Contributors to this volume: Alessio Ageno Jean-Pierre Antoine Sandro Caparrini Frans A. Cerulus Sidney D. Drell David Ritz Finkelstein Laszlo Grenacs Donal Hurley Giuseppe La Rocca Siegmund Levarie Bernd Lindemann Giulio Maltese Wolfgang K.H. Panofsky Orietta Pedemonte Patricia Radelet-de Grave Luigi A. Radicati di Brozolo Michael Vandyck Piero Villaggio Kim Williams
Two Cultures Essays in Honour of David Speiser
Kim Williams Editor
Birkhäuser Verlag Basel • Boston • Berlin
Editor: Kim Williams Kim Williams Books Via Cavour, 8 10123 Torino Italy e-mail:
[email protected] 2000 Mathematics Subject Classification 00B30
A CIP catalogue record for this book is available from the Library of Congress, Washington D.C., USA Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at . ISBN 3-7643-7186-2 Birkhäuser Verlag, Basel – Boston – Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. © 2006 Birkhäuser Verlag, P.O. Box 133, CH-4010 Basel, Switzerland Part of Springer Science+Business Media Cover illustrations: Giovanni Bellini, Sacra Conversazione, Pinacoteca di Brera, Milan Root diagram of SU(3) and its lattice gc. Diagram by Jean-Pierre Antoine. Translation from French to English for this volume by Sylvie Duvernoy. Translation from Italian to English for this volume by Kim Williams. Printed on acid-free paper produced from chlorine-free pulp. TCF ∞ Printed in Germany ISBN-10: 3-7643-7186-2 e-ISBN: 3-7643-7540-X ISBN-13: 978-3-763-7186-9 987654321
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Contents Introduction Luigi A. RADICATI DI BROZOLO. Foreword..........................................................................1 Kim WILLIAMS. Reflections on Interdisciplinarianism..........................................................3
The Sciences Jean-Pierre ANTOINE. David Speiser’s Group Theory: From Stiefel’s Crystallographic Approach to Kac-Moody Algebras .....................................13 David Ritz FINKELSTEIN. Whither Quantum Theory? .......................................................25 Laszlo GRENACS. The Direct Determination of the Induced Pseudoscalar Current (and about the slow metamorphosis of an institution) ........................................................39 Giuseppe LA ROCCA and Luigi RADICATI DI BROZOLO. In Praise of Asymmetry ...............45 Donal HURLEY and Michael VANDYCK. An Observation about the Huygens Clock Problem.....................................................................................59
The History of Science Frans A. CERULUS. Daniel Bernoulli and Leonhard Euler on the Jetski ..............................73 Giulio MALTESE. On the Changing Fortune of the Newtonian Tradition in Mechanics ....97 Patricia RADELET-DE GRAVE. Studies of Magnetism in the Correspondence of Daniel Bernoulli ......................................................................115 Piero VILLAGGIO. On Enriques’s Foundations of Mechanics............................................133 Sandro CAPARRINI. On the Common Origin of Some of the Works on the Geometrical Interpretation of Complex Numbers .................................................139
The Arts Siegmund LEVARIE. Architecture and Music ....................................................................155 Bernd LINDEMANN. An Unusual Sacra Conversazione by Giovanni Bellini......................159 Alessio AGENO and Orietta PEDEMONTE. Ancient Astrological and Musical Analogies in the Renaissance: Palladio’s Villa Rotunda and a Geometric Construction by Leonardo167
Nuclear Arms Sidney D. DRELL. The Gravest Danger: Nuclear Weapons and their Proliferation ...........181 Wolfgang K.H. PANOFSKY. Nuclear Arms Control ..........................................................189
Reference Bibliography of Works by David Speiser ..........................................................................195
David Speiser, 2005
FOREWORD This book is dedicated with love, gratitude and admiration to Professor David Speiser by a group of his friends on the occasion of his eightieth birthday. Let me say a few words about our dear friend David. He grew up in the highly civilised atmosphere of Basel, a state unto a state, un des lieux saints de notre civilization, as Lucien Fevre wrote echoing, one hundred years later, Jakob Burckhardt who had written “small states exist so that there may be some spot on earth where the largest possible proportion of the members of the state are citizens in the fullest sense of the word.” Beside having been born in such a blessed place, David has had the additional privilege of growing up in the intellectual and cosmopolitan milieu of a family where artists, scientist, musicians were frequent visitors. Moreover David’s uncle, Andreas Speiser, was an eminent mathematician who besides an influential book, Die Theorie der Gruppen von endlicher Ordung, wrote a delightful essay, Die Matematishe Denkweise, which introduced his gifted nephew to the mysterious relations between mathematics and the arts, a leit-motiv in all David’s thinking. I said in the beginning that this Festshrift is a token of our admiration for the breadth of David’s interests, which range from science to history to the arts. It is enough to look at the impressive list of his publications: a paper on the hexagonal symmetry in a Mycenean jewel; several papers and seminar talks on the irreducible representations of the group SU(3) needed to accommodate all elementary particles known at that time (1960-61) and on the importance of this group in the weak interaction transitions; papers on general relativity and quaternionic quantum mechanics, etc., etc. Then comes the impressive list of his papers on the history of science. Personally I owe to David all the little I know on the history of physics, which I learned from his series of inspiring lectures at the Scuola Normale Superiore on Galileo, Newton, Huygens, Maupertuis, etc. His monumental work on the Bernoulli papers is his monumentum aere perennius. Though I admire bronze monuments, I confess my preference for silver and gold: of this stuff is made his book on Petrus Peregrinus; his articles on the symmetries of the Leaning Tower and the Battistero in Pisa have been to me a real joy. However, what I have enjoyed most has always been talking to David, often disagreeing with him but always impressed by his wide-ranging knowledge and the brilliance of his opinions. We’ve been talking for more than forty years and still do it, unfortunately now only on the phone. Thank you David for our friendship. I said, David, you’ve been lucky to have been born in Basel and to have grown up in the stimulating atmosphere of your family: however il meglio mi scordavo. You have found the most charming, accomplished and, if I may add, patient wife, and that, in my opinion, may not be the least of your accomplishments. May you and Ruth enjoy many, many years of fruitful life in your buon ritiro of Arlesheim and may the Lord bless you. Happy birthday, David Luigi A. Radicati di Brozolo
REFLECTIONS ON INTERDISCIPLINARIANISM Kim Williams
We live in a time when there is much discussion about “interdisciplinarianism”. This is generally hailed as meaning that the age of specialization is coming to a close, and that groups, or cultures, that have been limited to a self-contained dialogue are now free to share and exchange knowledge across disciplinary borders. It is perhaps no coincidence that this desire expresses itself at a time when an equally great amount of discussion is being given to globalism, the breaking down of cultural and political boundaries to favor exchange of knowledge, goods, and wealth. David Speiser and I first met through correspondence in 1995, after I had published an article in The Mathematical Intelligencer on the geometry and the proportions of the pavement layout of the Baptistery of Florence. At that time, those of us who worked on theories and applications of proportion, geometry, symmetry and similar mathematical concepts in architecture despaired of finding a venue for publication. For the architectural journals our research was too mathematical (the attitude of the architectural historians was chilling: “When I see numbers, I just turn the page,” said one to me). For the mathematical journals, it was not mathematical enough. David had been working on concepts of symmetry and symmetry breaks in the Baptistery of Pisa, and enthusiastically wrote to establish contact with me. From the beginning, our correspondence was concerned with issues of interactions between arts and sciences, of which the interactions, or nexus, between architecture and mathematics is but one particular case. We have also discussed at length the differences between the disciplines of the scientist and the historian of science, as well as those between the architect and the historian of architecture. Heinz Götze, in a letter to me written on 30 March 1995, wrote: “Professor Speiser is one the few immense scientists who try to liberate the various sciences from their often rather narrow-minded point of view.” In his own field, physics, we find a first instance of David overcoming opposition and fragmentation: At the moment we consider inorganic science as organized and structured by four, maybe five, interactions between the various fields and quantum systems, namely the ‘strong’, the ‘weak’, possibly the ‘superweak’, which interact between elementary particles and nuclei, the electromagnetic and the gravitational interaction….Few, I guess, doubt that a unification of all five interactions will be realized one day. But: what are the deeper obstacles? Which path will the process of unification follow? How long will it take? Even such questions cannot be answered today. In such a situation one may be permitted to formulate a conjecture. Until today the sciences distinguish sharply between space-time on one side, and its content, the fields and quantumsystems on the other. In my opinion this distinction, which is taken for granted also by most philosophers, will disappear one day. There will be theories where this distinction cannot any longer be made” [Speiser 2003b, 491].
But characteristically, David hasn’t limited his observations to his own field. I have come across instance after instance in which he clearly outlines differences in disciplines and methodologies. One such instance concerned the differences between the scientist and the historian of the sciences: “…science and history are two radically different endeavors of the human spirit.
4 Kim Williams The essence of science lies in its property of being systematic since science ultimately always wishes to grasp the laws of nature, which it strives to uncover and to formulate in the simplest and most transparent form. But human history, and thus also the history of science, is the complete opposite of this: it is totally unsystematic, always complex and never simple nor transparent. So, for writing the history of science two different, indeed totally opposite, endeavors must simultaneously be at work in the same man. Thus, from the same man two almost irreconcilable gifts are requested—gifts from his intellect as well as from his heart. This confrontation, one might say ‘clash’, of the endeavor to systematize and to extract the universally valid from the documents which the historian finds before him, with the aim to determine the conditions under which this, always unique, discovery was made, under very special circumstances and by one distinct individual different from all others, and then to interpret its significance for the development of science, is the character of the history of science. It is its very essence, even its unique prerogative and also its characteristic charm” [Speiser 2003a, 39-40].
In one of our early exchanges of letters, I posed to him a question about methodology regarding investigations concerning both architecture and mathematics. On 2 February 1995 I wrote: Noted American [architectural] historian James Ackerman writes me to say…that he thinks my argument about the Baptistery is an interesting one, but that my method is faulty because I use a twentieth-century mathematician’s system to analyze an eighth-century monument. He suggests that I research mathematical tracts previous to the eighth century to find a method of the epoch upon which to base my claims. Of course, historians are primarily interested in a narrative, that is to say, what came before and what came after, in a linear progression. They wouldn’t want to admit an argument that takes elements out of sequence, no matter what such a leap across the ages might reveal about the edifice itself. I was wondering if you have encountered this reasoning before, and what you own attitude is in this regard.
David replied on 12 February: Did I “encounter this sort of reasoning by historians before”? And how! Many historians do not understand that mathematical truths are valid everywhere and always. The presentation is conditioned by time, but you are not obliged to go back to the language of the period, that is not your aim. Quite the contrary, the reader ought to be grateful to receive a presentation which he can understand and which he can follow. It would be awful if you had used the mathematical language of the period, and for this reason too I used in my appendices modern language. By the way, is James Ackerman a historian of science or of the arts? In neither case does he understand the meaning and importance of the use of mathematics in the arts.… In my mind the only relevant question is here: “was the geometry involved in the “sacred cut” accessible to people of the period?”
David’s capacity for synthesis is made evident again in his understanding of the relationships between built architecture and the painted architecture that is used to adorn it: Incidentally, when one manages to see and appreciate the built and the painted architecture as one single building, one realizes that one criticism often made of Baroque art is unjust: Baroque decorations, which are often felt to be overloaded and even bombastic, will be appreciated according to their just value if one realizes that they belong to one building only, which is, however, about twice as high as the purely architectural structure! [Caparrini and Speiser 2004, 12].
Interdisciplinarianismʊor lack ofʊin architecture In order to understand the implications and benefits of interdisciplinarianism, it is helpful to understand how the various divides in disciplines arose in the first place, for the forces behind the establishment of divisions were equally as pressing at one time as are the forces for conjunction today. The touchstone for this discussion is C.P. Snow’s Two Cultures [1993],
Reflections on Interdisciplinarianism
5
an essay originally given as a lecture at Cambridge in 1959, addressed the divide between the distinct cultures of literary intellectuals and scientists with which Snow, a scientist by training and a writer by vocation, was confronted. But Snow’s perception of a division between literary and scientific cultures is less difficult to understand than the current state of affairs within the single discipline of architecture, which is that we have practitioners who, if they don’t absolutely reject it, treat with the slightest regard one of their own most powerful tools—mathematics—and have divorced themselves from and left to the engineers one of mathematics’ most significant applications—structural mechanics. Indeed, of those professions related to the building arts, there are at least four distinct disciplines which only with difficulty manage to communicate: the architect, the engineer, the architectural historian, the architectural theorist. A similar situation exists in the discipline of engineering. As Jacques Heyman pointed out, engineers are “unlikely to know much about the theory and history of engineering” [2003, 11], or even to care; this is a direct result of the scant attention given to history and theory in the engineering curriculum. Thanks to their curriculum, architects have a greater familiarity with both history and theory, but their dedication to originality may mean that they regard both with an detachment equal to the engineers’. Indeed, by rationalizing the theory of the Renaissance, architectural theorists after the second world war, notably Rudolf Wittkower, encouraged an anti-historicism that continued until the rise of post-modernism in the late 1970s and the 1980s made historical forms popular again, albeit largely as decoration.
Developments in architectural theory The need to define disciplines is ancient; where the liberal arts were defined in classical times, the definitions of the seven mechanical arts arose in Middle Ages.1 In essence, modern disciplines are self-contained, self-referential, systems, the nature of each reinforced by scholarly research and, ultimately, by education. The reason for the present dilemma in which modern architects interested in poesis and significance have rejected the sciences in general and mathematics in particular has its roots in the gradual evolution in architectural theory during which mathematics lost its capacity as a container of meaning and became a mere technical tool; the widening gap between architecture as art and architecture as science has ultimately meant that architects who were concerned with issues of form or meaning rejected mathematics because it had come to be seen as a mere technical tool, or worse, because mathematics represented what had robbed architecture of its expressive capacity in the first place, while architects who employ mathematical concepts are often derided as mere technicians. In the 1400s, the mechanical arts were not highly esteemed, and were certainly less prestigious than the liberal arts. Painting, sculpture, and even architecture, were considered to be mechanical arts, and only then as subcategories of armatura. When Leonardo insisted that painting should not be regarded as one of the mechanical arts but rather as a science, he was not only attempting to raise the status of his own discipline, but was also questioning definitions of and boundaries between disciplines. Leonardo’s aim in his Trattato della pittura was to prove that painting was mental activity in order to establish it as a science. His 1
According to Hugh of St. Victor (1096-1141) the seven mechanical arts are cloth-making, metalworking, navigation, agriculture, hunting, medicine, and the theatrical arts (lanificium, armatura, navigatio, agricultura, venatio, medicina, and theatrica); painting and sculpture were considered to be subcategories of armatura. In classical times, Vitruvius included architecture among the liberal arts; cf. De architectura I, 1, 3ff. So did Marcus Terentius Varro (116 BC-27? BC).
6 Kim Williams argument was that in order for the painter to be able to recreate works of nature, he had to thoroughly understand what he was painting [cf. Clarke 1993, 127-128]; in other words, Leonardo justified his depictions of swirling water because he had undertaken scientific studies of fluids. He equally regarded his activities as architect (virtual though they were) as scientific, likening the architect to a physician: It is necessary for doctors who are the guardians of the sick to understand what man is, what life is and what health is, and in what way a balance and harmony of those elements maintains it, and how similarly when they are out of harmony it is ruined and destroyed, and whoever has a good knowledge of the aforesaid characteristic will be better able to heal than he who is lacking in it… The very same is required by an ailing cathedral—that is, a doctor-architect who well understands what a building is, and from what rules correct building derives, and from where such devices are drawn and into what number of parts they are divided… [Kemp 2004, 69].
Leonardo’s use of the concept of proportionality to describe sickness and health as well as sound building places him firmly in the context of Renaissance Platonism, according to which the harmonious disposition of parts determined the ultimate success of a creation, whether than creation be a painting, or a cathedral, or indeed, the successful cure of a malady. The mathematical concept of proportion was at once both mechanical and metaphysical: on the one hand, it provided the architect with a means to establish a precise relationship between parts; on the other hand, it permitted him to endow his creations with a cosmic harmony. In particular, the study and application of proportions to the architectural orders—in contemporary with the rise of architectural theory—was of the greatest importance to the architects of the fifteenth and sixteenth century. How that study evolved from the Renaissance through the nineteenth centuries tells the story how the progressive rationalization of architectural theory and its divorce from experience led ultimately to the mistrust with which the contemporary architect regards mathematics. Renaissance studies into the character and proportions of the architectural orders had two sources: the descriptions in Books III and IV of Vitruvius’s De architectura, and direct studies of Roman ruins scattered throughout Italy. At first, these studies were mainly visual; as time went on, they became more and more exact. This paralleled the development of architectural theory: in the 1400s, architectural treatises were more literary than practical; in the 1500s, they began to provide the practical information the architect required to design and build. Andrea Palladio, for instance, began his architectural education by copying the drawings of the orders by Raphael and Serlio, but was soon driven to undertake direct study in situ: to “see with my own eyes and measure everything with my own hands” [Palladio 1997, 3]. His enthusiasm for precision grew as well: “Finding [the remains of ancient structures] worthier of study than I had first thought, I began to measure all their parts minutely and with the greatest care” [Palladio 1997, 5]. The rise of orthagonal projection as a faithful means of representation meant that surveys could be published and compared; this had a direct effect on the development of theories of the orders. A plethora of architectural treatises dealing with classical architecture, and in particular the orders, appeared: Serlio (1537), Barbaro (1556), Vignola (1562), Palladio (1570), Scamozzi (1615). The first architectural theoretician to raise serious questions about the implications of the contradictions between observations about the Roman ruins and the formulations of proportional systems based on those ruins, or about contradictions between various theories of the orders presented in the many treatises of the sixteenth century was Claude Perrault. Although Perrault was trained as a physician, and had a thorough understanding of science and philosophy, his Ordonnance des Cinque Espèces de Colonnes, published in 1683, set forth a theory of the orders that was relied only superficially on the traditional theories of
Reflections on Interdisciplinarianism
7
proportions. According to Perrault, the contradictions of earlier theories were proof that there were no certain rules and rejecting the relationship between architectural and musical proportions, gave himself permission to determine a rational system of proportions based on whole numbers that was easily committed to memory and applied in practice. Alberto PérezGómez has called Perrault’s theory “protopositivistic” [Pérez-Gómez 1983, 32]: “Architectural proportion lost in Perrault’s system its quality of absolute truth” [PérezGómez 1983, 31]. Other French architectural theorists of the same period distanced themselves from Perrault’s theories, most notably François Blondel, who published the first textbook for the Academy of Architecture, Cours d’Architecture and was its first professor. Blondel, who like Perrault was a member of the Royal Academy of Science, was a great supporter of both the metaphysical as well as the technical roles of mathematics in architecture. The difference in points of view between Perrault and Blondel on the role of proportion in architecture in general and in the orders in particular would be the subject of debate for the next one hundred years. For instance, Jacques-Germain Soufflot, architect of the church of Ste.-Geneviève discussed the differences between Perrault and Blondel in Mémoire sur les Proportions d’Architecture of 1739 in which he sided with Blondel, maintaining that proportions of architecture, like those of music, were indeed found in nature. The major upheaval of the theory of architecture based on a conceptual rather than a metaphysical mathematics came in the beginning years of the nineteenth century. The theorist that had the most influence was Jacques-Nicolas-Louis Durand. …the reduction of architecture to a rational theory began to gain ascendancy toward the middle of the seventeenth century, culminating in the theories of Jacques-Nicolas-Louis Durand and his critics. Durand’s functionalized theory is already a theory of architecture in the contemporary sense: replete with the modern architect’s obsessions, thoroughly specialized, and composed of laws of an exclusively prescriptive character that purposely avoid all reference to philosophy or cosmology. Theory thus reduced to a self-referential system whose elements must be combined through mathematical logic must pretend that its values, and therefore its meaning, are derived from the system itself. This formulation, however, constitutes its most radical limitation since any reference to the perceived world is considered subjective, lacking in real value” [Pérez-Gómez 1983, 4]
How far had theory come from Leonardo, whose belief in experience was unshakeable: To me it seems that those sciences are vain and full of error which are not born of experience, mother of all certainty, first-hand experience which in its origins, or means, or end has passed through one of the five senses [Pedretti 1995].
