DEPARTMENT OF PHYSICS UNIVERSITY OF JYVASKYL¨ A¨ RESEARCH REPORT No. 1/2008 A METAGEOMETRIC ENQUIRY CONCERNING TIME, SPACE, AND QUANTUM PHYSICS BY DIEGO MESCHINI Academic Dissertation for the Degree of Doctor of Philosophy To be presented, by permission of the Faculty of Mathematics and Natural Sciences of the University of Jyv¨askyl¨a, for public examination in Auditorium FYS-1 of the University of Jyv¨askyl¨aon 9th February, 2008, at 12 o’clock noon arXiv:0804.3742v1 [gr-qc] 23 Apr 2008 Jyv¨askyl¨a, Finland February 2008 “Aristippus, the Socratic philosopher, shipwrecked on the coast of Rhodes and having noticed sketches of geometric figures, it is said may have spoken thus to his companions: ‘We can hope for the best, for I see the signs of men.’ (Vitruvius, De architectura, Book 6, Preface)” Frontispiece to (Euclid, 1703). Translation by H. J. Santecchia. Reproduced from (Heilbron, 2000). To Markku, who showed me the rare art of flying keeping one’s feet on the ground Do space and time exist alone and As part and parcel of the world, Or are they merely illusions Our minds create and set unfurled? The very products of our brain Like, say, the feeling of disdain? Three hundred years of bitter history Have failed this issue to resolve: Cartesian vortices and whirlpools, Around which corpuscles revolve; Contraptions built out of material Not once upgraded from ethereal. Today the ether has not left us; It lives in new, geometric modes: As loops and foams, as fancy networks, As strings and membranes, links and nodes; For are these objects, say, these strings, The seat of any real things? Are even spacetime points ethereal? An argument about a hole: So wide to charm most varied thinkers, But deep enough to lose your soul; Of which philosophers are fond, But which it’s time to step beyond. Forgo the hole, the points, the ether. Instead use things you can detect; The simplest things than Man can think of, With sources Man can clearly affect. Find answers past the older helm Within a non-geometric realm. A basic secret of the cosmos It is our goal to better grasp. A highly ambitious undertaking; Might even make the Reader gasp, Or utter apprehensive shrieks: “A task to daunt the Ancient Greeks!” —Or utter cries of disbelief And hope this thesis to be brief. A wish that we cannot concede: Of much soul-searching there is need. Preface The prologue is the most emotive moment, where the tension accumulated during a year of work is released like a gunshot. Miguel Indurain, Cycle Sport Magazine The goal of this thesis is to contribute to the elucidation of an elemental, age-old problem, namely, the physical nature of space and time. The words of my supervisor, Dr. Markku Lehto, to whom I owe countless hours of illuminating revelations, perhaps describe the spirit of this thesis better than no other: We participate in one of mankind’s most challenging intellectual efforts. The goal is to understand the biggest secrets of the cosmos: the structure of space and time as well as the connection of this structure with that of matter. One may even hold these secrets to be “greater than life,” when one thinks that life can be explained as an emergent phenomenon, i.e. a phenomenon which arises (probably very complicatedly) via evolution from the primitive structure of space and time and its interaction with matter. The first step is to explain whether space and time can have in general an own structure. Or are space and time in the end only human sensations like, for example, the feeling “happy”?1 In the study of this problem, like in that of all problems enduring obsti- nately through the ages, it soon dawned on me that the question to focus on was not the immediately expected, “How do we go about finding a solu- tion?” but rather the more careful and measured, “What is the problem?” Moreover, it eventually became apparent that these two questions were one and the same; that an understanding of the essence of a problem was quite apt to reveal ill-founded attempts at solving it, as well as point implicitly to the path of a methodical resolution. This realization, together with mounting weariness of hackneyed, indul- gent accounts of the nature and prospects of quantum gravity—the discipline which currently occupies itself with the issue of space and time—made an 1Private communication, 2001. vi Preface unsympathetic, bare-bones analysis of this field of research as indispens- able as it was enlightening. It allowed me to disentangle few facts from a surfeit of clich´ed beliefs (including even the occasional extravaganza) that populate the quantum-gravitational world—a referent-orphaned no-man’s- land of theoretical research. In particular, I learnt from this analysis that the present work can be considered part of quantum gravity only in a very shallow, conventional sense, since it does not share with this field its stan- dard motivations, methods of enquiry, or desired destination. If quantum gravity is to be reached by either one of the three geometric roads currently