ordering found to be conjugate to the electron charge and the elec- process: being equivalent to one another, i-points can be tron charge is quantized, an interesting result. substituted to one another without disrupting the cluster There is more. A fifth representation of h arises from integrity the relation e2/2αc = h which connects h with the electron charge e, the fine structure constant α, a pure number, 5.2. A Visual Aid and the limit velocity c that occurs in Poincaré’s relativ- Let ABC designate an equilateral triangle supposedly istic electron dynamics. In this representation, the part that constitutes Leibniz’s diffusio factor is not easily representing an h-induced i-points cluster. “Distances” identified. In the relation e2/2αc = h, e can vary only if between any two i-points in this cluster are not defined. the fine structure constant α varies accordingly. Nevertheless, for the purpose of the demonstration, let us Other representations of the action element will ensue assume that for each i-point in the cluster distances are if one writes other known quantum mechanical relations defined with respect to the three arbitrarily selected in the form “something = h”. Each new representation i-points A, B and C which define the cluster “bounda- provides a particular insight into the complex and deep ries”. Let 1 designate an i-point chosen randomly within nature of the action element. the cluster. Consider the substitution that replaces the i-point 1 by the i-point 2 located at mid-distance between 7. Conclusions 1 and B. Let a new substitution replace the i-point 2 thus identified by the i-point 3 located at mid-distance be- In our fundamental relation lp = h, if l has the dimension tween i-point 2 and i-point C. Each subsequent substitu- of a length, then p has the dimension of a momentum, p = tion advances the original i-point midway in the direction mv, thus introducing in the scheme a (hidden) velocity v of A, B or C, the direction being chosen at random. and a (hidden) mass, m = p/v. Somewhat surprisingly, if pursued ad infinitum, this pro- At the 25th Solvay Congress held in Brussels in Octo- cess transforms the random distribution of i-points in the ber 2011, Rapporteur Juan Maldacena reviewed the pro- original cluster into a fractal distribution of i-points re- blems attached to current research in Quantum Space- sembling a Sierpinski triangle. This construction illus- Time physics. One of them relates of course to the need trates the fact that a primitive form of self-organization to identify the nature and origin of the mysterious dark can result from a (partially) random substitution process matter and dark energy which seem to pervade the known occurring within an undifferentiated substrate [9]. Universe [10]. Henri Poincaré formulated his action conjecture in the 6. Esoteric Representations of h wake of the First Solvay Congress held in the same city one hundred and two years ago. If, as he suggested, Max In Quantum Electrodynamics (QED), the electron action Planck’s action element h is indeed a veritable atome—a is written as xon in our proposed terminology—and if the scheme we Slp2 Ede  . have constructed in these pages to accommodate this conjecture has any kind of vraisemblance, then it would In this formula, besides lp and Ed, two additional rep- follow that vacuum space might indeed be the site of resentations of the action element occur: φσ and eχ, complex hidden activities ensuing from the presence in where φ measures an angle, σ the corresponding conju- its midst of xons acting on i-points to create clusters and gate angular momentum (an action), and where e repre- yield, in particular… some form of Dark Matter? sents the electron charge and χ designates a gauge function. 8. Acknowledgements In Scale Relativity, the mysterious gauge function  which occurs in this formula is identified with the loga- We wish to express our gratitude to Laurent Nottale and rithm of the resolution scale factor   ee . Recalling to Cédric Villani for their precious comments on certain that, in our notation, ε0 designates the transition scale aspects of this work and, retrospectively, to Louis de such that for   0 the system considered is funda- Broglie for his encouragements in the early phase of this mentally resolution-dependent, while for   0 , it be- research and to Professor Richard Feynman who inspired comes essentially resolution-independent, let us consider us to coin the word xon [11].

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