Leonardo’s frame of reference, rooted as it was in the observation and experience of the natural world, remained outside of itself; Durand’s was entirely self-referential. Our contemporary specialization and structure of architectural disciplines is based on the setup of theory following Durand’s model. Nineteenth-century positivism had a devastating effect on architecture, because it reduced architectural design to formal and stylistic matters, without regard for meaning and cultural content. Further weakening the power of the architect was the divorce of statics from design. The result of the devaluation of mathematics from a system of symbols and relationships that imparted meaning through symbolism to mere technical tools for calculation ultimately meant that architects who wished to deal with cultural content and significance despised mathematics as mere cold, unfeeling calculation. In the twentieth century, the connections between architecture and mathematics have been somewhat reinforced by the fairly recent perception of mathematics as a generator of
8 Kim Williams form. One has only to think, for instance, of the use of minimal surfaces by Frei Otto, or Buckminster Fuller’s geodesic domes. However, what is missing from such applications of mathematical concepts to architectural form is the exploitation of mathematics to impart meaning as well as form. The meaning inherent in the Renaissance use of proportions as a link between a natural, perceivable order and a classically designed building is entirely missing in current architecture. This is especially true of computer-generated designs. Computer software has provided the architect with the most powerful tool yet to generate, visualize, and realize form, and yet for the most part there is no meaning inherent in the form, except to communicate that we now have the power to generate such forms. We know a great deal about Greek culture and civilization from observing their architecture; two thousand years from now, what conclusions about our own culture and civilization will be drawn?
Attempts to bridge the gap Practitioners of any discipline whose ultimate frame of reference remains within itself are restricted to a dialogue with only those who are trained and formed within that frame of reference. This is our situation today, when in order to appreciate a work of architecture such as Gehry’s Guggenheim in Bilbao, the non-architect must first be told what the architect’s objectives are before he can appreciate the results on any level except the merely subjective (“I like the shape”). If architectural discourse is to go forward non-exclusively, it is particularly important to try to reunite the various disciplines within architecture and to reintegrate in a meaningful way disciplines such as history and mathematics into architectural education and practice. In order for this to happen, the educational curricula of architects as well as engineers will have to change. One educator who dedicated a great portion of this life to this problem was Edoardo Benvenuto, whose aim was to reintegrate the history of structural mechanics into the contemporary curriculum. Benvenuto expressed his own interest in history thus: Perhaps it isn’t so much a historic objective that sustains this interest as much as the awareness that a deeper knowledge and an attentive reconsideration of the past are the necessary conditions for a genuine advance in research.2
And further, The epistemological contrast between a formalized science, such as structural mechanics, and other, less rigid, sciences, appears in the abstract to constitute a contradiction that cannot be overcome, but if it is looked at in light of, respectively, the true development of scientifictechnical thought and of the constructive solutions over the course in the history of architecture, it can be seen instead as the traces of a dialectic relationship that can only be interpreted and experienced by mixing the two horizons of comprehension and the two cultures.3
2
Forse non è tanto un obiettivo storiografico ciò che sostiene questo interesse, quanto piuttosto la consapevolezza che un’approfondita conoscenza e un’attenta rimeditazione sul passato sono oggi condizione necessaria per un reale avanzamento della ricerca [Benvenuto 1988, 11; my translation]. 3 Il contrasto epistemologico fra una scienza formalizzata, come è la meccanica delle strutture, ed altre scienze dello statuto più duttile ha in astratto l’aspetto di una insanabile contraddizione, ma se è colto rispettivamente nel reale sviluppo del pensiero scientifico-tecnico e delle soluzioni costruttive sull’arco della storia dell’architettura diventa invece il segno di un rapporto dialettico, che può essere interpretato e vissuto solo mescolando i due orizzonti di comprensione e le due culture [Benvenuto 1981, xii; my translation].
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C.P. Snow wrote, “The number 2 is a very dangerous number: that is why the dialectic is a dangerous process. Attempts to divide anything into two out to be regarded with much suspicion” [Snow 1993, 9]. The discipline of architecture has been divided into two, and then into two, and then into two again until we have various disciplines so specialized that there is no longer much communication between them.The rare individuals with a mental aperture such as that of David Speiser work to transfer the frame of reference outside any single discipline, opening the way to greater understanding and dialogue.
Acknowledgments First of all, my heartfelt thanks go to David and Ruth Speiser for their very precious friendship and gifts of encouragement to a young struggler. I thank all of the contributors to this book for their patience with an editor who was really not prepared to take on such a project, but who did so with all good will.
Bibliography BENVENUTO, Edoardo. 1981. Le scienza delle costruzioni e il suo sviluppo storico. Florence: Sansoni. CAPARRINI, Sandro and David SPEISER. 2004. How Should We Study the Nexus of Architecture and Mathematics? Nexus Network Journal 6, 2 (Autumn 2004): 7-12. CLARKE, Kenneth. 1993. Leonardo da Vinci. London: Penguin Books. CORSANEGO, Alfredo. 1999. L’eredità culturale di Edoardo Benvenuto. Bailamme 25, 3 (December 1999): 145-153. HEYMAN, Jacques. 2003. Truesdell and the History of the Theory of Structures. Essays on the History of Mechanics in Memory of Clifford Ambrose Truesdella and Edoardo Benvenuto. Basel: Birkhäuser. KEMP, Martin. 2004. Leonardo. Oxford: Oxford University Press. PALLADIO, Andrea. The Four Books of Architecture. Robert Tavernor and Richard Schofield, trans. Cambridge, MA: MIT Press, 1997. PEDRETTI, Carlo. 1995. Leonardo da Vinci: Libro de pittura. Florence. PÉREZ-GÓMEZ, Alberto. 1983. Architecture and the Crisis of Modern Science. Cambridge, MA: MIT Press. SNOW, C.P. 1993. Two Cultures. Cambridge: Cambridge University Press. SPEISER, David. 2003a. Clifford A. Truesdell’s Contributions to the Euler and the Bernoulli Edition. Journal of Elasticity 70, 1-3: 39-53. SPEISER, David. 2003b. The Importance of Concepts for Science. Meccanica 38: 483-492.
The Sciences
DAVID SPEISER’S GROUP THEORY: FROM STIEFEL’S CRYSTALLOGRAPHIC APPROACH TO KAC-MOODY ALGEBRAS Jean-Pierre Antoine
1 Introduction : 1961–1962 David Speiser is born in group theory; as for the mythical Gaul Obelix, il est tombé dedans tout petit… . This is, of course, not surprising, with such illustrious close relatives as Hermann Weyl (through his wife Ruth) and his uncle, Andreas Speiser, both famous (among other things) for their achievements with groups! And indeed, one of the first manifestations of David’s taste and expertise for groups was his lectures at the Istanbul 1962 NATO Summer School “Group Theoretical Concepts and Methods in Elementary Particle Physics” [1]. Rumour has it that several future experts actually learned their basics about group theory from these lectures. This was also the starting point of a long-lasting collaboration I enjoyed with David in Louvain, as I shall explain below. However, before that, I have to recall the previous episode of David’s career, namely his stay at the Institute of Advanced Study, in Princeton, New Jersey. Besides getting acquainted with several celebrities, always abundant in that prestigious place, he was involved in the then hot topic of finding a sensible way for classifying the elementary particles. Two things were clear. On one hand, if one believes in a so-called global symmetry, a concept due to Lee and Yang, the basic particles have to be associated with a single unitary irreducible representation (UIR) of the basic group, yet to be found. Then there were eight baryons, that ought to be put together, p, n, Ȉ±, Ȉ0, ȁ, Ȅ-, Ȅ0. Thus one is led to ask the following question: Which groups have an irreducible representation of dimension 8? The group has to be compact, in order to have a UIR of finite dimension, it has to contain as a subgroup the product U(1) × SU(2), corresponding to the known conserved observables, hypercharge and isospin, and upon reduction to that subgroup, the eight baryons must recover their known quantum numbers. Since there were no additional conserved quantities, the group should have rank 2 (the rank is the dimension of any maximal abelian (Cartan) subgroup). Thus David started a systematic analysis of all simple or semisimple groups, in collaboration with Jan Tarski. The outcome was clear. Among all the possibilities, the simplest and most natural one was SU(3). The result was written in a Princeton preprint in 1961, but the paper was considerably delayed and thoroughly rewritten and was only finally published in 1963 [2]. In the meantime, Gell-Mann [3, 4] and Ne’eman [5], independently, had obtained essentially the same result and they published quickly. As a result, they got the honours, including a Nobel prize for Gell-Mann in 1969 (after the discovery of the ȍ©), and Speiser and Tarski’s systematic work was largely ignored. It is worthwhile to compare the two approaches. Gell-Mann’s reasoning in [3] was very simple. Guided by a previous model due to Sakata, he postulated that there were three elementary building blocks, with the quantum numbers of p, n, ȁ. In the limit where the global symmetry is exact, these three particles are undistinguishable, so that any unitary transformation among the three leaves the system untouched. This brings in U(3) as symmetry group. Factoring the U(1) which describes baryon number conservation (multiples of the 3×3 unit matrix), one is left with SU(3), and the basic particles are described by the
14
Jean-Pierre Antoine
elementary UIR of dimension 3, noted obviously as “3”, and their antiparticles by the conjugate representation 3 . In order to go further, Gell-Mann, following Cartan, systematically builds the larger representations by taking tensor products and decomposing them into irreducibles, such as 3 u 3 1 8 , etc. Actually, all this is done only at the level of Lie algebras, in terms of the 8 generators, called since then Ȝi, i = 1, . . . , 8. In the published paper [4], however, Gell-Mann goes much further, introducing the vector and axial vector baryonic currents with independent transformation properties under U(3) or SU(3). Thus he has the SU(3) × SU(3) current algebra, that governs the reactions between baryons, mesons and leptons, and also the classification of baryons into SU(3) supermultiplets, i.e., UIRs. All this constitutes the famous “eightfold way”, and it was an instant success. As for Ne’eman, he simply started from the 8-dimensional (adjoint) representation of the Lie algebra of SU(3) and used it in the form of a gauge transformation à la Yang-Mills, obtaining the correct assignment of the 8 baryons and the 8 mesons. By contrast, Speiser and Tarski do not specialize from the outset to a single group. Instead, they try to answer the basic question: What groups can be used as global symmetry groups? Here, a global symmetry group is a group such that all baryons belong to the same supermultiplet. They also consider the case of a partial global symmetry, where baryons are distributed among several supermultiplets. Then they search in a systematic way all possible groups, taking into account a number of constraints such as isospin and hypercharge assignments, or known decay processes. They cover both semisimple and nonsemisimple groups, and also connected and nonconnected groups, obtaining a complete classification in the former case. They also introduce the graphical representation of UIRs known as weight diagrams, a tool widely used by Wigner. For the convenience of the reader, they also provide a rather thorough mathematical appendix on Lie groups and their unitary representations. This is, of course, a solid piece of work, largely mathematical, but it lacks the brilliance of Gell-Mann’s style and physical thoroughness. Thus it is not so surprising that Speiser and Tarski’s paper did not obtain the recognition it deserved.
2 The Istanbul lectures and their aftermath Now let us go back to Istanbul and its consequences. At that time, in 1962, I had just finished my MSc (Licence) in Louvain under the supervision of Frans Cerulus, a long term friend and thesis mate of David. My work was centred on the Sakata model, a sort of ancestor of the quark model, which involved a heavy dose of SU(3) representations. My source of information was very limited, namely Boerner’s textbook and some brief excursion into H.Weyl’s masterly papers on group representations—all in German! Thus Cerulus had the idea of sending me continue my studies with someone more expert than himself on group theory. He contacted two colleagues, P.M. Matthews in London, who was not available during that summer, and Speiser, then a Privatdozent at the University of Geneva, who invited me. This was just after the Istanbul school, so David was full of ideas and we started to look for a geometrical reformulation of Weyl’s celebrated character formula, but within the global approach due to Hopf and Stiefel. I still remember him coming day after day from the crystallographic lab with lattice models, made of wooden balls and iron wire, a very concrete realization of the beautiful mathematical constructions! We’ll have more to say about this below. Very soon after my return to Belgium, David was invited to Louvain, at the initiative of Frans Cerulus, who had thus renewed his contact with him. Of course, he gave again his Istanbul lectures, and my first task as a young assistant was to translate these into French (a good way of learning the material in a lively context!). The next year, David was appointed a
David Speiser’s Group Theory
15
professor at the Université Catholique de Louvain, he moved to Belgium for some thirty years, and I started to work with him on a PhD thesis — which turned out not to be based on group theory as the main language, but rather on distributions (upon a suggestion from V.Bargmann, to whom David had introduced me in Zürich). The Istanbul lectures had three aspects. First they contain a lengthy introduction to Lie groups and their topological properties. Then there is a systematic overview of the so-called global method of Hopf for the study of compact Lie groups and its extension to representations due to Stiefel. Finally comes the application of these mathematical results to the global symmetries of strong interactions and hadron classification. Previous authors had used exclusively the infinitesimal approach, following Cartan and Weyl (see [6] for an up-todate treatment). In the latter language, a given (simple) Lie algebra is characterized by its roots. The root vectors are the nonzero eigenvectors of the characteristic equation of the group, which is constructed in terms of the infinitesimal generators. Take a simple compact group of order m and rank l. Then its Lie algebra possesses a (Cartan) basis hi , eD , i 1, , D r1,r2, ,r 1 m l such that the commutation relations take the form
^
2
>hi , h j @ >hi , eD @ with nDE z 0 iff J
`
0,
>eD , eD @
>eD , eE @
D i eD ,
D i hi
(2.1)
nDE eJ
D E is a root. Thus a root D is a vector in Rl, kD is a root iff k =
±1, and the set of all roots is invariant with respect to the reflection in the hyperplane perpendicular to the pair rD . All those reflections generate a finite group, called the Weyl group W. In the particular case of SU(3), with Lie algebra A2, in Cartan’s notation, we get six nonzero roots, the tips of which draw a regular hexagon (Fig. 1).
Figure 1: The root diagram of SU(3) and its lattice g
c
Next come the representations. A UIR of a compact group is completely determined, up to equivalence, by its character, i.e., the trace of the representation matrices. For classical simple groups, the character of the representation D is given by Weyl’s formula (more details are given in Section 3 below) F D
X D
'
.
(2.2)
16
Jean-Pierre Antoine
Here both X and ǻ are homogeneous sums of exponentials, alternating under the action of the Weyl group W. Then Ȥ(D) is expressed in terms of weights, simultaneous eigenvectors of the (representatives of the) commuting generators ^! i ` , thus also vectors in Rl. Altogether they build a set of points in Rl, each with a certain integer multiplicity, namely the tips of the weight vectors. This set, called the weight diagram of the representation, is also invariant under W, and the number of points it contains, counting multiplicities, equals the dimension of the representation. In a nutshell, the weight diagram is nothing but a geometric picture of the character. Thus enumerating the UIRs of the group reduces to drawing all possible weight diagrams. All this goes back at least to Wigner and has been exploited systematically, both for its own sake and in the context of hadronic physics, for instance in [7]. The theory discussed so far is usually formulated in the infinitesimal approach of Cartan and Weyl and requires rather tedious calculations for UIRs of larger dimensions. The major contribution of David Speiser in this context is to reformulate the whole theory using the global approach of Hopf and Stiefel and then to provide a very simple method, entirely geometrical, even graphical, for computing all weight diagrams of UIRs of all classical groups. The first part is covered in the Istanbul lectures, the second one is contained in three papers based on the latter [8, 9]. Let us first have a glimpse of the Hopf–Stiefel global method (see [1, 8] for the references), without proofs, of course. We discuss the general case, but all concrete exemples and all figures refer to SU(3), since this is the most important case for hadronic physics. Given a semisimple compact Lie group G of order n, we consider abelian connected compact subgroups of G, called toroids and, in particular, maximal toroids. Any two of these are conjugated and they are all direct products of l factors SO(2), where the invariant l is called the rank of G (the maximal toroids are the subgroups generated by the Cartan subalgebras of the Lie algebra of G ). Then, according to Hopf, every element of G belongs to (at least) one maximal toroid. An element of G which belongs to only one maximal toroid is called regular, and singular otherwise. Because G is a Lie group, thus a manifold of dimension n, a neighbourhood V (e) of the identity of G may be mapped homeomorphically ~ onto a neighborhood V of the origin 0 of a Euclidean space Rn. Then, given a maximal ~ ~ toroid T, the intersection VT = V (e) ŀ T is mapped onto an open set VT V of dimension ~
l. Next we can continue VT to the image of T, then to the image of the universal covering group of T, which is isomorphic to Rl. Thus every element of T is represented by an infinite point lattice in Rl, in particular, the identity e is represented by a lattice ge, with origin 0. The image of the singular elements of T consists of m families of (l©1)-dimensional hyperplanes, where m 12 n l . The set of all these hyperplanes constitutes the Cartan– Stiefel diagram ī. The essential property of ī is that it is invariant under a reflection in any of the hyperplanes of which it is composed. The points of maximal intersection, i.e., belonging to one hyperplane of every family, represent the center of G. Since G is semisimple, its center is a discrete subgroup, the image of which is a lattice gc that contains ge as a sublattice. The finite discrete group generated by the reflections in the hyperplanes passing through the origin 0 of gc is the group W of Weyl. W is the crystal class of gc and it divides Rl into #W fundamental domains Di (mathematicians call them Weyl chambers); these are pyramids with apex at 0 and l edges, thus simplexes in Rl. Finally, the root vectors are the 2m vectors orthogonal to the m hyperplanes passing through 0 and twice as long as
David Speiser’s Group Theory
17
the distance to the next parallel hyperplane. Fig. 1 shows the particularization of all this to SU(3), for which n = 8, l = 2 and m = 3. The dotted lines represent the 3 singular hyperplanes through 0, which divide R2 into 6 fundamental domains Di. There are six roots ^r D i , i 1, 2, 3` . The Weyl group W generated by the reflection in the 3 singular hyperplanes is the symmetric group S3, of order 6, and it simply permutes the fundamental domains Di among themselves. Next one can introduce coordinates in Rl that induce a lexicographic order among the points (vectors) of gc. In the case of SU(3), the vertical axis separates R2 into a positive region on the right and a negative region on the left. In particular, the three roots ^D i , i 1, 2, 3` are positive, their opposites ^E i D i ` are negative. Among the positive roots, l (the outermost ones) cannot be written as linear combinations of other ones; they are called elementary or fundamental roots and they will play an essential role in the construction of weight diagrams. For SU(3), we see on Fig. 1 that the elementary roots are D1 and D 2 . We denote by U
1 2
m
¦i 1D1
half the sum of all positive roots and call D0 the
fundamental domain containing ȡ. Next we define an affine coordinate system with basic vectors v
1 2
Ȝ1,
…
,
Ol g c
¦i p1Oi { p1 ,, pl
such
that
every
vector
v g c has
the
form
with pi integer, and D0 is defined by pi t 0, i , while the j-th
face of D0 corresponds to pj = 0, j = 1, … , l. Then Weyl has shown that
1 m ¦ D1 2i 1
U For SU(3), Fig. 1 shows that U
D3
O1 Ol .
O1 O2
(2.3)
1,1 .
3 Weight diagrams, Speiser’s “Rubber Stamp Rule”, and all that Now we proceed to the construction of weight diagrams. According to Weyl, each UIR D is uniquely characterized by a lattice point Ȝ strictly inside D0, and any such point Ȝ defines a unique UIR D(Ȝ). The corresponding character may be written in the form (2.2), namely Ȥ(Ȝ) = X(Ȝ)/ǻ, where X Ȝ
¦ H w e i wȜ,M , O g c D0 .
(3.1)
wW
In this expression, (·,·) is the inner product in Rl, the Π k are group parameters in the toroid T and İ(w) = ±1 is the parity of the element w W . (For simplicity, we use the following shorthand [10], called a formal exponential ; to be mathematically precise [6], these objects are elements of the group ring over gc, but we will keep the simple physicist’s language): e i N ,M { e N , N g
c
(the expression e(ț) is denoted by e ț in [11] and [ț] in [8]). Notice that e (ț) e (țƍ) = e (ț + țƍ) and e(ț)©1= e(©ț), that is, we have a sort of logarithmic calculus that represents the vector addition in gc.
18
Jean-Pierre Antoine
Under the action of W, the expression (3.1) is an alternating sum of exponentials, called the characteristic of the representation (called a “girdle” in [7]). As for ǻ, it is given by Weyl’s formula below, but it is also the characteristic of the identity representation, i.e., ǻ = X(ȡ), since Ȥ(ȡ) = 1. The equality between these two forms is known as the denominator identity: m
ǻ
Di
m
Di
( e 2 ) e( 2 ) i 1
e U 1 e E i i 1
(3.2)
¦ H w e wU .
wW
Here the products extend over all positive roots ^D i ` , resp. all negative roots ^E i
D i ` .
