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) Is Dark Matter Made up of Xons? 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 con- cept of quantum jumps. sidered Planck’s action element to be the true fundamen- tal 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.