envisioned, this work treads none of them. After two and a half millennia of bitter and fruitless struggle—the last three and a half centuries of which within the context of modern science—the problem of the physical nature of space and time continues to be nothing short of an opinionated battlefield, a vicious display of parry and repost, a twisted labyrinth with no exit anywhere to be seen. We believe that unravelling this age-old problem requires a return to a physics based on ob- servations, together with the conceptual overthrow of the geometric notions in terms of which the philosophically and mathematically minded debate continues exclusively to rage, i.e. via a thorough departure from all forms of geometric thought and language, an idea we condense in the name “meta- geometry.” Focused on metageometry, however, we notice with disquiet the omi- nous fact that human beings carry with them an innate craving for geo- metric modes of thought and expression. This is not only revealed by an overview of physical theories, or even of scientific theories, but, without fear of overstatement, one may even say of all human endeavour. Human beings’ fascination for geometry easily grows to the point of enslavement; and some- times even to the epitomical extreme of having Geometry become a modern religion—physicists’ incarnation of God. While a solution to the ontological spacetime problem cannot be found through philosophically minded, labyrinthine discussions void of fresh phys- ical content, neither can it be found through the hit-and-miss invention of new spacetime structures off the top of one’s head; and especially not when these procedures are based solely, or mostly, on pure mathematical insight. In physics, mathematics is a good servant but a bad master. Because it is a physical question that we have in our hands, its clarification requires strong physical intuition, the identification of relevant physical principles, and a firm anchorage in observations and experiments as an unconditional starting point. It is in view of all this that this work deals with the question, “What do observations, quantum principles, and metageometric thought have to say about space and time?” It consists primarily in the development of new, ele- mental physical ideas and concepts and, therefore, does not need to abound in complicated technicalities. Obstinate problems require conceptually fresh starts. ✸ ✸ ✸ vii Literally speaking, this work would have been impossible without the wisdom and extraordinary guidance of my supervisor, Dr. Markku Lehto. Not only did he gently and unselfishly show me the way through the maze- like passageways of quantum-gravity research, unobtrusively fostering at all times the independent development and expression of my own thoughts, but also showed me the difference between physics, mathematics, and philoso- phy. In so doing, he taught me by example (the one truly effective teaching method) the meaning of genuine physical thinking—a surprisingly difficult activity rarely witnessed in frontier theoretical physics. To him go my infi- nite gratitude and admiration. From a financial point of view, the making of this thesis was assisted by the support of different organizations. I thank the Centre for International Mobility (CIMO) for sponsoring the beginning of my Ph.D. studies; and the Ellen and Artturi Nyyss¨onen Foundation, the Finnish Cultural Foundation, and the Department of Physics, University of Jyv¨askyl¨a, for allowing their continuation and completion. The research leading to this thesis was carried out during the period September 2001–July 2007 within the context of the last-mentioned Depart- ment, where I found ample freedom of action and better working conditions than I could have dreamt of. I particularly want to thank its administra- tive staff for its efficiency and helpfulness and my roommates, Ari Peltola, Johanna Piilonen, and Antti Tolvanen for pondering and assaying with me, sometimes lightheartedly, sometimes in all seriousness, the exactions of life in the brave new world of academia. Last but not least, the actual material creation of this thesis was made not only possible but, moreover, enjoyable by the typesetting system LATEX— an essential tool, almost on a par with the differential calculus, of the mod- ern man of science; a tool which deserves more attention in our typeset-it- yourself world of amateur desktop publishing than is usually paid to it in the sciences and humanities curricula. Man does not live by bread alone. I would also like to thank other people who have indirectly contributed to the fulfillment of this project; they are: my parents, Stella Maris and Omar, for bringing me up within the framework of an excellent family life and education—therein lie the seeds of all other achievements; Kaisuliina, for her companionship and warmth in the joys and commonplaces of everyday life alike; and Ulla and Teuvo, for unconditionally furnishing the context of a second family in a foreign, yet marvellous, land.