In order to obtain explicitly the character Ȥ(Ȝ), one has to perform the division of X(Ȝ) by ǻ, a tedious operation. The result is again a sum of exponentials (called a formal character in the mathematical literature [6]):
F O
¦ J k e i N ,M { ¦ J N e N ,
(3.3)
N
where ț are lattice points representing weights and Ȗț is the multiplicity of ț. By construction, Ȥ(Ȝ) is invariant under W: this is the weight diagram to be computed. Among the weights, there is a highest one ȁ, defined by Ȝ = ȁ + ȡ, which has always multiplicity 1. Thus every lattice point ȁ Щ D0 (boundaries included) is the highest weight of a unique UIR, and the corresponding characteristic is X(ȁ + ȡ). For the case of SU(3), every lattice point Ȝ = (p1, p2) strictly inside D0 defines a UIR, of dimension d(p1, p 2 ) 12 p1 p 2 p1 p 2 . Fig. 2 shows the set of all UIRs of SU(3) of low dimension, each of them being represented by its dimension d(p1, p2) next to the corresponding point (p1, p2). In addition, these representations fall into three classes, which form a group isomorphic to the cyclic group Z3, but we will not pursue this point.
Figure 2: The low-dimensional UIRs of SU(3); each UIR is indicated by its dimension
The novel idea in [8] is twofold. First, instead of dividing X(Ȝ) by ǻ, we multiply X(Ȝ) by 1/ǻ, and then the whole operation is performed in a purely graphical way. The first step is to compute 1/ǻ. First we consider the roots as independent and work in R . From (3.2) we get m
David Speiser’s Group Theory
1 ǻ
m § f m § f · · e U ¨ ¦ e E i ki ¸ e U ¨ ¦ e k i E i ¸ ¨ ¸ ¨ ¸ i i 1 © ki 0 i 1 © ki 0 ¹ ¹ f f § m f f § m · · e U ¦ ¦ ¨ e k i E i ¸ e U ¦ ¦ e¨ ¦ k i E i ¸. ¨ ¸ ¨ ¸ k1 0 k m 0 © i 1 k1 0 k m 0 © i 1 ¹ ¹ m
1 E i 1 e 1
e U
The last term
19
m
¦i 1 k i E i
(3.4) (3.5)
represents an arbitrary point in the lattice generated by the
negative roots ȕ1, … , ȕm. Thus the whole expression represents all points of one of the 2m “octants” of Rm, each with multiplicity 1, and 1/ǻ is the same figure shifted by ©ȡ. This expression may still be simplified by introducing the operation “summation along a vector”, as follows. Given a point ț in a lattice with basis ȣ1, …, ȣm and a figure F(ț) attached to ț, we call “sum of F(ț) along ȣț” the figure obtained by superposition of all figures congruent to F modulo ȣț: f
¦ F N ¦ F mXN N XN
m 0
In this language, we may write 1 ǻ
¦¦ e U E1
(3.6)
Em
Next we take into account the linear relations among the negative roots, which are all linear combinations of the elementary ones. This may be considered as a projection from Rm on Rl, an operation that conserves coincidence properties and thus has for only effect to increase multiplicities. If, in (3.6), we sum only along the elementary roots, we get a simplex (pyramid with l faces) in gcңRl, all points with multiplicity 1. Then each additional summation will add multiplicities on the interior points. This construction is exemplified for SU(3) in Fig. 3, for which we have 1 ǻ
¦E §¨© ¦E ¦E e U ·¸¹ . 3
1
2
c
The expression in parentheses is a sector in g whose points have all multiplicity 1, and the sum along ȕ3 = ©ȡ yields the superposition of infinitely many copies of the same, shifted to the left by íȡ,í2ȡ,í3ȡ, … . Having thus obtained the expression (3.6) for 1/ǻ, it is easy to compute the weight diagrams of all UIRs. Given a characteristic F O ¦wW H w e wO , the same reasoning as above yields the corresponding character as F O X O
1 ǻ
¦ ¦ X O e U , E1
Em
where the expression in parentheses is the characteristic X(Ȝ) shifted by ©ȡ.
(3.7)
20
Jean-Pierre Antoine
Figure 3: The 1/ǻ of SU(3)
Taking now into account the fact that the corners of X(Ȝ) have alternating multiplicities 1, it is then easy to show [8] that only a finite number of terms of 1/ǻ contribute to Ȥ(Ȝ), the other ones cancel out. The result is a W-invariant polyhedron, whose corners constitute the orbit under W of the dominant weight ȁ = Ȝ í ȡ. Consider the case of SU(3) for simplicity. For any lattice point (m, n), we denote by F(m, n) the diagram consisting of (m, n), its five equivalents under W and all points of the same sublattice located inside or on this hexagon, all with multiplicity 1. Given a characteristic X(p1, p2)( p1 p2), the structure of the corresponding weight diagram is simply (+ means again superposition) F p1 , p 2 F p1 1, p 2 1 F p1 2, p 2 2 F p1 p 2 ,0 ,
(3.7)
that is, a sequence of nested hexagonal shells of uniform multiplicity increasing steadily from 1 on the outermost one. This is, of course, a direct consequence of the structure of the 1/ǻ described in Fig. 3. Fig. 4 shows the weight diagram Ȥ(3, 2) of SU(3) corresponding to the representation with highest weight ȁ = (2, 1), of dimension 15. Finally, this graphical method also yields immediately the decomposition into irreducibles of direct products of representations. Given two UIRs D(Ȝ1) and D(Ȝ2), their direct product decomposes into UIRs D(Ȝ j), where each of these may occur several times: D O1
D O2
D O j . j
(3.8)
David Speiser’s Group Theory
21
Figure 4: The weight diagram of the representation 15 of SU(3), with highest weight (2,1). The basis vectors are Ȝ1 = (1, 0) and Ȝ2 = (0, 1)
Correspondingly, we have for the characters (where the rhs means superposition of the weight diagrams): F O1 F O 2
¦ F O j . j
Dividing the two sides by ǻ, we obtain F O1 X O 2
¦ X O j j
or, equivalently, F O1 O 2
¦O j .
(3.9)
j
Translated graphically, (3.9) yields the so-called Speiser’s rubber stamp rule : “stamp” the weight diagram Ȥ(Ȝ1) on the lattice point Ȝ2; every lattice point Ȝj thus obtained yields one summand in (3.8), with the multiplicity of Ȝj times the sign İ(w) of the fundamental domain wD0 to which it belongs, the points on the boundaries (singular hyperplanes) having multiplicity 0. Fig.5 shows the decomposition of 8
3 in SU(3):
8
3 15 6 2 u 3 ҷ3 15 6 3 .
There are indeed two elements 3 corresponding to the multiplicity 2 of the weight (0,0) of 8, plus another 3, with multiplicity ©1, in the domain adjacent to D0.
22
Jean-Pierre Antoine
Figure 5: Decomposition of the direct product of SU(3) UIRs 8 × 3
4 From classical groups to Kac-Moody algebras The beauty of the graphical method described in the last section is that it is perfectly general. It applies to any classical group and allows to compute—or rather draw—the weight diagram of any UIR. To give an example, when studying UIRs of SU(4), I could without effort compute explicitly the weight diagrams of all UIRs of dimension up to 1000 (these are three-dimensional polyhedra, since SU(4) has rank 3 [12]). And, as an additional bonus, the formulas are very simple and the whole method is iterative, hence rather easy to program on a computer. But there is more! Namely, writing the geometrical procedure in algebraic form, as we did above in (3.2) or (3.3), one obtains formulas that extend verbatim to the whole class of infinite dimensional Lie algebras called Kac-Moody algebras. A quick look at the textbooks by Kac [10] or Wakimoto [11] will convince the reader of this fact, of course totally unexpected in 1964! This is not the place for an extended discussion on Kac-Moody algebras. Suffice it to say that most of the concepts of finite dimensional Lie algebras extend to this case: root systems, Weyl group generated by reflections, fundamental domains or Weyl chambers, etc. The novel fact is that now everything becomes infinite. There are infinitely many roots, but the root diagram is still invariant under the Weyl group W, now an infinite-dimensional discrete group generated by reflections. These roots are of two types, the so-called real roots, as in the finite dimensional case, and the imaginary roots. Real roots have multiplicity 1, but not necessarily the other ones. Hence the Weyl denominator 1/ǻ takes the following form: 1 ǻ
e U
1
1 e D mult D mult D e U 1 e D e 2D . D !0 D !0
This formula is identical to (3.4) and could be given the same geometrical interpretation. However, since we are now in a space of infinite dimension, algebraic formulas are more practical. In the finite dimensional case, the formulas described in the preceding section were, of course, known to mathematicians for a long time (see, for instance, the standard textbook of Humphreys [6]). The point we want to emphasize here is that these formulas,
David Speiser’s Group Theory
23
and the corresponding graphical method developed in [8], remain perfectly valid for the infinite dimensional Kac-Moody algebras. This came as a surprise when, years later, I was systematically reading Kac’s volume [10]. A good example of the efficiency of David Speiser’s approach, based on what he calls experimental mathematics !
5 Outcome What came out of the graphical method? The original papers [8] were quite successful, they are even quoted in Humphreys’ textbook [6], but the generality and ease of application of the method were not perceived by physicists. However David’s real motivation was particle physics. For some time, he pursued the problem of particle classification, together with several collaborators, such as Bob Oakes from Stanford [12], Anne and Pierre de Baenst [13], G. Iommi and myself. He tried SU(4), SU(8), the reduction from SU(3) to SO(3), etc. But then his interest shifted to Classical Mechanics and General Relativity. Of course, he always remained, and still is, fascinated by symmetries. But he chased them in different contexts, such as Mycenian jewelry or the Battistero in Pisa. Thus we are back in the other culture!
References [1] D. Speiser, Theory of Compact Lie Groups and some Applications to Elementary Particle Physics, Group Theoretical Concepts and Methods in Elementary Particle Physics, NATO Summer School Istanbul 1962, pp. 201–276; F. Gürsey, ed. New York: Gordon and Breach, 1964. [2] D.R. Speiser and J. Tarski, Possible schemes for global symmetry, J. Math. Phys. 4 (1963): 588– 612. [3] M. Gell-Mann, The Eightfold Way: A theory of strong interaction symmetry, Caltech report CTSL-20 (1961). [4] M. Gell-Mann, Symmetries of baryons and mesons, Phys. Rev. 125 (1962):1067-1084. [5] Y. Ne’eman, Derivation of strong interactions from a gauge invariance, Nucl. Phys., 26 (1961): 222–229. [6] J.E. Humphreys, Introduction to Lie Algebras and Representation Theory. New York–Heidelberg– Berlin: Springer, 1972. [7] R.E. Behrends, J. Dreitlein, C. Fronsdal, and W. Lee, Simple groups and strong interaction symmetries, Rev. Mod. Phys., 34 (1962): 1–40. [8] J-P. Antoine and D. Speiser, Characters of irreducible representations of the simple groups. I. General theory. II. Application to classical groups, J. Math. Phys. 5 (1964): 1226–1234; 1560– 1572. [9] D. Speiser, Fundamental representations of Lie groups, Helv. Phys. Acta 38 (1965): 73–97. [10] V. Kac, Infinite Dimensional Lie Algebras—An Introduction. Boston–Basel–Stuttgart: Birkhäuser, 1983. [11] M. Wakimoto, Infinite-Dimensional Lie Algebras. Providence, RI: Amer. Math. Soc., 2001. [12] J-P. Antoine, D. Speiser, and R.J. Oakes, SU(4) mass formula and particle classification schemes, Phys. Rev. 141 (1966): 1542-1553. [13] A. de Baenst–Vandenbroucke, P. de Baenst, and D. Speiser, Induction procedure for the reduction SUn ĺ SOn, Proc. Roy. Irish Acad. 73 A (1973): 131-150.
WHITHER QUANTUM THEORY? David Ritz Finkelstein
1 Goal The work described in this paper grows out of the same project as earlier papers with David Speiser, to whom this report is affectionately dedicated. The long-range goal was and remains a finite quantum-logical theory of space-time and the forces of nature. Quantum theory uniquely reconciles discreteness with continuity: namely, discreteness of the values of the variables of a physical theory with continuity of the symmetry group of the theory, as in the theory of spin. If quantization can be extended to space-time, the resulting theory can be finite and yet have continuous invariance groups in accord with experiment. Hartland Snyder [29] first quantized space-time, apparently inspired by ideas of Eugene Wigner. Roger Penrose formulated a spin-net approach to quantum space-time around 1960. Attempts to quantize space-time continue to the present day [21, 7, 5, 18, 26]. The main unsolved problem obstructing progress is how to formulate a quantum dynamics for a field over a quantum space-time—briefly, a q/q field theory—corresponding to a given quantum field theory over classical space-time (q/c field theory). Two significant changes in my own strategy during this project have made the problem seem solvable.
2 The process of physics First, let us renounce the search for the Final Theory. Our task is not to create the universe but merely to extend our understanding of it, in some directions of our own choosing among the many directions that are possible at any time. The discoveries in the last century of special relativity, general relativity, quantum theory, and the standard model seem like unique earthquakes in the world of physical ideas, but one should expect such iconoclasms to continue indefinitely, for familiar heuristic reasons like the following two: 1. Any working physical theory bases itself on absolutes that come from outside the theory; for example, general relativity bases itself on but does not account for the existence of space-time and matter, and quantum electrodynamics takes the electron as given. In any sharp (that is, maximally informative) experiment, we study under high resolution only a miniscule part of the cosmos against the unresolved background of most of the cosmos, including ourselves and our instruments. The absolutes assumed in any theory are idealizations emerging from this background. They are subject to revision when we shift attention from foreground to part of the background, when these constructs are studied under higher resolution. 2. Furthermore, we can only know these alleged absolutes—like anything else—from experimental experience, that is through an interaction that significantly changes us or our instruments. But what changes us must itself change, in reaction. The apparent absolute therefore loses its absoluteness under closer study. The Final Theory is a latter-day equivalent of the Philosopher’s Stone and the Holy Grail. There are theological arguments for it but no empirical evidence. When Einstein, for example, joked about the Old One not playing dice with the universe, he may have been expressing a conviction that a Final Theory existed, but he was ruefully acknowledging at the same time that this conviction had a quasi-theological nature and no empirical support.
26
David Ritz Finkelstein
We suppose that we will always be in much the present situation, with a theory or two in our hands that work well, and many better theories waiting under the cabbages. Physics is the process, it seems, not one of its products. Therefore the process deserves more professional study than it has received. To be sure, much of the physics process is unpredictable within physics because it depends on what phenomenon nature throws at us next, and which people decide to study next; “Physics is a social science,” Victor Weisskopf said. One part of the process seems to follow a regular law, however, which we sketch next.
3 The stabilization of physics Since 1998 I have given priority to small changes in the present physics that stabilize it, in the following sense of Segal [27]. Modern physical theories have several Lie algebras in their foundations, defined by commutation relations among their infinitesimal generators. The work of Inönü and Wigner directed worldwide on attention toward small changes in the commutation relations, both successful ones of the past [17] and speculative ones for the future [15], but seem to have left open the question of which changes of this kind were likely to lead to useful physics. Irving Segal earlier provided an important strategy for such explorations [27]. He divided the continuous groups into stable and unstable groups, or, synonymously, into regular and singular groups. Stable groups are unchanged up to isomorphism by small changes in their commutation relations. Since all measurements have error bars, only the stable groups can be empirical; the unstable are faith-based. The stable groups are essentially the simple Lie groups. The group of a stable quantum theory may be semisimple at first, but then a single quantum measurement suffices to reduce the semisimple group to a simple subgroup. Any compound (non-semisimple) group in present physical theory is an avalanche waiting to happen and easily triggered. At the same time, we must moderate this strategy with the recognition that stability is a relative concept. A simple Lie algebra is stable among the Lie algebras but not among the quantum algebras which are not co-commutative, nor among the non-associative algebras. Segal made a significant nontrivial choice when he adopted stabilization within the manifold of Lie algebras as his strategy. This embraces the previous iconoclasms but some day it may be necessary to go deeper. Indeed, the Segal strategy already automatically brings in non-cocommutative gauge groups, as we discuss in section 14. Simple groups have finite-dimensional representations, in which all operators have bounded discrete spectra. It is only the compound, singular groups that force infinitedimensional representations upon us. The infinities of present physical theory all come from its unstable compound groups. So the drift of physical theory toward stability is also a move toward finiteness. Physical theory drifts toward stable and therefore simple groups for Darwinian reasons. Compound groups, being unstable, do not survive experimental test as long as simple ones. This is why regularization (and not say its opposite singularization) predominates in past iconoclasms. Sometimes regularization can be accomplished by the inverse process to group contraction. More generally, to regularize a group we must append generators as well as change the commutation relations. Then to take the singular limit we must not only stay
Whither Quantum Theory?
27
near the identity in some dimensions of the algebra, as in group contraction, but must also freeze out the new generators. In the past, changes in the algebra commutation relations—briefly, flexings of the algebra—have taken two forms: global and local.
4 Flexing In flexing, the instability of the theory drives an abrupt increase in regularity, stability, simplicity, and finiteness, through the introduction of curvature, non-commutativity. This does not increase the number of variables in the historical precedents of special relativity and quantum theory, and more generally it may increase the number of variables merely quadratically, as is required for regularizations of quantum theory. Saller discusses the regularization of the Heisenberg group H(1) [24, 25] and Shiri-Garakani applies this process to the harmonic oscillator [28]. Baugh gives economical regularization variables for the Heisenberg groups H(n) for every dimension n [3].
5 Localization Localization replaces global variables by new local variables attached to every point of space-time, as in gauging. Displacements of these local variables in different space-time directions do not commute; this is the non-commutativity resulting from localization. This process increases the number of variables exponentially. The familiar process of gauging is a singular form of localization and introduces instability; general relativity, electrodynamics, and the standard model are singular cases of localization. But localization is regular when the underlying space-time and the overlying field variables are both regular.
6 Unstable statistics Symmetry groups are not the sole source of instability, perhaps. In some places the unstable Heisenberg Lie algebra lg H(1) (the algebra with three generators p, q, i and the Lie-products p ' q = i, i ' p = i ' q = 0) enters not as an algebra of symmetry-group generators but as the commutation relations of the creators and annihilators of quanta, defining the passage from a one- to a many-body quantum theory. This passage was aptly called quantification before the advent of quantum theory, by the Scottish philosopher-logician William Hamilton, and we retain his term to maintain continuity. Quantification is studied in higher-order logic and higher-order set theory. When the Heisenberg associative algebra of p and q arises as a Bose-Einstein theory of quantification, what is unstable is the statistics. The insight of Segal remains: Simple is stable. The drift toward simplicity is ultimately driven by a Darwinian selection of stable algebras over unstable. Aesthetics typically favors unstable singular theories over stable regular ones. Given three points, we fit them with a straight line if we possibly can, and with a circle only if we have too. Perhaps we will always be more comfortable with singular theories than regular ones. Like the geocentric theory, a singular theory makes us special. “Special” is indeed one of the intended meanings of “singular,” just as one of the meanings we intend for “regular” is “generic.” In that case there will always be instabilities to trigger future iconoclasms, simply because we uncritically build instabilities into our first theories of any domain of experience. We should therefore allow for future iconoclasms as we allow for future earthquakes.
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We propose that a better quantum theory is founded entirely on stable algebras, regularizations of the unstable algebras of present theory. We use a technique of algebra regularization to morph unstable physical algebras into more stable ones that may be equally physical or even more so. The space-time quantization problem now appears as a special case of the regularization problem, which is better posed and more tractable. We are to find regular quantum correspondents for the higher-order singular classical logical processes underlying dynamics and field theory, especially the concept of function itself. This resumes a historical progression. Special and general relativity and quantum theory substantially regularized some of the singular groups of physics, and reduced the degree of singularity of others, but left some singular groups standing. Perhaps they have stood so long because the next step is so big. The main surviving unstable Lie algebra, the ubiquitous Heisenberg lg H(n), infests the continuum theories of space and time, the canonical commutation relations for position and momentum, gauge theory, the Heisenberg and Schrödinger dynamical equations, and Bose-Einstein statistics. All of these must change together to stabilize the standard model and gravitation. We use this instability to predict and hopefully to trigger some of these impending iconoclasms. They lead to a finite quantum theory of time, described here. Each iconoclasm (in our present sense of the term) is both conservative and radical: Conservative, in that it has but small measurable consequences in the domains of prior experience; Radical, in that it relativizes old absolutes and restructures our view of the cosmos. The change in thinking that constitutes an iconoclasm passes over the physics community like a shock wave.
7 Algebraic instability and regularization A semisimple group is stable in the sense that a sufficiently small neighborhood of the group in the algebraic manifold of structural tensors contains only isomorphic groups. Nonsemisimple (henceforth, compound) groups are generally not stable in this sense. They are different in structure from almost all the groups in their neighborhood. In this sense, simple groups are regular groups. They are just like all the other groups in their neighborhood. There is an easily remembered test for group regularity. Groups are singular when their Killing metrics are singular [27]. Similarly Clifford algebras are singular when their metrics are singular. All unstable algebras have radicals (maximal solvable subalgebras). Eliminating the radical makes the algebra stable. (The converse statement is false. The Lie algebra defined by a ǻ b b is all radical, yet stable, due to the paucity of two-dimensional Lie algebras. I owe this counterexample to James Baugh. ) Regarded as inferences from experiment, compound groups have 0 probability relative to infinitely many nearby simple groups. Commutativity destabilizes. Flat sails luff in the least breeze.
8 Regulants Some of the major metamorphoses of physical theories in recent centuries have had the effect of slightly changing the relativity group of physics so that a compound group becomes simpler, less compound, and loses at least part of its radical. They do this by introducing a non-commutativity whose scale is set by some small parameter that becomes a fundamental
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constant of the next physics [27, 17, 13, 2]. We call this fundamental constant the regularization constant of this process, or briefly, the regulant. Some regulants together with the singular limits that restore commutativity are k o 0 (statistical mechanics), c o f (special relativity), R o f (de Sitter space-time), G o 0 (general relativity), h o 0 (quantum theory), e o 0 (electrodynamics), and g o 0 (chromodynamics). (For the Boltzmann regulant k, the non-commutativity is that of adiabats and isotherms.) In most cases the non-commutativity that is introduced regularizes some singular algebra of the prior theory, an effect that we call algebraic regularization. In these cases the regularized physical theory is more fundamental, more relativistic, more unified, simpler, stabler, more operational, and more finite than its singular limit, and agrees better with experiment. When we call this process of theory-repair “deformation,” we build in our self-defeating natural preference for singular theories. Some who study deformations of singular theories even keep both the singular and the regular structures in the physical theory, defeating our main purpose, regularization. We are reforming the theory, not deforming it. It is the singular theory that is deformed. The very variations against which a theory is unstable define regularizations that restabilize the theory. Thus Einstein’s space-time theory is stabler than Galileo’s relative to variations in the speed of light, which has the singular value in Galileo’s relativity. Heisenberg’s quantum theory is stabler than Newton’s classical theory relative to variations in Planck’s constant, which has the singular value 0 in Newton’s theory. Anholonomy, inexactness, and nonintegrability are instances of such non-commutativity. Among the historic algebraic regularizations, neither Heisenberg’s stabilization of classical phase space and Einstein’s stabilization of Galilean space-time fully stabilized the unstable theories from which they sprang. The localization by general relativity (G regularization) eliminated the radical of special relativity, the translation subgroup, but at the same time it introduced the more serious instability of the canonical group of the gravitational field. which requires regularization today. Likewise only a second regularization eliminates the residual instability of quantum theory [27]. The most economical regularization of quantum theory is the finite quantum theory that we discuss here.
9 Regulators The regularization of quantum theory differs from those of Galilean relativity and classical mechanics, whose groups could be simplified merely by slightly changing their structure constants. It breaks the familiar pattern of group contraction because the unstable Heisenberg algebra is not the contraction of a stable algebra of the same dimension. Einstein did not need to enlarge the Galileo algebra to form the Lorentz algebra, and Heisenberg did not need to enlarge the Poisson algebra to form the Heisenberg algebra, but to simplify the Heisenberg algebra we must first enlarge it. For H(1), at least one new variable r must be introduced. For H(n), we must introduce O(n2) new variables. For brevity we call these new operators introduced for the sake of regularization, and frozen out in the singular limit, regulators. The regulators must be frozen in the vacuum to account for the success of the singular theory. In general, algebra regularization uncovers degrees of freedom that are frozen and hidden in the singular theory. Specifically, one-dimensional space regularization requires us to introduce one regulator r with maximum eigenvalue r 1 , so that we can replace the Heisenberg commutation relations qp pq i! of prior quantum theories by the SO(3)invariant Segal relations
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(1) qp pq i!r , rq qr i! cp , pr rp i! ccq The Heisenberg algebra is represented by differential operators or infinite matrices. The Segal algebra can be represented in a finite-dimensional Clifford algebra of real 2 n u 2 n spin matrices. The structural constants include a finite small quantum of time or chronon !! c , a quantum of energy E !T , and a large pure number N 1 ! c! cc . We attribute T the relative constancy of the ambient r to the same condensation that forms space-time and splits space-time from momentum-energy. Hopefully the variable r provides a Higgs effect much as the variable i did in quaternion quantum theory [12, 1]. The finite quantum oscillator has essentially the same bounded, uniformly spaced spectra for its three basic variables p, q, r, except for scaling, which is provided by the three quantum constants ! c , !cc , ! ccc . The usual quantum theory now appears to use a vanishingly small sector of a much larger Hilbert space. This sector is spanned by eigenstates of the regulator r near r 1 . There must be a great many such eigenstates for the singular quantum theory to be a good approximation; another regularization parameter N must be large. In general the equipartition principle and the Heisenberg uncertainty relation do not hold in the finite quantum theory [28]. New principles take their place which reduce to the old ones in the singular limit. We approach the regular quantum theory by two stages, one for the stationary (timeindependent) theory, which is straightforward, and one for the dynamical theory, which takes more thought and speculation. The stationary theory uses first-order quantum logic, but our dynamical theory is a quantum theory over a quantum theory, a q/q theory, and uses a second-order quantum logic that has not been experimentally tested.
10 Stationary quantum theory We illustrate the method with a toy linear harmonic oscillator [28]. This can be regarded as a spatial oscillator, like a particle in a potential well, or a field oscillator among many. Obviously the infinite zero-point energy of canonical quantum field theories must become finite under regularization. Regularization changes the total zero-point energy because it makes the number of oscillators finite, and because it changes the zero-point energy of each oscillator. M. Shiri-Garakani examined how it affects the zero-point energy of each oscillator [28]. Every finite quantum linear harmonic oscillator is isomorphic to a dipole rotator with Hamiltonian of the special form H
1 1 K x L x 2 K y L y 2 2
2 , K x
P2
P
, Ky
Q2
O
(2)
The classical oscillator and the singular quantum oscillator have continuous coordinates and momenta. The position and momentum variables of the finite oscillator are quantized with finite, uniformly spaced, spectra, with spacing P, Q respectively, and maximum values lP, lQ. To emulate singular quantum oscillators, which have infinitely many states, finite quantum oscillators must have many states, l Ҍ1. In the classical theory all oscillators are isomorphic up to scale. The quantum of action ! does not break this scale invariance. All singular quantum linear harmonic oscillators are isomorphic up to scale. The constants ! , ! cc finally break this scale invariance. The finite
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quantum linear harmonic oscillators fall into three broad classes, which we term soft , medium, and hard, according to the dimensionless ratio K y K x : N 2 of maximum possible potential energy to maximum possible kinetic energy. Medium oscillators (ț ~1 ) have many low-lying states with nearly the same zero-point energy and level spacing as the singular oscillator. In those states they resemble rotators nearly polarized along the z-axis with Lz ~ ± l. They resemble the singular quantum oscillator in obeying the Heisenberg uncertainty principle and the equipartition principle when they are in their low-lying energy levels. The soft and hard oscillators do not resemble the singular quantum oscillator at all. Their low-lying energy states correspond to rotators with ț ~ 0 or ț ~ . Their 0-point energy is infinitesimal compared to the singular quantum oscillator. They grossly violate both the uncertainty principle and equipartition in all their states. Soft oscillators have frozen momentum p ~ 0, their maximum potential energy being too small for even one quantum of momentum. Hard oscillators (kinetic ҋ potential) have frozen position q ~ 0, their maximum kinetic energy being too small for even one quantum of potential energy. These quantum freezings of degrees of freedom resemble but extend the original ones by which Planck obtained a finite thermal distribution of cavity radiation. Even the zero-point energy of a similarly regularized field theory will be finite, and can therefore be physical. In the soft and hard oscillators the energy cannot be completely exchanged between kinetic and potential forms. The zero-point energy is much smaller than !Y 2 . In the medium cases the singular theory is a good approximation. This is just what is needed to make the total energy of the oscillators of a field theory finite without disrupting the present experimental agreement. The finite quantum theory of the stationary oscillator supplements Planck’s quantum constant ! with two Segal quantum constants ! c , !cc . As an inference from experimental data, Heisenberg’s theory that ! c ! cc 0 has probability 0 relative to Segal’s theory that ! c , ! cc z 0 . We may take the regulants of the finite quantum stationary oscillator theory to be T ĺ 0 and N ĺ , shown with their singular continuum limits. The extension of the T homotopy from time and energy to relativistic space-time and momentumenergy is obvious and requires no further regulants. It includes the Snyder regularization [29]. It is instructive to contrast the regularized quantum theory with lattice quantum theories. The continuous groups of the standard continuum theory are compound and admit no useful finitedimensional representation. The standard lattice approximation replaces the continuous groups by discrete groups. The regular quantum theory makes an arbitrarily small curvation that changes the group to an orthogonal or unitary group, still continuous, which then admits exact faithful finite-dimensional representations.
11 Clifford logic Our only regular candidate for the higher-order quantum logic needed for quantum dynamics over a quantum space-time is a higher-order Clifford algebra. This is a Clifford algebra C with an additional linear monadic isometric operation Ț : CĺC, corresponding to
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the classical process of unit-set formation: {a} = Ța. This means that our simple groups will have to be orthogonal groups. We therefore use Clifford algebra as quantum logic and set algebra [9]. To formulate the interpretation we begin with a Clifford algebra over the binary field ^0, 1` , used as an algebraic classical logic. In the logical interpretation, 0 represents nothing, 1 represents the empty set, and the top element ȖT = Ȗ1 … ȖN (the product of all the generators) represents the universal class. The Clifford product ab represents a symmetric union, the union less the intersection, which we sometimes write as a ҵ b. Addition modulo 2 represents a POR (or partial OR) operation, the original partial operation that Boole and Peirce wrote as a + b, except that we give the sum the default value 0 when Boole left it undefined. Binary Clifford algebra admits powers as well as sums and products, though we do not develop them here. 2
We interpret real Clifford algebra as a quantum logic guided by binary Clifford logic. 0, 1, and Ȗ still represent nothing, the empty set, and the universal set. Clifford algebra elements now represent input-output operations (“state vectors”). Clifford addition represents quantum superposition. Clifford multiplication represents a symmetric union a ҵ b, now non-commutative. For higher-order logic we use the hierarchy of Clifford algebras over Clifford algebras, generated by adjoining the monadic operator Ț to the usual operators of Clifford algebra. This Ț strengthens the usual Hilbertspace quantum kinematics much as the brace operation {a} strengthens Boolean algebra. Real Clifford algebras are already regular, so we do not modify them during regularization. In particular algebraic regularization respects the real Dirac spin algebra (the Majorana algebra) and the Fermi-Dirac algebra. The real form of the complex Clifford-algebraic statistics of Nayak and Wilczek [20] are also stable under regularization. We regularize higher-order theories by morphing their commutation relations within a higher-order Clifford algebra. This Clifford algebra replaces the usual Hilbert space [6, 33]. The quantum correspondent of the set exponential of classical dynamics is the Clifford exponential of a finite quantum dynamics. To incorporate classical finite set theory too, and accommodate singular limits, we construct the infinite-dimensional Clifford algebra over itself, C = Cliff C, by iterating Clifford algebra formation and taking the limit: R Cliff: R = C0 ĺ C1 ĺ C2 ĺ … C = lim C n no0
Cn
(3)
n
We designate the generator of Cliff V associated with a vector v ЩV by Țv (Peano’s symbol for the unit set {v}). We adjust the sign in the definition of Cliff so that Ț is an isometry: (4) Lv v 1 is not the unification of any Clifford element: Ț†1 = 0. Then C1 is the Study or pseudocomplex number system, C2 is the pseudoquaternion algebra, C3 is the sedenion or Majorana-Dirac spin algebra of special relativity, and C = C includes an infinitedimensional fermionic algebra. C is singular. In any given regular physical theory we use only a finite-dimensional subalgebra C N ңC. N ĺ is a singular limit.
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12 Dynamical quantum theory The quantum kinematics of the harmonic oscillator, we have seen, is destabilized by its Heisenberg algebra. Then the quantum dynamics is doubly unstable, since it has two Heisenberg algebras. Besides the quantum kinematical Lie algebra of (q, p) discussed already, there is the spatial Lie algebra of (t, d/dt). Its elements are not quantum observables; this non-commutativity exists in classical physics too. It occurs in all working field theories today, c/c or q/c. To regularize a dynamical theory we must first express it algebraically. The oldest form for the basic law of q/c quantum dynamics is Heisenberg’s Equation (5) UX dX dt
i!>H , X @
Ut
The variable X that appears in this law does not designate an observable but an observable-valued-function-of-time X(t). So does H = H(t) in general. The observablevalued-functions-of-time form a much larger and even more singular algebra than the observables. Let us call them dynamicals for short, and their algebra the dynamical algebra. This has to be regularized, not just the algebra of observables. We therefore turn our attention from regularizing the algebra of observables to regularizing the algebra of dynamicals. The concept of observable is based on a particular split of the cosmos into agent and patient (two useful terms of nineteenth-century philosophers). The quantum relativity of Heisenberg takes the patient as an absolute. Here we encounter a still greater relativity than the quantum relativity. The patient is no longer absolute but relative; to be sure, within an absolute quantum cosmos. Bohr at first objected vigorously to the concept of the quantum cosmos, since it violates strict operationality; no one in the cosmos can “see” it. But later he suggested that such extensions might ultimately be necessary for further progress. Nowadays the quantum cosmos is a commonplace in quantum speculations. Only after we divide the cosmos into agent (which includes us) and patient can we extract internal quantum observables of the usual sort and external classical nonobservables like time from the algebra of dynamicals. This split must be made in exactly the right place. We do experiments with apparatus that has an enormous number of variables, knowing but a handful of them. Yet with these blunt instruments we produce sharp states, quanta on which we have near maximal information. This is possible only if we separate system from instruments at exactly the right place. It depends on condensations in the instrument that do not include the system. Dynamicals still appear in more covariant form of dynamical law like the Symanzik Equation (6) ª GS >A@ º « GA j » Z > j @ 0 ¬ ¼ for the generating functional Z of the Green’s Functions (7) G x1 x m 0 | TA x1 A x m | 0 in which the dynamical field variable A = (A(x)) is the operator A x i!
G Gj x
(8)
involving functional differentiation with respect to the source variable j = (j(x)) [22]. The appropriate algebra for this theory is generated by dynamicals A(x), j(x) (not observables) subject to
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David Ritz Finkelstein
>A x , j y @
i!G x y , for all the world like a Heisenberg commutator algebra H f of infinite dimension.
(9)
The c/c problem is always handled with higher-order logic. The functions from c time space T to a c or q phase space X. constitute the set-exponential XT of higher-order logic. The q/q dynamical theory thus requires a regular correspondent of higher-order quantum logic and the set exponential [9, 3]. The c set-exponential leaves finite sets finite but converts tame infinities into wild infinities. As a reminder of this fact we recall that 20 = C, the cardinal number of the continuum. Sums of 0 non-zero terms can converge; sums of c non-zero terms cannot. To avoid producing the latter when we exponentiate, we must avoid the former in the exponent. Our main problem is to express the q/q idea of a field with given quantum field values and quantum space-time. The corresponding c/c theory would have a field phase space F , a space-time T, and a space of dynamicals U = FT . In the quantum theory the spaces F, T, U are quantum spaces associated with quadratic spaces HF, HT, HU. Schematically speaking, the H
state of the regularized field theory would be described in the Hilbert space H U H F T , if that can be defined, corresponding to the classical field phase-space FT . The usual singular quantum field theory will then appear as a semiclassical limit, in which the field preserves its quantum aspect but the space-time is treated in a classical approximation. We can easily define this regular exponential in the case where HF is itself a Clifford algebra H F 2 f Cliff f over some vector space f, as if the field were a fermionic aggregate; and only in that case have we been able to define it at all. Then one defines H U : Cliff f
H T 2 f
H T . In the correspondence limit of classical space-time, this
agrees with a classical exponential. This process accounts for multicomponent fields F by adjoining extra variables f to space-time. This resembles the procedure used by Connes in his theory of the standard model [26]; here we quantize the space-time variables as well as the unitary ones. We may think of the extended space f
H T as the quadratic space of statevectors of a quantum probe with which we explore the field. We first indicate how a generic Clifford element can be regarded as a state-vector for a binary quantum field (one with the values 0 and 1) over a quantum space-time T. This corresponds to the special case where HF=R and f is the one-point linear space ^0` . For this system H U 2 H T . We designate vectors in a basis of HT by J ' s . Each J k in an orthonormal basis ^J k ` for HT represents a disjoint cell in T space. The Clifford unity element 1 H U represents the null set, and a binary field that vanishes everywhere. Then LJ 2 T (a first-grade Clifford element of unit length), represents a point-set with one point represented by J , and it also represents a binary field with value 1 on that point and 0 on every disjoint point J c A J . Binary one-point fields supported by disjoint points anticommute. H
If Į is any complex number, DLJ 2 H T represents the same field. The coefficient Į gives quantum probability amplitudes in quantum superpositions like DLJ D cLJ c .
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A Clifford product like LJLJ c 2 HT with J A J c represents the point-set with two points represented by Ȗ and Ȗƍ. It also represents the binary field that is 1 on those two points and 0 elsewhere. The extension to Clifford elements of any grade is obvious. The grade of a Clifford element gives the number of space-time points in the support of the binary quantum field. An inhomogeneous Clifford element is a quantum superposition of psi vectors representing fields supported by various number of points. First-grade cliffors correspond to single-time dynamicals, and their higher-grade products correspond to manytime dynamicals. We must represent a more general kind of field than binary, with quantum values represented by a Clifford algebra F = 2f , by special configurations of a binary field over f
HT . The extension to still higher orders results in the functor Cliffn, the n-fold iteration of Cliff.
13 Correspondence principle The space HU and its physical interpretation define the kinematics of the q/q field theory. We must give a “dynamical vector” < H U to specify the dynamics, assigning a probability amplitude < ) to each history ) of the quantum field. In the regular theory this probability amplitude and the Feynman sum over histories become well-defined finite mathematical objects. The operators t , E : i!d dt correspond to specific infinitesimal elements of the group of the quadratic space HT [28, 3]. These are their q correspondents. It is natural to extend their action to the basis vectors I m
W n H U by letting them act on just the factor IJn. We
^
`
define their action on the generic element of HU, which may have any degree in the variables I m
W n , so that they are derivations of the Clifford algebra HU. This defines finite regular
^
`
quantum correspondents for all the singular space-timeenergy-momentum operators. The regular q/q algebra Endo HU corresponds to the singular algebra of dynamicals of the q/c field theory. These regularizations are necessary but not enough to regularize a given q/c dynamics like the standard model. To formulate the regular q/q dynamics, we must make a q/q correspondent of the action principle that works the q/c theory. We need a regular q/q correspondent not merely for the q/c algebra of dynamicals, given above, but also for a specific singular q/c field operator ȥ(x) itself, a generator of that algebra. This problem has obstructed the development of our theory until now. A Grassmann field variable ȥ represents quantum creation or annihilation and is not Hermitian. Then the Hermitian sum ȥ(x) + ȥ†(x) (used as basic field variable by many) “toggles”’ the quantum at x. That is, it creates a quantum at x if there is a vacancy at x, and annihilates one if there is not. So does the difference ȥ(x) – ȥ†(x), with a change in sign of the quantum phase. In the Clifford-algebraic theory it is convenient to work with stable togglers obeying I 2 = ±1 rather than with unstable annihilators and creators obeying ȥ2 = 0. For any vector a H T with non-zero norm, L a toggles a space-time point described by
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David Ritz Finkelstein
a . The Clifford elements a
generate the many-quantum set algebra of quantum space-
time.
The single q/q field-variable operator \ x Endo H U corresponding to the q/c scalar field variable ȥ(x) is therefore \ x L itself.
14 Quantum gauge groups Gauge groups have a local exponential form GT in c/c gauge theories. In the regularized theory, T is a quantum space-time, and the exponential is defined by GT = 2gT . There is no simple product gT of a classical space g and a quantum one T . Simplicity implies that g and therefore the gauge group G too are quantized by regularization. Co-commutativity of G is lost. The group parameters are not commutative. In fact quantum gauge fields have dealt with quantum groups from their inception in quantum electrodynamics. A flux is a group parameter for the transport of a gauge field probe or generalized charge around the flux. In quantum gauge field theory the fluxes do not commute; therefore the group parameters do not commute; therefore the group is a quantum group. However existing quantum gauge field theories harbor one H(4) in the underlying manifold, and an H() in the canonical group of the gauge fields. Since economical regularizations for these singular groups are now known, it seems straightforward to regularize the quantum gauge theory.
15 Experimental implications Regular quantum field theory should inherit all the experimental successes of the prior singular quantum field theory, since its predictions can be made to come as close as we like to those of the prior theory, for suitably small regulants and suitably frozen regulators. A correspondence principle for vanishing regulants gives experimental meaning to all the variables of the finite quantum theory. We look to extremely large energy E ~ !/T for significant finite quantum corrections that fix the chronon T. To regularize the quantum dynamics of a particle in space-time one must perform all the homotopies introduced by de Sitter, Snyder, and Segal at once. If one of them is omitted, the theory remains singular. Naturally this regularization modifies the Poincaré group cosmologically, but this seems harmless, and the local Lorentz group survives intact. In general, stable groups like the Lorentz group and SU(3) are preserved exactly in the finite quantum theory of space-time. The differential-manifold theory of space-time is now an approximation to finite quantum space-time in limiting cases where the chronon T is negligible and the space-time condensate persists, an ether that defines no rest-frame. The finite regular ether presumably breaks the Segal group SO(3) in the way that a ferromagnet does, by a spontaneous condensation. The standard-model forms of the locality and uncertainty principles break down in finite quantum theory at high energy ๆ !ȱ/ T2, when r can approach 0 and spacetime can disorder itself. When that happens the unbroken Segal group invariance leads to an interconversion of time and energy differences at the rate of exchange E = WT with fundamental power
Whither Quantum Theory?
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37
! T 2 , just as one interconverts x and y coordinate differences by a mere rotation, or
time and space differences by a boost. W is at least one Higgs mass per Higgs time, which amounts to some kilograms of energy per second; many would expect the fundamental power W to be one Planck mass per Planck time.
16 Acknowledgments This report describes work done with James Baugh, Andrei Galiautdinov, Mohsen ShiriGarakani, and Heinrich Saller.
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Physics 21 (1982): 489-503. [10] Finkelstein, D., Quantum Relativity (Heidelberg: Springer-verlag, 1996). [11] Finkelstein, D. R. and A. A. Galiautdinov, “Cliffordons,” Journal of Mathemetical Physics 42 (2001): 3299-3314. [12] Finkelstein, D., J.M. Jauch, S. Schiminovich and D. Speiser, “Principle of general Q-covariance,”
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[21] Penrose, R. Angular momentum: an approach to combinatorial space-time, pp, 151– 180 in Quantum Theory and Beyond, ed. T. Bastin (Cambridge: Cambridge University Press, 1971). [22] Rivers, R. J., Path Integral Methods in Quantum Field Theory (Cambridge: Cambridge University Press, 1987). [23] Saller, H., “Symmetry reduction from interactions to particles,” International Journal of Theoretical Physics 40 (2001): 1151-1171. [24] Saller, H., “Harmonic analysis of space-time with hypercharge and isopsin,” hepth/0312084. [25] Saller. H., “Basic physical Lie operations,” International Journal of Theoretical Physics, in press. hepth/0410147. [26] Scheck, F., W. Werner, and H. Upmeier, “Non-Commutative Geometry and the Standard Model of Elementary Particle Physics,” Proceedings of the Hesselberg Conference, March 14-19, 1999. [27] Segal, I. E., “A class of operator algebras which are determined by groups,” Duke Mathematics Journal 18 (1951): 221. [28] Shiri-Garakani, M., “Finite Quantum Harmonic Oscillator,” Ph. D. Thesis, School of Physics, Georgia Institute of Technology. [29] Snyder, H.P., “Quantized space-time,” Physical Review 71 (1947): 38. [30] Stückelberg, E. C. G., “Quantum theory in real Hilbert space,” Helvetica Physica Acta 33 (1960): 727-752. [31] Wheeler, J. A., “The elementary quantum act as higgledy-piggledy building mechanism,” pp. 2730 in Quantum Theory and the Structure of Time and Space, vol. IV, ed. L.Castell, M. Drieschner, and C. F. v. Weizsäcker (Munich: Hanser, 1973). [32] Wilczek, F. and A. Zee, “Families from spinors,” Physical Review D25 (1982): 553. [33] F. Wilczek, “Projective statistics and spinors in Hilbert space,” hepth/9806228 [LANL]. [34] Wiman, A., Mathematische Annalen 47 (1898): 531 and 52: 243.
THE DIRECT DETERMINATION OF THE INDUCED PSEUDOSCALAR CURRENT (AND ABOUT THE SLOW METAMORPHOSIS OF AN INSTITUTION) Laszlo Grenacs
Introduction The Centre de Physique Nucléare (CPNL) of the Université Catholique de Louvain at the beginning of the 1960s: low-energy nuclear physics studies and measurements of parityconserving and parity-violating beta decay observables constituted the major part of the experimental works. However, a couple of seniors, besides their low-energy programs, participated in intermediate- and high-energy experiments performed abroad, and obtained significant results. One instance was the measurement of the longitudinal polarization of e+ and µ+-decay. A group of fellows—experimentalists—, including Laszlo Palffy, Jean Lehmann and myself, certainly influenced by the example of our seniors, initiated an ambitious scientific project motivated by particle physics: the direct determination of the socalled induced weak currents in nuclei. Their significance lies in that they reflect symmetries of weak interactions and the determination of the helicity of the muon-type neurino. One of the induced currents is the weak magnetism (WM), a piece in the weak vector currents, a cornerstone term for the critical verification of the Conserved Vector Current (CVC) theory. Another one is the induced tensor in the axial vector current, a second class current (SSC). This term, if it existed in nature, would destroy the charge symmetry of weak interactions. The third term is the induced pseudoscalar current. It results from the imperfect preservation of the chiral symmetry of strong interaction. Relationships between the ingredients regarding this matter are treated in the Partially Conserved Axial Vector Current theory. To verify this theory one had to measure the induced pseudoscalar coupling. The nomination of David Speiser as a professor of theoretical physics at our university coincided with these times. His offices, where he gave seminars about physics subjects at the frontier, were also located in CPNL. One of them was about his just-fresh personal discovery: the SU(3) is the group of symmetry of strong interactions. Another treated the SU(3) octet of weak and electromagnetic currents, and the algebraic relations among them. We questioned him about that matter, hoping to understand the place of the induced currents in the SU(3) of currents. David Speiser was always accessible for discussions, unless he had important teaching duties and research work with his students. Also, he had organizational responsibilities in our department. He created the Licence Spéciale, a oneyear pre-doctoral program with a diploma. The students had to follow lectures specialized with a broad spectrum and undertake research work. This institution is still standing. David Speiser aimed to transform our department made of parts into a department of collaborating theorists and experimentalists of CPNL. That project had no chance for realization in a short term, due chiefly to the effects of tradition. For further discussions, we turned to Robert Brout, invited professor, and gifted students of David Speiser—Jacques Weyers, Jean Pestieau, Pierre De Baenst—and, later, to Claude Leroy. David Speiser gave the task of introducing us into the world of PCAC to Pierre De
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Laszlo Grenacs
Baenst. The present recollection is dedicated to the master and his student in acknowledgment. Other experimentalists of CPNL approached our theorists as well. This was a sort of mini-peregrination. The experimentalists questioned; the theorists seldom, or rather never. They did not seem to appreciate the intellectual content of an experiment. The ice was broken by Claude Leroy. Although a theorist, he participated on equal footing in our beta decay experiments aimed at determining directly the weak magnetism and the second class induced tensor. He also participated in high-energy experiments at CERN conducted by physicists of Louvain. A bit later, experiments conducted at our Cyclotron on the radiative capture of a neutron by a proton (Group LIMAL) were performed, in collaboration with the theory of J. Pestieau and J. Govaerts, which was a deep and productive one. Such interactive talks acted positively on the minds of the members of our department. Theory and experiment came closer.
Experiments Implementation of direct methods. In the mass triade A=12 (fig. 1) there exists a sufficient number of independent observables, accessible to experiments, which allow independent determinations of the induced weak current contributions. This experiment became possible due to novel techniques involving nuclear orientations, polarization and alignment, that we succeeded in developing. For this purpose, and the performance of the experiments themselves, we had the 3 MeV electrostatic accelerator of CPNL at our entire disposal, thanks to an intervention of Pierre Macq.
Fig. 1. Reactions to investigate induced weak currents
Beta decay. The operator of an induced interaction is proportional to the product of the momentum transfer operator qµ with a spin operator. Thus, the corresponding amplitude in the case of stretched coupling j=3/2 of the involved angular momenta of one and one-half units, interfering with the leading Gamow-Wave of j=1/2, can lead to a rank-two density
The Direct Determination of the Pseudoscalar Current
41
proportional to the energy E of the electron. This is the density which builds up the alignment term (~P2) in the angular distribution of beta rays: ~ 1+ĮEAP2. Here Į is contributed by the couplings of induced currents only and A is the alignment of the parent. The determination of induced currents via the measurement of the alignment term is reminiscent of direct determination of g-factor anomalies. The use of zero-rank or rank-one densities to determine induced currents, an indirect procedure used in the past, lead to confusing conclusions regarding SCC and CVC. We succeeded in producing 12B with alignment A during an interval of time, then with A during a subsequent interval, alternately. The difference of the respective beta ray counting rates over their sum yields Į. The measurement of Į+, in the decay of aligned 12N, following the same procedure, was performed in the second half-time of the project with the participation of the experimental team of ETH-Zürich. The operation was led by Professor Telegdi. Our faculty already named him to receive the recognition deserved for highest merits. The difference a a yields the sum of charge-dependent induced currents contribution: WM and SCC! This sum is exhausted by WM alone, determined directly in muon capture experiments described below. Thus, there is no room for SCC.
The capture reaction µ + 12Cψvµ+12B and the induced pseudoscalar coupling [1,2] There are two independent polarizations of 12B in the capture reaction 0 ψ1 One of them is measured in the frame of the captured µ, polarization of 12B averaged over the direction of the unobserved vµ, called average polarization Pav.
Fig. 2. Measurements of the average polarization of 12B in the capture of polarized µ by 12C
42
Laszlo Grenacs
Fig. 2 is a pictogram of the experimental disposition. The spin direction of µstopped in the graphite target, is indicated by an arrow, along the beam direction (z-axis). When only the intrinsic spin of the neutrino plays a role (the pure Gamow-Teller case), the polarization of 12B is given by spin-geometry considerations, Pav=2/3. In that case the reaction amplitude ai leading to the M=0 and M=1 substates of 12B are the same: a0/a1҂ X=1. The assymmetric emission of beta rays by 12B, indicated by the arrows in fig. 2, stands for this particular case, in a pedagogical assumption that the captured µis fully polarized. The induced pseudoscalar, spin-singulet in spin space feeds the M0 amplitude only, thus affecting X and therefore Pav. To determine Pav (x), we went to the muon-producing Accélérateur Linéaire de Saclay. The arrangement was similar to that in fig. 2, adding a longitudinal decoupling magnetic field for the preservation of the spin of 12B slowed down in the graphite target. We obtained, as a final result, Pav (x) = +0.49±0.04. The positive sign means that the polarization of 12B points along the forward muon beam resulting from pions in flight. Originally, this experiment was proposed by Jackson, Treiman and Wyld to determine the helicity of the muon-type anit-neutrino from ʌ-decay. The above result means that the helicity of vµ is positive, like that of ve, and is close to unity. The above result yields X=0.24 ± 0.08. There is a generous effect of induced currents. Another polarization of 12B is measured in the neutrino or equivalently in the recoil frame of this nucleus, called the longitudinal polarization, PL, a pseudoscalar quantity. In the Gamow-Teller case this polarization is also 2/3. Figs. 3a and 3b show the arrangement to measure PL.
Figs. 3a (left) and 3b (right). Measurement of the longitudinal polarization of 12B
The target is made of a spin-destroying D layer (aluminum), the graphite layer and a spinpreserving P layer (silver). The graphite layer is thin enough to let the 12B recoil escape from it after the capture of µ by 12C; the D and P layer are just thick enough to slow down the recoiling 12B. When vµ is emitted along the beam direction, 12B recoils in the D layer, where it is depolarized. However, when vµ is emitted against the beam, 12B recoils into P, where it remains polarized. The global polarization12B in this composite target is P= Pav /2 + PL /4. In the arrangement shown in fig. 3b, the order of layers is inverted; the global polarization is now P= Pav /2 – PL /4. Their sum yields Pav, and their difference PL /2. Fig. 4 shows the polarizations observed with the two orderings. Note the relatively small effect of the scalar Pav, due to the small (1/5) polarization of µ at the instant of its capture. A (CH2)n target in which 12B is depolarized serves to verify the zero of the set-up. The ratio R(x) = Pav / PL §
The Direct Determination of the Pseudoscalar Current
43
0.5, which can be read from this figure, is also a very large effect due to induced contributions. This ratio has an enormous advantage of experimental points of view: it is independent of the background, geometrical effects, spin relaxation of 12B in the P layer, etc.—a pure experimental number. From the observed ratio R we have x=0.28 ± 0.08. The average is then X=0.26 ± 0.08. From the relation between PL and the helicity of the muonic neutrino, we obtain hv = –1.0 ± 0.1.
Fig. 4. Polarizations of 12B with various stacks and stack orientations. Squares (points) obtained by F(B) detectors. The results, as expected, are equivalent
Substitution of X2 into the rate of this reaction leads to FM/FA, the WM form factor vs. the axial vector. The largest uncertainty is due to the factor F scaling the axial vector form factor vs. the momentum transfer. A source of information derives from 12Cψ12C* (e, eƍ) measurements, F = 0.75 ± 0.01. This leads to FM/FA= 4.3 [3]. From nuclear physics calculation F = 0.79 (FM/FA =3.0). The average F=0.77 leads to FM/FA = 3.6 ± 0.6, close to the Gell-Mann effect (FM/FA)CVC = 3.88 ± 0.12, obtained from the ī = 37.0 ± 1.1 eV width of the analogue M1 12C*ψ12C transition. There remains the induced pseudoscalar coupling to evaluate. Fp/Fp(PCAC)=1.0 ± 0.15.
The result is:
In conclusion of the long adventure described herein, only left handle, first-class conserved vector and (partially) conserved axial vector currents exist.
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Laszlo Grenacs
Conclusion Visiting the Nuclear Physics Institute of Louvain-la-Neuve, continuator of CPNL, which is still a low-energy institution, the visitor realizes that a very large population of its members work part-time or full-time oin a project for the detection of the Higgs boson at CERN. The collaboration between theory and experiment cannot be closer. There is a satellite-born experiment to detect gamma rays in our galaxy, and other actions on the frontier. The day of David Speiser’s arrival in our centre polarized our world, the direction always pointing towards frontiers. The radius of that world became larger and larger, including at present, our institute. There is a crowd which thanks David Speiser for his generous actions. I feel that in their names and in mine, I can say loudly: Bon Anniversaire Monsieur Speiser.
References [1] P.A.M. Guichon and C. Samour, Nucl. Phys. A382 (1982): 461. [2] L. Grenacs, Ann. Rev. Nucl. Part. Sci. 1985: 35, 455. [3] The result in Equ. 60 of ref. [2] corresponds in fact to F=0.75 from (e, eƍ) in 12Cψ12C*.
IN PRAISE OF ASYMMETRY Giuseppe C. La Rocca and Luigi A. Radicati di Brozolo
1 Introduction “Symmetry is the vital element of art,” wrote the famous mathematician Andreas Speiser in his booklet Die Mathematische Denkenweise, and Hermann Weyl reiterated this concept in his essay, Symmetry : “beauty is intimately connected with symmetry.” From a different point of view, the Russian mathematician Yu. I. Manin went as far as saying that symmetry “is the deepest of physical and mathematical ideas.” The reader may ask why we have chosen Asymmetry as the subject of this article we dedicate with admiration to a recognized expert on symmetry and a most dear friend of one of us. If we want to discuss asymmetry it is certainly not because we differ from the statements of the great authorities we quoted above. We only want to show that asymmetry is not devoided of positive value and we will base this assertion on Pierre Curie’s dictum, c’est la dissymétrie qui crée le phénomène. Now to create phenomena means to extend knowledge, to produce information, to help us understand nature. Does asymmetry destroy beauty? Not necessarily. The slight asymmetry of the regard de Vénus is a mysterious humanizing addition to her charm. And to Hans Castorp of Mann’s Zauberberg appeared as a revelation in his nightmarish dream during the snow storm that “the perfect icy regularity of the snow flake is a frightening element hostile to life [which explains] why the builders of the old temples secretly introduced some small asymmetries in the order of the columns”. In a brilliant and slightly paradoxical essay,1 Freeman Dyson contrasts two styles of doing science: abstract and concrete, unifying and diversifying, two complementary attitudes, typified by two cities, Athens and Manchester, and by two great scientists Einstein and Rutherford. More precisely: Unifiers are people whose driving passion is to find general principles which will explain everything. They are happy if they can leave the universe looking a little simpler than they found it. Diversifiers are people whose passion is to explore details. They are in love with the heterogeneity of nature... [and] they are happy if they can leave the universe a little more complicated than they found it.2
In their approach to symmetry we could call the mathematician Felix Klein a “unifier” and the physicist Pierre Curie a typical “diversifier.” Klein’s point of view in his famous Erlanger Programm was to show that the structure of a geometry is determined by the group of transformations that leave invariant its proposition, i.e., by its symmetry. To some extent geometries can thus be ordered according to their generality: for example in going from the Euclidean to the projective group the concept of Euclidean length is lost, but a more general, unifying point of view is gained. Curie instead investigated the new phenomena that appear 1
F.J. Dyson, Infinite in All Directions (New York: Harper & Row, 1988). We quote from the Perennial Library edition, New York, 1989. 2 Dyson, l.c., p. 43-44.
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Giuseppe C. La Rocca and Luigi A. Radicati di Brozolo
as a consequence of restricting a symmetry group to one of its subgroups. His problem is not mathematical, but strictly phenomenological: how should symmetry be violated to make possible the observation of a phenomenon. Extending the symmetry means disregarding the differences between certain elements of a set (e.g., the difference between circles and ellipses). On the contrary, restricting the symmetry allows to observe finer details (phenomena) that lose their objective significance from a more symmetrical point of view. Klein the unifier sacrificed detailed information for the sake of a deeper grand unification. Curie the experimentalist knew that information can only be gained by giving up the abstract elegant world of perfect symmetry. As Dyson says, unifiers may shed light on the state of the very early universe when the laws of physics were perfectly symmetric; diversifiers investigate the necessary conditions for the discovery of new asymmetrical phenomena as Curie himself did when he discovered piezoelectricity.
2 Curie’s Principle Pierre Curie (1859-1906) was a French experimental physicist fully immersed in the positivistic culture of the nineteenth century. He was fascinated by the optical and electromagnetic phenomena—“effects” as they were then called—that were being discovered in his time: piezoelectric effect, Wiedemann’s effect, Hall’s effect, magneto-optical effects, etc. He was the first to realize that symmetry or, better, the breaking of it, was the key to understand the origin of these phenomena: Je pense qu’il y aurait intérêt à introduire dans
l’étude des phénomènes physiques les considérations sur la symétrie familières aux cristallographes.3 Curie’s interest in symmetry goes back at least to 1880 when, in collaboration with his elder brother Jacques, he discovered the phenomenon of piezoelectricity. In the biography of her late husband,4 Marie Curie wrote that this discovery was not accidental but came as a result of an extensive experimental study of the symmetry properties of crystalline matter. During the following years he published several papers on the theory of symmetry. In one of them,5 he pointed out the different behaviour under coordinate inversion of the electric and magnetic fields. There he stated that if one considers only electromagnetic and induction phenomena it is only possible to conclude that their parity, i.e., their behaviour under inversion, is opposite. To decide the absolute parity of the two fields one must consider their interaction with matter for example the “production of electricity from chemical reactions, the electric phenomena of tourmaline, the rotation of the polarization direction by the magnetic field.”6 Among the experimental physicists of his time, Curie was an exception: he had learned group theory from C. Jordan, and he knew the work of Bravais and that of the mineralogists, Fedorov and Schonflies. He could therefore define in a precise mathematical language the concept of symmetry and of its invariants. Rather than in the abstract problem of the origin of the principles of physics, Curie, the diversifier, was interested in finding the necessary conditions for the existence of specific phenomena. By analysing several examples he established the following principle: what is
3
P. Curie, “Sur la symétrie dans le phénomènes physiques,” Journal de Physique III (1894): 393. Marie Curie Skolodowska, Pierre Curie (Paris: Payot, 1924). 5 P. Curie, l.c. 6 P. Curie, l.c. Curie implicitly assumes that electric charges are invariant under space inversion. 4
In Praise of Asymmetry
47
necessary [for the existence of a phenomenon] is that certain elements of symmetry do not exist. It is asymmetry that creates the phenomenon. In modern times Curie’s point of view has been expressed in a particularly lucid form by T.D. Lee in his essay Symmetries, Asymmetries and the World of Particles : All symmetries are based on the assumption that it is impossible to observe certain basic quantities which we call ‘non-observables’. Conversely, whenever a non-observable becomes an observable we have a symmetry violation.
Among the examples Curie analysed, the most illuminating one is perhaps the Hall effect,7 namely the creation of an electric field by the combined effect of an electric current and a magnetic field. Let us take an orthogonal frame in three-dimensional space with origin O, the z-axis parallel to a current j and the x-axis parallel to a magnetic field B. Both j and B break the symmetry O(3) of empty space, i.e., its isotropy, but the subgroups which leave invariant the polar vector j and the axial vector B are different.8 The subgroup of O(3) that keeps j fixed is the group Oz 2 O3 , i.e., the group of real orthogonal matrices A(ʌ, d) with determinant d= r1 which keep the z-axis fixed: A I , d
§ cos I ¨ ¨ d sin I ¨ 0 ©
sin I d cos I 0
0· ¸ 0¸ 1 ¸¹
The matrices A(0,-1) and A(ʌ ,-1) represent reflections in the planes (z,x) and (z,y) respectively that contain the current j. The symmetry under reversal of the direction of the yaxis is incompatible with the existence of an electric field (a polar vector) parallel to y. The invariance group of the axial vector B along the x-axis is instead SOx 2 u ^E , I ` where SOx 2 is the group of orthogonal matrices C(T ) with determinant equal to 1 §1 ¨ C T ¨ 0 ¨0 ©
0 cos T sin T
0 · ¸ sin T ¸ cos T ¸¹
that keep the x-axis fixed and E and I are the identity and the inversion in the origin. The configuration of a current j along z and a magnetic field along x is left invariant by the group Oz 2 SOx 2 u ^E , I `
^E , Rx `
where Rx is the reflection in (z,y). Rx
7
A S ,1 C S u I .
P. Curie, l.c. One of us (L.A.R.B) has discussed the Wiedemann effect in an article: “Remarks on the early developments of the notion of symmetry breaking,” in Symmetries in Physics, M.G. Doncel, ed. (Barcelona: 1987). 8 See E.P. Wigner, Gruppentheorie und Quantenmechanik (Leipzig, 1931). We quote from the English edition, Group Theory and its Applications (New York: Academic Press, 1939), p. 198 ff.
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Giuseppe C. La Rocca and Luigi A. Radicati di Brozolo
The intersection of the invariance groups of j and B does not contain the reflection in the plane (z,x) and therefore the presence of an electric field E (a polar vector) in a direction orthogonal to both j and B is not excluded. As Curie noticed the condition is not sufficient as the observation of the electric field E along y depends on the sensitivity of the measuring apparatus and the symmetry principle does not specify the strength of the electric field. Only a knowledge of the constant that multiplies the vector zˆ B ( zˆ is the unit vector along the x-axis) allows to determine the value of E.9
3 Examples of Curie’s Principle Let us apply Curie’s Principle to the symmetries described by the group Z 2 ^E, I ` , where E is the identity element and I 2 =E. For example I may represent the reflection of the space coordinates, usually denoted by P, or the exchange of particles with antiparticles, C or their product CP. Invariance under Z2 means that the two states u and uƍ=Iu are indistinguishable, whereas they become distinguishable if the Z 2-symmetry is violated. Consider for example a stationary state of an electron in a central non-Coulombian potential. The invariance under O(3) =SO(3)u Z 2 prevents the existence in a stationary state of an electric dipole moment. To test whether the weak interactions violate P invariance Lee and Yang10 suggested trying to detect a pseudoscalar quantity as for example the helicity s k , i.e., the product of the spin of a particle times its momentum, originated in a weak process. The experiments confirmed not only the violation of P invariance but also that of C and later even of CP.11 Consider for example the weak decays of positive and negative ʌ-mesons (spin zero) into
P-mesons and neutrinos. S o P v
S o P v 9
As a matter of fact, the appearance of the electric field can be easily understood on the basis of the laws of electrodynamics which also fix the value of the proportionality constant. We can imagine the current j as due to the motion of particles of charge q and number density n with velocity v, namely j =nqv. Then, the electric field E is precisely the one required to counterbalance the deflecting Lorentz force exerted on the current by the magnetic field B, therefore E v B c B j n q c . The material dependent proportionality constant 1/(n q c) is usually determined by employing a conductor shaped as an elongated bar. A current source makes a known current flow along the bar and perpendicular to a known external magnetic field. In such conditions, a voltage difference develops between opposite faces of the bar in the direction perpendicular to both the current and the magnetic field, and this is the Hall voltage which is measured as a function of the current and the magnetic field. This steady state situation is achieved when the current is turned on during a very rapid transient in which small opposite charge densities accumulate on those faces due to the deflecting Lorentz force until the latter is perfectly balanced. Hall effect measurements, in particular, can be used to determine the concentration and type (positive or negative) of the elementary charge carriers and have been of paramount importance in the study of semiconductors. 10 T.D. Lee and C.N. Yang, Phys.Rev. 104 (1956): 254. 11 C.S. Wu, E. Ambler, R.W. Harpes and R.P. Hudson, Phys.Rev. 105 (1957): 1413; R.L. Garwing, L.M. Lederman and M. Weinrich, Phys.Rev. 105 (1957):1415; J.J. Friedman and V.L. Telegdi, Phys.Rev. 105 (1957): 1681.
In Praise of Asymmetry
49
where Q and Q denote neutrinos and antineutrinos. It is found that in the rest system of the pions the momenta k of the leptons are alligned with their spins s k v
m
S
sv
o
P
kˆ s v
1 2
sP
k v
m
sv
S
o
P
kˆ s v
12
sP
The helicity k s is a pseudoscalar quantity: its measurement shows, according to Curie’s principle, that weak interactions violate space reflection invariance. The decays of ʌ+ and ʌare related by charge conjugation. The sign difference of the helicities in the two processes shows that weak interactions violate also charge conjugation invariance. The product CP of space inversion P times charge conjugation C exchanges the two processes: we could therefore suppose that weak interactions though they violate both P and C nevertheless preserve the product CP. That this is not so has been shown in 1964.12 It had been suggested in the beginning that CP violation could be due to a new “superweak” interaction. However, recent beautiful experiments have conclusively shown that CP violation is a ubiquitous feature of all phenomena controlled by weak interactions.13 If one accepts the validity of the invariance under CPT (T denotes time reversal) the violation of CP implies the violation of microscopic time reversibility, i.e., the existence of a time arrow in the structure of the fundamental equations of physics. The violation of space inversion symmetry has an interesting effect on the electronic states of crystals. Let E s k denote the energy of these states where k is the Bloch wave-vector and s the spin (the band index n is for simplicity not indicated). Time reversal (T) invariance implies E s k E s k as T reverses both k and s. If the crystal is invariant under space inversion P, it follows that E s k E s k since P reverses k but not s. Therefore, the combined action of P and T (PT) leads to Es k E s k : for every k the levels with spin up and down are degenerate. On the contrary, in crystals in which P symmetry is violated because of the asymmetrical arrangement of their atoms, in general E s k z E s k , i.e., the spin degeneracy for a given k is removed. The spin-orbit interaction in the crystal is the mechanism responsible for this effect which is presently attracting much interest as the possibility of distinguishing the two spin states could lead to the development of semiconductor spintronic devices. 12
J.H. Christenson, J.W. Cronin, V.L. Fitch and R. Turbay, Phys.Rev.Lett. 13 (1964): 138. See for a review of their and other people’s work, see M.S. Sozzi and I. Mannelli, “Measurements of direct CP violation,” la Rivista del Nuovo Cimento 26 (2003): 2. 13
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Giuseppe C. La Rocca and Luigi A. Radicati di Brozolo
A striking example of Curie’s principle comes from the comparison of the simplicity of the hydrogen spectrum with the complex structure of the alkali spectra. Both spectra arise from the transitions of a single electron moving in a central potential. In the hydrogen case the Hamiltonian is invariant under the group SO(4) which is characteristic of the Coulomb potential Vc (r)=D/r where r is the distance from the nucleus. All other central potentials, as for example those describing the motion of the valence electron of the alkali atoms, are invariant only under the group O(3), a subgroup of SO(4). Reducing the symmetry from SO(4) to O(3) gives rise, in accordance with Curie’s principle, to the richer phenomenology of the alkali spectra. More specifically: the n2-degeneracy of the n-th level of hydrogen is removed when V(r) differs from Vc (r); the n-th level of hydrogen splits accordingly into a set of (n, Ͱ) levels with Ͱ=0,1,…, n–1.14 It should be remarked that the hydrogen levels are not in general eigenvectors of the space reflection operator I (x ĺ –x, y ĺ –y, zĺ –z), whereas the (n, Ͱ) levels of the alkali atoms have a definite parity.15 Hence the orbital angular moment Ͱ and the parity w are the observable quantities (phenomena) which appear as a result of breaking the SO(4) symmetry of the hydrogen atom. The fact that parity and the orbital angular momentum do not need to be good quantum numbers for hydrogen, contrary to the case of the alkali atoms, is reflected in the way the spectral lines split under the action of a constant external electric field E0, i.e., in the Stark effect. To be definite, let us consider the transition from the n=1 to the n=2 levels of hydrogen induced by ultraviolet light polarized along E0, taken to be parallel to zˆ . The presence of the electric field partly removes the fourfold degeneracy of the n=2 level. In particular, the Ͱ=0 (even parity) and the Ͱ=1, Ͱz=0 (odd parity) states are totally mixed to form two states with a permanent electric dipole oriented, respectively, parallel or antiparallel to E0. The latter states are still eigenstates of the hydrogen hamiltonian corresponding to the n=2 level and have neither a definite parity nor a definite orbital angular momentum. They suffer opposite energy shifts which are linear in the magnitude of the electric field, whereas the remaining two states of the n=2 level having Ͱ=1, Ͱz =r1 remain unperturbed and degenerate. On the contrary, in the alkali atoms the Ͱ =0 (even parity) and Ͱ =1, Ͱz=0 (odd parity) states, i.e., the 2S and 2Pz states, are not degenerate. In the presence of E0, they can only partly mix with each other leading to an energy shift which is only quadratic in the magnitude of the electric field.
4 Spontaneous Symmetry Breaking In his papers on symmetry Curie does not mention the possibility that symmetry could be broken spontaneously without the intervention of external asymmetrical causes, as an electric current and a magnetic field in the case of the Hall effect. On the contrary, he explicitly
14
n 1
Notice that
¦ 2" 1
n 2 . Here (2Ͱ+1) is the number of levels with the same Ͱ but different
" 0
components Ͱz. 15
A detailed treatment of the hydrogen atom can be found, for example, in W. Thirring, Lehrbuch der
Mathematischen Physik, vol. 3, chap. 4.
In Praise of Asymmetry
51
states: Lorsque certains effets révèlent une certaine dissymétrie cette dissymétrie doit se retrouver dans les causes qui lui ont donné naissance.16 Though Curie was aware that large symmetry breaking causes can produce small, perhaps undetectable asymmetrical effects, he overlooked the possibility that large asymmetrical effects might arise from symmetrical causes. It seems that it was difficult for him to realize that “a symmetric problem need not have stable symmetric solution.”17 This fact, though in principle well known, was often overlooked. Indeed the symmetrical solution of a symmetrical problem which always exists is the easiest to find: it is much more difficult to prove its stability for all values of the parameters that enter the equations of motion and their boundary conditions. The symmetry change caused by the change of a scalar parameter is often the source of what Birkhoff calls “symmetry paradoxes”. Today we would say that these paradoxes originate from spontaneous symmetry breaking. Spontaneous symmetry breaking does occur frequently in nature and indeed some examples of abrupt symmetry changes had been extensively discussed in the mathematical literature before 1894. To quote just a few that occur in classical physics: the buckling of symmetrically loaded rods (Euler’s elastica), the onset of turbulence in infinitely long circular pipes, the triaxial ellipsoidal shapes of fast rotating fluid masses in selfgravitating equilibrium (Jacobi’s ellipsoids) and the even more asymmetrical shapes discovered by Curie’s colleague Henri Poincaré (Poincaré’s pears). The equations governing these problems are all nonlinear and depend upon a set ȁ of control parameters. When these parameters reach some critical values (bifurcation points),18 new solutions may appear which in general are invariant only with respect to some subgroup H of the invariance group G of the equations. To determine which of these solutions is stable is in general a difficult problem. That an equation invariant under a group G may have a solution which is invariant only under a subgroup H G seems at first sight odd. However one must bear in mind that from an H-invariant solution a whole orbit of solutions is generated by the action of the group G: the orbit of these G-equivalent solutions is of course a G-invariant concept. Each element x of the orbit is only invariant under a subgroup H x G (Hx is called the stabilizer of x, Hxx
x ): if y is a different element of the orbit, y = gx, g Щ G, then H y
gH x g 1 , i.e.,
all the stabilizers of the orbit are conjugate in G. All elements x of the orbit of solutions are a priori equally probable and so are their invariance groups H G . Which of them will appear at or above the bifurcation point can not be predicted in a G-invariant way. The choice will be dictated by some small undetectable G-breaking perturbation. In his book Science et Méthode Henri Poincaré wrote: Une cause très petite, qui nous échappe détermine un effet considérable que nous ne pouvons ne pas voir...; il peut arriver que de petites différences dans les conditions initiales en engendrent de très grandes dans les phénomènes finaux: une petite erreur sur les premières
16
P. Curie, l.c., p.127. G. Birkhoff, Hydrodynamics (Princeton: Princeton University Press,1950), p.30. 18 See: D.H. Sattinger, “Bifurcation and Symmetry Breaking,” Bulletin of the American Mathematical Society 3 (1979): 778. 17
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Giuseppe C. La Rocca and Luigi A. Radicati di Brozolo
produirait une erreur enorme sur les dernières. La prédiction devient impossible et nous avons le phénomène fortuit.19
In particular at a bifurcation point, the small, undetectable effect of fluctuations can remove the degenerary of a symmetrical set of solutions with conjugate stabilizer and cause the appearance of a particular unsymmetrical state.
5 Ellipsoidal Figures of Equilibrium Let us illustrate these general considerations in the case of the equilibrium configuration of a homogeneous self-gravitating fluid in rigid rotation around a fixed axis. The problem was first discussed by Newton in connection with the earth’s rotation under the assumption that the departures from the spherical shape are small. Soon afterwards (1742) the problem was solved in the general case by MacLaurin who showed that for any angular velocity, i.e., for arbitrary ellipticity, the equilibrium figure is an axially symmetrical ellipsoid. In view of the axial symmetry of the hydrostatic equilibrium equation for a rigidly rotating fluid mass, the axial symmetry of MacLaurin’s solution was accepted as a natural, and probably (Lagrange) necessary, consequence of the problem’s symmetry. It was Jacobi (1834) who first showed that for a sufficiently large value of the angular velocity there exist an infinity of non axially symmetric solutions.20 The equilibrium of a homogeneous incompressible self-gravitating fluid in rigid rotation around a fixed axis with angular momentum J is governed by the equation:21 grad p x M x
0
(1)
where x =(x1, x2, x3) are the coordinates in the rest frame of the fluid, with x3 in the direction of J and x1 and x2 the coordinates in the equatorial plane; p(x) is the pressure and M x M g x M c x is the sum of the gravitational and the centrifugal potential, the latter depending on the square of the angular momentum J 2. The boundary condition is p x
S x
0
(2)
where S(x) = 0 is the boundary surface of the fluid. We take S x t 0 inside the fluid. With condition (2) we obtain an equation for S :
M >S @ x, J 2
const.
(3)
which depends on the scalar parameter J 2. Using the expressions for ijg and ijc we easily verify that the ellipsoids
19
H. Poincaré, Science et Méthode (Paris: Flammarion, 1908), p. 68. For a brief history, see S. Chandrasekhar, Ellipsoidal Figures of Equilibrium (New Haven: Yale University Press, 1969). 21 We follow the paper by D.H. Constantinescu, L. Michel and L.A. Radicati, “Spontaneous symmetry breaking and bifurcations from the Maclaurin and Jacobi ellipsoids,” Journal de Physique 48 (1979): 147. 20
In Praise of Asymmetry
Fig. 1. Ellipsoidal equilibrium solutions. Polar eccentricity squared (a) and equatorial eccentricity squared (b) versus angular momentum squared. Solid curve: Maclaurin sequence; dashed curves Jacobi sequence. The dots indicate the Maclaurin-Jacobi bifurcation and the next bifurcation on each branch.
53
54
Giuseppe C. La Rocca and Luigi A. Radicati di Brozolo
3
xi2
i 1
Hi
S x 1 ¦
satisfy equation (3). Here H1 , H2, H3 are the squares of the three semiaxes of the ellipsoid ordered and normalized as follows: H 3H 2 H 1 1
H 3 d H 2 d H1 ,
In terms of the H’s we define the polar (e2) and equatorial (K2) eccentricities: e2
1
H3 , H1
K2
1
H2 H1
These are plotted against J 2 in fig.1a and fig.1b. As fig.1a shows, an axially symmetrical solution exists for any value of J 2 whereas (see fig.1b) triaxial Jacobi ( K 2 t 0 ) ellipsoids exist only for J 2 t J c2
0.384436 .22 In presence of viscosity the MacLaurin ( K 2
0 ) solution
23
becomes unstable. Each MacLaurin ellipsoid is invariant under the group D 2h of the equilibrium equation whereas a Jacobi triaxial ellipsoid is only invariant under the subgroup D2h generated by the reflections in the symmetry planes. We thus see that symmetry breaking phenomena can be caused by the change of a symmetrical parameter J 2, a confirmation of Garret Birkhoff’s statement.24
6 Symmetry and Information Curie’s assertion that asymmetry creates phenomena means that asymmetry brings to light differences between certain elements of a set that from the point of view of a larger symmetry are equivalent, i.e., objectively indistinguishable. Thus, by violating the symmetry of a set we produce information, that is, we decrease the original entropy. A simple example may clarify the connection between symmetry breaking and the gain of information.25 Consider two equal empty vessels connected by a pipe. The geometry is such that the system is invariant under the group Z2 whose elements are the indentity and the rotation by ʌ around the vertical axis. By filling the vessels with two different monoatomic gases at the same pressure and temperature and with the same number of atoms the Z2 symmetry is broken and an order is established which allows the two vessels to be distinguished. If the two gases are now allowed to mix the symmetry is restored while the disorder, as measured by the entropy per particle, increases by ln2. As a second example consider a system of free magnetic dipoles with spin 1/2 in an external magnetic field. At infinite temperature the entropy per dipole is maximum (equal to ln2) and the system is
22 23
The angular momentum squared is measured in units
12 25
Gm 3 a1 a 2 a 3 1 3 where ai are the semiaxes.
See Chandrasekhar, l.c., p. 95. See G. Bertin and L.A. Radicati, “The bifurcation from the Maclaurin to the Jacobi sequence as a second order phase transition,” Astrophysical Journal 206 (1976): 815. 25 L.A. Radicati di Brozolo, “Chaos and Cosmos,” in: Symmetries in particle physics, I. Bars, A. Chodos, Chia-Hsiung Tze eds. (New York: Plenum Press, 1984). 24
In Praise of Asymmetry
55
invariant with respect to the reversal of each dipole. As the temperature is lowered the symmetry is broken until at zero temperature the system reaches a completely ordered state. A similar situation occurs in ȕ-brass which is a binary alloy of approximately 50% Cu and 50% Zn.
Fig. 2. Crystal structure of ȕ-brass in the ordered (top) and disordered (bottom) phases
At zero temperature, Cu and Zn are arranged, respectively, on the sites of two interpenetrating cubic sublattices, each Zn atom being located at the center of the cube formed by the eight nearest Cu atoms, and vice versa (see Fig.2, top). This completely ordered low temperature phase corresponds to a simple cubic lattice with two different atoms, one Cu and one Zn, per unit cell. Such a phase has a vanishing configurational entropy as the knowledge that the site at the origin is occupied, say, by Cu gives a complete information on which atoms occupies any other site, even at very large distances. As the temperature increases, Cu and Zn begin to interchange their positions, i.e., some Cu atoms can be found on sites of the Zn sublattice and vice versa (see fig.2, bottom), with a corresponding loss of information on the geometrical arrangement of the atoms, i.e. an increase of configurational entropy. At a temperature Tc ~ 470qC , Cu and Zn get completely mixed, i.e., at any site of either sublattice the probabilities of finding a Zn atom or a Cu atom become equal. As a consequence, in this phase the crystallographic symmetry of the alloy, i.e., the symmetry of a “virtual” crystal in which each site is occupied by a half Zn and half Cu “average atom,” is higher and corresponds to a body-centered cubic lattice with only one “average atom’’ per unit cell. In the high temperature phase, the higher symmetry is due to the fact that, on average, each site of the crystal is equivalent to any other one, the distinction between the two sublattices occupied at low temperature, respectively, by Cu and Zn having disappeared. In other words, the knowledge of which sublattice belongs to Zn and which one to Cu is totally lost. As the temperature decreases, the symmetry changes abruptly at T=Tc when the difference between the occupation probabilities of Cu and Zn on each sublattice deviates continuously from zero. In fact, the order-disorder transition in ȕ-brass is a second order phase transition and precisely that difference plays the role of the order parameter. As soon as it is different from zero, the two sublattices become distinguishable, information on which
56
Giuseppe C. La Rocca and Luigi A. Radicati di Brozolo
one is predominantly occupied by Cu or Zn is gained and the symmetry of the high temperature phase is broken. As a consequence, new Bragg peaks appear (labeled by g in Fig. 3) in the X-ray diffraction spectra (or even more clearly in the neutron diffraction spectra) of ȕ-brass, the intensity of which is proportional to the difference between the scattering form factors of Cu and Zn and to the difference in the occupation probability of the two sublattices by, respectively, Cu or Zn. These intense and sharp Bragg peaks are due to the long range order of the low temperature phase in which Cu and Zn alternate regularly in space (or at least the difference in their occupation probability persists) at all distances making the two sublattices distinct. The intensity of the Bragg peaks typical of the low temperature phase approaches zero when the temperature increases up to Tc. In the high temperature phase, in which the two sublattices are no longer observable, such Bragg peaks are not present. Their intensity can be taken as a direct measurement of the order parameter of the transition. Above Tc, in general, only much broader and much less intense spectral features are observable in place of the Bragg peaks labeled by g in Fig.3. These weaker features are due to the diffuse scattering induced by short range order, i.e., by the correlations in the positions of Cu and Zn atoms that still occur among nearest and next-nearest neighbours, but rapidly vanish at large distances.
Fig. 3. Schematic diffraction spectra of ȕ-brass. From top to bottom the temperature is lowered, the crystallographic symmetry changes at T=Tc from the high symmetry phase to the low symmetry one. The Bragg peaks labelled by h are allowed in both phases, whereas those labeled by g are typical of the asymmetric phase, only diffuse scattering being observable above Tc in their place.
In Praise of Asymmetry
57
7 Coda We began this article by quoting Hermann Weyl: “beauty is intimately connected with symmetry’’ and we ended up by showing that the symmetric state of brass (and the conclusion is of general validity) is a completely disordered, chaotic state to which the concept of beauty hardly applies. Of course, Weyl knew that empty space, for example, or in general a homogeneous system, is more symmetrical that any of the beautiful works of art he showed in his book. But he never thought of grading the beauty of works of art according to the “degree” of their symmetry. His primary interest was to show that their beauty reflects the abstract, rational elegance introduced in space by the action of a group. The point of view of Curie (who, as far as we know, Weyl never mentions) was more prosaic: his aim was to establish the condition for the origin of phenomena which he discovered to be the violation of an undifferentiated Ursymmetrie which contains in potentia all symmetries. Perhaps, Curie would have preferred to speak of Chaos, the formless, chaotic, immense vacuum where according to Hesiodus morphogenesis took place:26
Instead, the more idealistic Hermann Weyl, and also our good friend David Speiser, prefer to ignore the initial obscure and turbulent morphogenetic process and are happy to contemplate the graceful dance of the Heliconian Muses.27
26 27
Hesiodus, Theog. 116, LBL edition. Hesiodus, Theog. 1-4, LBL edition.
AN OBSERVATION ABOUT THE HUYGENS CLOCK PROBLEM Donal Hurley and Michael Vandyck
1 Introduction In February 1665, Huygens discovered that two pendulum clocks attached to the same support synchronise themselves spontaneously. In his own words: … horologiorum duorum … miram concordiam observaveram, ita ut ne minimo quidem excessu1 alterum ab altero superaretur. … Pendebant horologia ex suo quodque tigno 3 circiter pollicum crassitudine quorum extrema sedibus duabus pro fulcris innitebantur [1].
Moreover, each pendulum soon settles in an oscillatory movement that is half a cycle out of phase with respect to the other one: Horum reciprocationes ita componebantur ut simul semper accederent ad se mutuo ac simul recederent; alia ratione ire coactae non manebant sed illuc ultro revertebantur, ac deinde invariatae permanebant [2].
In the hope that spontaneous synchronisation would improve the accuracy of timekeeping at sea, Huygens investigated this phenomenon in detail, and, by the first of March, he had correctly identified its cause as being the capacity of the support to transfer vibrations between the pendula: Utrique horologio pro fulcro erant sedes duae quarum exiguus ac plane invisibilis motus pendulorum agitatione excitatus sympathiae praedictae causa fuit, coegitque illa ut adversis ictibus semper consonarent [3].2
What Huygens seems not to have done, however, was to design a single clock with two pendula, especially arranged for the latter to be close together, for instance, one behind the other. The possibility of building a twin-pendulum clock would have to wait until the early 1780s before being put in practice by the French clockmaker Antide Janvier and, slightly later, by Abraham-Louis Breguet, who was of Swiss origin. In addition, after gaining experience with twin-pendulum clocks, Breguet produced pocket watches with two balancewheels.3 The concept of twin-oscillator clocks and watches lay then dormant for almost two hundred years. In the year 2000, however, Stephan Gagneux and Beat Haldimann designed and built a twin-pendulum clock [8–11] after being inspired by one of Breguet’s clocks [6], which Haldimann had seen at an exhibition. Furthermore, around the same time, a watch with two balance-wheels was built by François-Paul Journe [7, 12].4 (There too, Breguet’s work provided the inspiration [7].) The two-century gap was thus bridged, and, since then, the 1
Our emphasis. For more history, including the role of other scientists, see [4, 5]. 3 For details, see [6, 7]. 4 The fact that Stephan Gagneux lives in Basel, and that the Gagneux-Haldimann clock, as well as Journe’s chronometer, were first exhibited at the Basel Fair is certainly worth emphasising in a Festschrift dedicated to David Speiser. 2
60
Donal Hurley and Michael Vandyck
horological world has witnessed several other endeavours along the lines of time-keepers based on spontaneous synchronisation of twin-oscillators.5 During the “period of dormancy”, roughly between 1800 and 2000, theoretical investigations of Huygens’s observation seem to have been scarce (to the best of our knowledge), with the notable exceptions of Korteweg’s classical work [5] and of that of Bleckhman [13]. On the other hand, at the National University of Ireland, Cork, we had always retained an interest in the Huygens problem, mainly from the point of view of a particular aspect of Classical Mechanics, which will be specified hereafter. This had led us to a generalisation of the Huygens problem, which seems not to have been discussed in the literature. More recently, the original Huygens problem has attracted a good deal of attention [14, 15]. However, the aforesaid generalisation still remains unpublished; yet it contains an important feature that is absent from the original problem. Moreover, this feature may have a certain relevance for the behaviour of the Gagneux-Haldimann clock of the year 2000. Given the well-known contributions of David Speiser to the field of History of Science, and his long-standing interest in the foundations of Mechanics (classical and relativistic), it is proper and fitting that we honour him, on the occasion of his eightieth birthday, by introducing here what we shall call the ‘Generalised Huygens Problem’. It is not our aim to present a complete study of this question; we rather wish to draw attention to the existence of this generalisation. We leave to the reader the pleasure of adapting the powerful methods of [14, 15] (when appropriate) to treat our generalisation. In the forthcoming Section 2, we shall define the generalised Huygens problem, and set up its proposed mathematical description. Then, in Section 3, we shall display a simple extreme case, which will emphasise the new feature present in the generalised problem. Finally, we shall give a few indications on how the general case may be handled.
2 Fundamental Equations For our purposes, it is not necessary to repeat the whole analysis of the original Huygens problem. It will suffice to recall [5, 14] that, in the original problem, one considers two pendula attached to a horizontal beam, which is allowed to move longitudinally, i.e., parallel to its own length. Let the displacement of the beam with respect to a certain origin be denoted by Y . The beam is supposed to be subjected to the horizontal linear restoring force FB = íkBY , where kB is the horizontal stiffness constant. The problem under investigation is then to determine the motion of the coupled system of the beam and the two pendula. On the other hand, the Gagneux-Haldimann twin-pendulum clock of the year 2000 has its two clockworks side by side, and its two pendula are hanging “one behind the other on a single bracket between them” [8]. (See the photographs in [6, 8, 9].) In this geometry, the two suspension points of the pendula suffer two different displacements when the bracket undergoes a deformation, because the deflection of the bracket may (ideally) be considered as vanishing at the point where the bracket is attached to the case of the clock, whereas it is maximal at the free extremity ͭ of the bracket. Moreover, ͭ may, a priori , move both vertically and horizontally.
5
For recent developments, see [7].
An Observation about the Huygens Clock Problem
61
Fig. 1. Geometry of the generalised Huygens problem
Let the yˆ -axis be horizontal and pointing positively to the right (fig. 1). Let the xˆ -axis be vertical and pointing positively downwards. Let the undeflected bracket be a horizontal segment, perpendicular to the ( xˆ , yˆ ) plane, with its point of attachment located at the origin. Let also the coordinates of the extremity ͭ of the bracket be denoted by (X, Y ). They are non-vanishing when the bracket is deflected. By using the equations of the “beam-bending” problem, it is established (and analysed critically) in the Appendix that the coordinates X i , Yi , 1 d i d 2 , of the suspension point of Pendulum i may be related to (X, Y ) by Xi
D ic X , Yi
Di Y ,
(2.1)
where D i and D ic are constants lying between zero and one. To recover the original Huygens problem, in which the beam moves exclusively along the yˆ -axis, and the suspension points of the two pendula experience the same displacement as the beam, it suffices to restrict attention, in (2.1), to the special case
D ic
0 , D1
D2 1.
(2.2)
Henceforth, we shall call ‘general’ the Huygens problem for which no restriction is imposed on D i and D ic . Let us now assume that the deflection of the bracket is so small, compared to its length, that the pendula still move in vertical planes. It is then clear that a point Pi situated at a distance di along the rod of Pendulum i has the coordinates xi yi
X i d i cos T i
D ic X d i cos T i
(2.3)
Yi d i sin T i
D i Y d i sin T i ,
(2.4)
when Pendulum i makes an angle și with respect to the vertical. The rest of the treatment is straightforward, and need only be sketched. For the Lagrangian L, we adopt the form
62
Donal Hurley and Michael Vandyck
L
L pendula Lbracket 1 ¦ mi x i2 y i2 ¦ mi gxi 2 i i 1 1 2 2 P X Y k A X 2 k B Y 2 PgX , 2 2
(2.5)
(2.6)
in which g, mi, µ, kA and kB denote, respectively, the acceleration of gravity, the mass of Pendulum i, the mass of the bracket, and the two stiffness constants of the bracket. The first term on the right-hand side of (2.6) expresses the kinetic energy of the pendula. Consequently, in full generality, the constants di present in x i2 and y i2 , through (2.3) and (2.4), must be taken as the radii of gyration. The second term describes the gravitational potential energy of the pendula, so that the constants di present in xi are the distances between the suspension point and the centre of mass of each pendulum. The remaining terms model, phenomenologically, the bracket as a two-dimensional anisotropic harmonic oscillator6 in the gravitational field. In addition to L, we need the generalised dissipative forces ĭ. They may be derived either from the virtual work įW that they produce [16], or (equivalently) from Rayleigh’s Dissipation Function [16] for viscous friction. If the air is at rest with respect to the ( xˆ , yˆ ) axes, one writes: ĭ X GX ĭ Y GY ¦ ĭ i GT i { GW i
ī XGX YGY ¦ J i x i Gxi y i Gy i ,
(2.7)
i
where ī and Ȗi are the constants of viscous friction of the bracket and the pendula. To diminish the amount of algebraic labour, let us now concentrate on identical, simple pendula of length ȟ. It is elementary to replace (2.3), (2.4) with d i " in (2.6), (2.7), and to express Lagrange’s equations, to find the linearised equations of motion as being
2 X X : A2 X
g~
WA
m 2 Y Y : B2Y M WB zi
2
W
§
(2.8)
2
¦ D i2 ¨© zi W
z i Y 02 z i Y
i
2 Y
W
· zi ¸ ¹
0, 1 d i d 2 .
0
(2.9)
(2.10)
In (2.8)–(2.10), the constants IJA, IJB and IJ denote the relaxation times of the bracket and the pendula, whereas ȍA, ȍB and Ȧ0 stand for the angular frequencies of the undamped bracket and the pendula. (They are easy to relate to the quantities ī, Ȗ, m, µ, Įi, D ic , g, ȟ, kA and kB.) Furthermore, M, g~ and zi are defined by
6
Some of the brackets used by our students for their experiments performed in 2002 were deliberately supported by springs.
An Observation about the Huygens Clock Problem
63
(2.11)
M { P m¦ D i2 i
§ · g~ { g ¨¨ P m¦ D ic ¸¸ i © ¹
§ · ¨¨ P m¦ D ic2 ¸¸ i © ¹
(2.12) (2.13)
z i { "T i D i .
It is enlightening, in order to improve our physical understanding of the problem, to multiply (2.10) by mĮi, which yields m
d2 dt
D iY "T i 2 m d D iY "T i mgT i 2 W
dt
(2.14) 0.
The interpretation of (2.14), as Newton’s Second Law for the pendula, is quite clear, because the combination D i Y "T i represents the (linearised) displacement of Pendulum i along the yˆ -axis, with the contribution D iY from the motion of the bracket duly taken into account. The form (2.14) also justifies intuitively why, in (2.10), the velocity Y of the bracket appears together with the relaxation time IJ of the pendula. It would be incorrect to attempt to obtain the equations of motion by expressing first Lagrange’s equations in the absence of friction, and then adding “by hand” a term proportional to the velocity, in the hope of describing dissipation. Such a naive procedure would yield, instead of (2.8)–(2.10), the equations7
2 X X : A2 X
WA
2 Y Y : B2Y
WB
zi
2
W
zi Z02 zi
g~
(2.15)
m ¦ D i2 zi M i Y, 1 d i d 2.
(2.16) (2.17)
We shall state later some of the consequences of (2.15)–(2.17), and compare them with those of (2.8)–(2.10). Note that one ingredient is still missing from (2.8)–(2.10), namely the influence exerted by the clockworks on the pendula, which may be considered as described by external driving terms due to appear on the right-hand side of (2.10). We have methods [18] to incorporate this influence, but we prefer not to complicate the matter unnecessarily by introducing any driving terms here, because the new feature that we would like to point out occurs even in the absence of such terms. Moreover, owing to the fact that the system (2.8)–(2.10) is linear, it may be solved by elementary methods. As stated in [14], a reasoning based on the undriven problem is “instructive, has the essential ingredients, but is incomplete since it ignores the energy input altogether.” The
7
This aspect of the Huygens problem provides a realistic example, much appreciated by students [17], illustrating the importance of the method of Virtual Work in a dissipative situation.
64
Donal Hurley and Michael Vandyck
reader might like to adapt to the generalised Huygens problem the treatment of the driving forces that was employed in [14] for the original Huygens problem. At this stage, it is already clear that the vertical movement of the bracket decouples from the pendula, and may be ignored in what follows. Furthermore, if the two equations (2.10) are replaced by their sum and their difference, and use is made of the change of variables (2.18)
2Ȉ { z1 z 2 , 2ǻ { z1 z 2 u { Y 0t ,
c{
d , du
(2.19)
Q { Y 0W ,
one obtains Ȉ cc
2 2 Ȉ c Ȉ Y cc Y c Q Q 2 ǻ cc ǻ c ǻ Q
(2.20)
0
. 0.
(2.21)
The interest of keeping the constants Įi arbitrary in all this development will now become obvious when the change of variables (2.18), (2.19) is performed in (2.9). Indeed, with the notation (2.22)
Q B { Y 0W B , G { Y 0 ȍ B 2
E { 1 m D12 D 22 M
P M H { m D 12 D 22 M ,
(2.23)
the equation for Y reads Y cc
2 1 Yc Y QB G
E 1 §¨¨ Ȉcc
2 · § 2 · Ȉ c ¸¸ H ¨¨ ǻ cc ǻ c ¸¸ Q ¹ © Q ¹ © § 2 · E 1 ¨¨ Ȉcc Ȉc ¸¸ Hǻ, Q ¹ ©
(2.24) (2.25)
where, for the last step of (2.25), we have employed (2.21). When D1 D 2 1 , the parameter H , defined in (2.23), vanishes. Then, the equations (2.20), (2.21) and (2.25) reduce to those of the original Huygens problem. In this special case, the horizontal motion of the bracket couples to the conjunction mode Ȉ of the pendula but not to their opposition mode ǻ, as it is well known [14]. The situation is crucially different when D1 z D 2 : As a result of (2.23), the parameter H no longer vanishes, and the function ǻ acts as a driving term on the right-hand side of (2.25). Such a phenomenon does not occur in the original Huygens problem. It is important to emphasise that, in general, the driving term ǻ is transient, because it follows from (2.21) that ǻ is given by ǻ u e u Q A cos ru B sin ru , r 2 { 1 Q 2 ,
for two arbitrary constants A and B.
(2.26)
An Observation about the Huygens Clock Problem
65
In order to illustrate the effect of the ǻ-term in (2.25), we are going to study, in the next section, an extreme case where the influence of ǻ is predominant. Later, we shall show how to adapt this approach to render it applicable in general. REMARKS
x
Although we shall constantly refer to the functions Ȉ and ǻ as describing the conjunction mode and the opposition mode, it is important to realise that the terms ‘conjunction’ and ‘opposition’ must be understood in a generalised sense, which befits the generalised Huygens problem (and reduces to the ordinary sense in the original Huygens problem). Indeed, if one returns to the definition (2.13), (2.18) of Ȉ and ǻ, one sees that these quantities depend on Įi, so that, in general, it is not true that ș1 and ș2 have opposite signs when Ȉ vanishes. For future convenience, let us introduce new functions, S and D, by putting 2S { " T1 T 2
D1 D 2 Ȉ D1 D 2 ǻ 2 D { " T1 T 2 D1 D 2 Ȉ D1 D 2 ǻ.
(2.27) (2.28)
The functions S and D represent “conjunction” and “opposition” in the ordinary sense, and coincide with Ȉ and ǻ (respectively) if D1 D 2 1 .
x
If the equations (2.15)–(2.17) are used instead of (2.8)–(2.10), it follows that (2.20), (2.21) and (2.25) are replaced by the set 2 Ȉ c Ȉ Y cc 0 Q 2 ǻ cc ǻ c ǻ 0 Q § 2 1 2 · Y cc Y c Y E 1 Ȉ cc H ¨¨ ǻ ǻ c ¸¸ G QB Q ¹ ©
6 cc
(2.29) (2.30) (2.31)
On the other hand, for identical compound pendula, the correct analogues of (2.20), (2.21) and (2.25) read
6 cc
2 2 Ȉ c Ȉ Y cc cY c Q Q 2 ǻ cc ǻ c ǻ Q 2 1 Y cc Yc Y QB G
(2.32)
0 0
(2.33)
E 1 §¨¨ Ȉ cc ©
· ª º 2 2 cȈ c ¸¸ H «ǻ 1 c ǻ c» , Q Q ¹ ¬ ¼
(2.34)
where c denotes the ratio of the radius of gyration of the pendula to a length characteristic of the friction. (The value of c for simple pendula is c = 1.) It is enlightening to see (2.29)–(2.31) emerge from (2.32)–(2.34) in the non-physical limit c ĺ0, the other parameters remaining unchanged. This is all we wish to say about the case of compound pendula and about its relationship with (2.29)–(2.31). In what follows, we shall return exclusively to (2.20), (2.21) and (2.25).
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Donal Hurley and Michael Vandyck
3 Special case Consider the case where the pendula themselves are undamped (Q = ), but the bracket experiences friction (QB < ). In this limit, the effect of ǻ is greatly exacerbated, because, by virtue of (2.26), the influence of ǻ on the right-hand side of (2.25) is no longer transient, but permanent. To decouple the equations of motion (with Q = ), we begin by adding (2.25) to (ȕ ©1) times (2.20), and we find Ȉ in terms of Y and ǻ as being
1 E Ȉ
EY cc
2 1 Y c Y Hǻ . QB G
(3.1)
When this expression for Ȉ is inserted in (2.25), and use is made of (2.21), the resulting equation, which determines Y , is
EGY cccc
2 Y ccc Y c 1 G Y cc Y QB
0.
(3.2)
It would be simple, but tedious, to solve (3.2) exactly. In the present note, however, it will suffice to treat the problem perturbatively at first order in Q B1 . This is elementary because, when QB is infinite, the characteristic polynomial of (3.2) is a biquadratic equation. The final answer for Y reads Y u e u Q A cos r u B sin r u e u Q A cos r u B sin r u ,
(3.3)
in which A± and B± are integration constants, whereas Q± and r± are known in terms of the parameters ȕ, į and QB . One has, for instance, Q# Ѥ r Q B
1 G F , G 1 G 1 # F 2 E
>
F { 1 4EG 1 G 2
@
12
.
(3.4)
It is not excluded, a priori , that Q+ or Qí might turn out to be negative (or even complex), but, for the values of ȕ and į that apply to the experiments that our students performed in 2002, both Q+ and Qí are real and positive. Consequently, when u tends to infinity, Y tends to zero. In this limit, (2.26) and (3.1) imply that Ȉ becomes Ȉo
H
ǻ
E 1 D 22 D 12 A cos u B sin u . o 2 D 1 D 22
(3.5) (3.6)
In terms of the function S, the result (3.6) may be reformulated as S o D1D 2
D 2 D1 A cos u B sin u . D12 D 22
(3.7)
Therefore, if Į1 and Į2 are different, the conjunction modes Ȉ and S do not disappear entirely when u tends to infinity.
An Observation about the Huygens Clock Problem
67
A similar method may be employed when both Q and QB are finite. In the next section, we shall provide a few indications about the treatment of this more general case.
4 General Case Let us again begin by adding (2.25) to (ȕ © 1) times (2.20), now without assuming that Q is infinite. The analogue of (3.1) reads then
1 E Ȉ
EY cc 2>1 QB 1 E Q @Y c Y G Hǻ .
(4.1)
When (4.1) is inserted in (2.25), and use is made of (2.21), the following equation is obtained for Y :
EGY cccc 2G >1 QB 1 2E Q @Y ccc ®1 G 2 G QB 1 Q Y c Y
0,
¯
4G >1 QB 1 E Q@½¾Y cc Q ¿
(4.2)
which should be compared with (3.2). The solution of a differential equation of the kind of (4.2) is always a linear combination of trigonometric functions multiplied by exponentials. Therefore, at least for some values of the parameters, the function Y is expected to decay exponentially. Given that, by virtue of (2.26), the function ǻ also decays exponentially, (4.1) predicts that the conjunction mode Ȉ decays exponentially as well. This observation is not surprising, because the presence of dissipation in both the bracket and the pendula renders inevitable that the whole system tends, asymptotically, to a state of rest. It would be feasible to calculate the relaxation times for Ȉ and ǻ, so as to determine which of the modes decays faster, but our method based on undriven oscillations shows here its limitation. To obtain genuinely useful information on the generalised Huygens problem, beyond the case Q = , it is necessary8 to incorporate a driving mechanism, as in [14].
5 Conclusion In this note, our purpose was simply to point out how Huygens’s original problem of the spontaneous synchronisation of two pendulum clocks attached to the same beam may be generalised to take into account circumstances in which the suspension points of the two pendula may receive unequal movements from their support. In this case, the motion of the support couples not only to the conjunction mode of oscillation of the pendula (as in Huygens’s original problem), but also to their opposition mode. In the limit of frictionless pendula attached to a damped support, the system is then seen
not to settle purely in the opposition mode, but a certain residual conjunction is exhibited.
8
There is a conjecture that one may, perhaps, be tempted to make. It arises from noting first that the driving mechanism, in the original Huygens problem, prevents the opposition mode ǻ from decaying (for some values of the parameters). Moreover, in the (undriven) general Huygens problem, the assumption Q = produces exactly the same effect on ǻ. The case Q = , in the (undriven) general Huygens problem, may thus be more realistic than one might have thought initially, and there is some plausibility in the belief that a behaviour of the kind (3.6), (3.7) is possible. A detailed investigation of this matter is essential before a firm conclusion can be reached.
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Donal Hurley and Michael Vandyck
This has only been proved here, by virtue of (3.6), (3.7), in the absence of any driving mechanism transferring energy to the pendula. The reader might like to investigate whether the potentially new phenomenon described above may be overshadowed by the influence of a driving mechanism, or whether it may be due to an unrealistic assumption made in our mathematical treatment.
6 Appendix Consider the classical problem of the weightless cantilevered beam, as described in standard textbooks. Here, we shall adopt the treatment of Feynmann in his Lectures on Physics [19]. Let a weightless horizontal beam B be partially embedded in a thick vertical wall W, so that a known length L of B is projecting out of W. A force F is applied vertically downwards to the free extremity F of B. Then, the vertical deflection z that B experiences at a distance x from W is given by [19] (6.1)
kFx 2 3L x ,
6z
where k is a constant involving Young’s modulus and the cross-section of B. (The radius of curvature of B is assumed to be much larger than the thickness of B.) If Z denotes the deflection of the free extremity F , it follows from (6.1) that 3Z
kFL3 ,
(6.2)
which enables one to eliminate kF from (6.1), and obtain Zx 2 3L x L3 .
2z
Consequently, a point P situated, along the beam, at a distance x P a known constant cP , is deflected by the amount 2zP
Z 3 cP c P2 .
(6.3) c P L from the wall, for
(6.4)
This result may be rewritten as zP
DPZ
D P { 3 c P c P2 2,
(6.5) (6.6)
and it establishes that the deflection of an arbitrary point along the beam is proportional to the deflection of the free extremity. It was used in Section 2 under the form (2.1). If one wishes to assess the practical applicability of (6.5), (6.6), one must distinguish carefully the main statement (6.5) from the evaluation (6.6) of ĮP . It is clear that the latter relies on the details of the curve (6.1), and thus on all the hypotheses underlying the ‘beambending’ problem. For instance, the beam is partially embedded in a thick wall, which means that one of its extremities is rigorously immobile. In the case of a twin-pendulum clock, it is conceivable that the bracket supporting the pendula, instead of bending significantly, may remain rather rigid, and may partially communicate the motion of the pendula to the case of the clock, which may then oscillate (microscopically) from side to side. In other words, (6.6) might not be accurate in all
An Observation about the Huygens Clock Problem
69
circumstances, because the hypothesis that one of the extremities of the bracket is rigorously immobile may not be well satisfied. An oscillating clock case would bring us back, very largely, to the original Huygens problem. Be that as it may, the original problem is compatible with (6.5), as we have shown in Section 2, provided appropriate values are adopted for the parameters D i and D ic in (2.1). A realistic situation would probably involve a certain amount of bending of the bracket, combined with a certain amount of oscillation of the clock case, so that the values of D i and D ic in (2.1) would correspond neither exactly to the predictions of (6.6), nor exactly to those of the original problem. At any rate, as soon as any bending takes place, the suspension points of the two pendula receive different movements from the bracket, which is precisely what (6.5) is allowing for. Consequently, one may consider (6.5), and more generally (2.1), as a reasonable prototype of such a situation. In practice, however, the values of D i and D ic depend crucially on the details of the suspension system of the pendula, and they may be so close to those of the original Huygens problem that the potentially new feature contemplated here may not be observable. From the available photographs, it seems that the bracket of Janvier’s twin-pendulum table regulator [6] is supported in its centre, and has one pendulum on each side, whereas the Gagneux-Haldimann clock [6, 8, 9] of the year 2000 has a bracket supported at one extremity only, with one pendulum closer to the supported extremity than the other one. Therefore, our generalisation might have a certain relevance for the latter clock, as opposed to the former. Note that Stephan Gagneux’s second clock, which was exhibited at the 2003 Basel Fair, has a completely redesigned bracket, which appears to be supported at the two extremities (at least inasmuch as one can judge from a photograph [7]). One would thus expect to be much closer to Huygens’s original situation than for the first clock.
7 Acknowledgements It is a pleasure to thank Dr. T. Treffry (British Horological Institute) for material and information about the behaviour of twin-pendulum clocks. The authors also benefitted from discussions with Prof. M. Mortell, Prof. C. O’Sullivan, Dr. P. Cronin (NUIC) and Dr. P. Delaney (National Microelectronics Research Centre), as well as with Messrs. M. Schulver and R. Murphy (NUIC). Finally, Mr. R. Gillen (NUIC) is gratefully acknowledged for processing Fig. 1.
References [1] Huygens, C. 1665. Œuvres complètes de Christiaan Huygens, vol. 17 (The Hague: Martinus Nijhoff), p. 183. [2] Huygens, C. 1665. Œuvres complètes de Christiaan Huygens, vol. 17 (The Hague: Martinus Nijhoff), p.184. [3] Huygens, C. 1665. Œuvres complètes de Christiaan Huygens, vol. 17 (The Hague: Martinus Nijhoff), pp.185, 186. [4] Huygens, C. 1665. Œuvres complètes de Christiaan Huygens, vol. 5 (The Hague: Martinus Nijhoff), pp. 241, 243, 244, 256. [5] Korteweg, D. 1906. Les Horloges Sympathiques de Huygens. Archives Néérlandaises, Série II, tome XI : 273–295. [6] Delfs, T. 2004. Dual Oscillators. Horological Journal 146 : 288–290. [7] Delfs, T. 2004. Dual Oscillators. Horological Journal 146 : 323–325. [8] Gagneux, S. 2000. Double-pendulum clock. Horological Journal 142 : 149.
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[9] Gagneux, S. 2000. Double the excitement with a double-pendulum clock. Horological Journal 142: 384–387 [10] Gagneux, S. 2001. Double pendulum. Horological Journal 143 : 39. [11] Gagneux, K. 2001. New double-pendulum clock. Horological Journal 143 : 57. [12] Randall, A. 2002 F.P. Journe’s Chronomètre à Résonance. Horological Journal 142: 92–94. [13] Bleckhman, I. 1988. Synchronization in Science and Technology. New York: ASME Press. [14] Bennett, M., Schatz, M.F., Rockwood, H. & Wiesenfeld, K. 2002. Huygens’ clocks. Proc. Roy. Soc. Lond. A 458: 563–579. [15] Klarreich, E. 2002. Discovery of Coupled Oscillation Put 17th-Century Scientist Ahead of his Time. SIAM News 35, 1. [16] Goldstein, H. 1980. Classical Mechanics (Reading, Massachusetts: AddisonWesley), pp. 23–24. [17] The National University of Ireland, University College Cork, Summer Examinations 2004, 3rd Science (honours) and B.Sc. (combined honours)–Physics, Classical and Relativistic Mechanics, Question 4. [18] Hurley, D. and Vandyck, M. 2004. The Pendulum Clock as a Dynamical System. (unpublished.) [19] Feynman, R. 1965. The Feynman Lectures on Physics, vol.2 (Reading, Massachusetts: AddisonWesley), pp. 38.13–38.17.
The History of Science
DANIEL BERNOULLI AND LEONHARD EULER ON THE JETSKI Frans A. Cerulus
Preliminary My friendship with David Speiser goes back to 1952, when I came to Basel and joined him as a doctorand of professor Fierz. I felt at first a little lost in the foreign town with an unfamiliar language, where some people mistook my origin “Belgien” for Berlin. David became my most important guide to the microcosm of the Rhenan city. I learned soon, however, that he could not be trusted to convey faithfully the general Basel way of thinking; he displayed an uncommon independence of judgment and a most rare combination of a mathematically oriented mind with a deep understanding of art and culture. Some years later our lives ran again closely parallel when we were both at the university of Louvain where we could build up a theory group of enthusiastic young people. When the university underwent its mitosis each of us found himself as the nucleus of a new theoretical cell but our friendship continued. Many years later, when we had left active teaching and were living a day’s journey away he invited me to enter the grand project of the Bernoulli edition. He gave me a ticket into the fascinating world of the history of ideas of which he had become an expert. The work on Bernoulli and the warm hospitality of David and Ruth have given me some of my happiest moments of the last years.
1 Introduction Ours is the jet age. Jetplanes and rockets are simply part of our world; we even call our upper classes the jetset. But just how old is the jetmotor? The ancient Chinese, who invented fireworks, knew the principle. Hero of Alexandria (first century AD) described a toy moved by the reaction of a steamjet. And what about the neolithic fisherman who threw a stone backwards from his small boat and felt himself moving forward? Somewhere between the discovery of the principle in the dim past and the construction of the first motor for an aeroplane1 we can tag the first mathematical treatment of jet propulsion: Chapter XIII of the book Hydrodynamica written in Latin by Daniel Bernoulli (1700-1782) and printed in 1738 ([1]; [2, pp. 398-424]). It is a famous book; authors of textbooks like to refer to it (few have read it). Less well known but more elaborate, correcting even the former, is a memoir by Leonhard Euler (1707-1783) of 1743 ([3]; [4, pp. 190228]). These texts, until the end of the twentieth century, were hidden in antiquated books or issues of old journals which only a few libraries preserve in their vaults. Moreover the main texts are in Latin and use a now-obsolete mathematical notation. Due to the modern critical editions of the works of Euler and the Bernoulli family those texts have now become widely 1
In the 1930s, independently by Frank Whittle and Hans von Ohain.
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Frans A. Cerulus
available. We should recall here that David Speiser contributed to the first and was a tireless general editor of the second. The present essay is meant as a small complement to those and attempts to explain to the present-day reader in a familiar language the timeless ideas of the founding fathers of jet propulsion. For an historical essay we start unconventionally: first we bring a brief theory of the jetski which, because of its extreme concept, is a most instructive example of propulsion by a reaction motor. In the historical sections proper we shall then illustrate a number of points by referring to that example.
2 The jetski as model 2.1 What is it?
Fig. 1: L. Euler (left) and D. Bernoulli (right) on the jetski
The word “jetski” was originally the brand name of a new fun-boat made by the Kawasaki Motor Company. Most of this kind of boats are manufactured by firms producing motorcycles or snowmobiles. Their riders find similar thrills on the water as bikers on the road (fig.1). A jetski is a small shallow craft, two to three meters long, carrying a powerful engine which drives a pump that sucks in water at the bottom and propels it in a fast jet in the air, to the rear (fig.2).
Daniel Bernoulli and Leonhard Euler on the Jetski
75
Fig. 2: Sketch of the axial flow pump in a jetski: the multi-cylinder two-stroke engine is to the right but not shown
By reaction the jetski is driven forward and glides over the water at high speed. It is in fact much jet and barely a boat. On the sporty models the driver is standing up, on more sedate models he is seated and there is room for one or two passengers. 2.2 The idea of the jetski We shall simplify to the utmost and consider only a jetski moving straight on level water at a steady speed. In that case the forward thrust by the jet is exactly equal to the resistance of the water. This last force is not a simple matter, because the boat offers a different section to the water at different speeds: its bow rises with increasing speed until only a small surface stays in the water as the boat skis along. We shall assume that in the steady regime this small surface stays constant and, for the range of speeds considered, the resistance is proportional to the square of the velocity; this kind of law was assumed by Bernoulli and Euler2 and will serve to illuminate their works, even though a naval architect might want a more elaborate theory. So we shall take for the force resisting the advance of the boat: FR
1 AU v 2 2
(1)
where ȡ is the density of the water; the parameter A is an area related to the “wet surface” of the boat; the factor 1/2 is traditional and used by Bernoulli. In addition the water, in our model, is a fluid without viscosity, just as in the models of the eighteenth century. We refer the movements to the x-axis of a reference system at rest with respect to the sea or lake; the jet is spouting in the positive direction, the boat moves in the negative direction. We shall assume an engine of W watts, which produces a jet, streaming into the air, with a velocity of c meter/second (referred to the boat) through a nozzle of area S, propelling the boat with a velocity ív.
2
It was derived by Bernoulli for the resistance due to motion; it is equally valid for the friction but for the wake formation it can only be assumed for sufficiently small intervals.
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Frans A. Cerulus
The two parameters W and A are the only input and the goal is to find the optimal choice of S (and hence of c) in order to maximize v, i.e., to find the most efficient way of using the engine’s power. It is essentially a one-dimensional problem in point mechanics and the two conservation laws of energy and momentum should allow to find the solution. 2.3 Theory Per second a mass of water m SUc leaves the nozzle having the velocity cív with respect to the sea. The same amount is taken in, at velocity zero. The change of momentum per second, i.e., the force on the water, is then SUc c v ; the reaction force on the ship is equal and opposite to this and that in turn cancels the reaction force of the waters (1), i.e., (2) 1 2 2
SU c c v
AU v
The motor power is used to accelerate the mass Sȡc in one second to the velocity cív and to move the boat at velocity ív against the force (2); this requires the following amounts of power: (3) 1 2 W jet
2
Wboat
SU c c v
SU c c v v
(4)
and the sum of the two is the total power W:
(5)
1 AU v 2 c v 2W 2
(6)
W
1 SU c c 2 v 2 2
Combining (2) and (5) we obtain:
which can be solved for c as a function of v: c v
4W 1 v AU v 2
(7)
Substituting this c in (2) gives for the area of the nozzle: S v
1 v2 A 2 c v c v v
(8)
2.4 An example Taking W = 109000 watts and A = 0.0080 m2, which are typical figures for a fast jetski, and ȡ = 1000 kg/m3 as the density of water, we show two plots of the exit velocity and the nozzle area as functions of v, the seaspeed (figs. 3 and 4). To reach a higher speed, one must pick a larger nozzle and a slower exit velocity
Daniel Bernoulli and Leonhard Euler on the Jetski
77
Fig. 3: Plot of the nozzle velocity c (v) vs. the seaspeed (in m/s). The maximal velocity in the example is 30.1 m/s.
Fig. 4: Plot of the nozzle area S (v) vs. the seaspeed, in m2; at the maximal velocity S becomes infinite
2.5 The maximal speed There is a theoretical limit to the velocity v, viz. when v = c, which yields also the highest possible efficiency, which can be defined as: (9) 2 vc Wboat , v W
a function of
v c
1
c
which grows monotonically from 0 to 1 as v increases from 0 to c.
Turning to (2) and solving for v c
v c
we obtain · S §¨ A 1 2 1¸ , ¨ ¸ A© S ¹
showing that the efficiency is really a function of effective cross section of the boat.
S A
, the ratio of the nozzle area to the
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Frans A. Cerulus
The theoretical maximum itself does depend on the ratio
W AU
, as can be seen from (7): if
c= v , then v max
3
2W . AU
(10)
Of course, that value can never be reached because in the limit for c (v ) ĺ v the nozzle area becomes infinite, as is clear from (1) or (5). In practice one should try to come as close as possible, because the curve c(v) stays near zero until close to its asymptote for v = vmax. It is then a matter of practical engineering, with a real fluid, how large a nozzle one can afford. Within our hypotheses, in the example above we obtain vmax =30.1 m/s; choosing as working point v = 29.0 m/s (i.e., 104.4 km/h), one gets c =35.8 m/s, S = 128 cm2 and an efficiency of 0.895.
3 The Hydrodynamica 3.1 The basics We said already that the principle had been known for ages. Newton himself experimented with globes filled with wet gunpowder; the gases produced by the burning powder, escaping through a small hole, pushed the globe in the opposite direction of the flames.3 Daniel Bernoulli recalls this example in the introduction of chapter XIII of his Hydrodynamica, which is believed to be the birth certificate of jet propulsion.4 We shall report the original arguments of Bernoulli, go through his correspondence in subsequent years and see how his ideas developed, to arrive at the essays both he and Leonhard Euler wrote for the 1753 competition of the Paris Académie des Sciences. When presenting the ideas of those authors we shall translate them into the language of modern mechanics and therefore, as a rule, not take over the original notations. In notes and remarks we shall point out the merits and, sometimes, the shortcomings of their approach. The first draft of the Hydrodynamica was written in St Petersburg during Bernoulli’s stay there as a member of the Academy of Sciences, from 1725 to 1733, when he returned to his home town of Basle, to become a professor at the University. In Basle he reworked his manuscript—he may have added chapter XIII—and succeeded finally in having it published in Strasbourg in 1738. The main theoretical tool in those investigations is the conservation of energy (as we now say) in the guise of the equivalence of motion and height. This means the following: a mass m falling from rest over a distance a does this in a time t such that a
1 2 gt , 2
(g is the acceleration of gravity, on the average and at sea level equal to 9.81 m/sec2.) It attains in this time a velocity (11) v 2 ga . 3 4
Principia [5] Lib.II, Prop. XXXVII, in fine.
Actually it was not, as was discovered by G.K. Mikhailov (see [2] p.445). It was J. Allen who proposed the idea in 1730; his text is reprinted in [2, pp. 449-463].
Daniel Bernoulli and Leonhard Euler on the Jetski
79
This is equivalent to mv 2
2amg . For Bernoulli and his contemporaries this meant that the vis viva, i.e., mv2 (a measure of the “power of motion”) is equal to the weight of the mass, i.e., mg, times double the distance travelled, 2a. A force, viz. the weight, transported over a distance means work done and the last formula means: the vis viva of a moving mass of velocity v is equal to the work done by lifting the mass over the distance 2a, where a is the distance needed, in a free fall, to reach the velocity v. Both Bernoulli and Euler take in fact the height as their basic variable. In the Hydrodynamica, Ch. XIII, for example, the jet velocity is denoted 2 A : A is the height and the argument assumes a choice of units making g = 1. Euler calls the velocity ¥a, assuming g 12 .
It would take a century, generations of students battling with factors 2 in exercises, and the spreading of the use of algebraic numbers before this was generally rewritten as 1 mv 2 amg 2
0
and interpreted as
x x
1 mv 2 is the kinetic energy acquired in the fall; 2
íamg is the loss of potential energy in the fall;
and the rule is: in a free fall the sum of kinetic and potential energy is constant. The choice of the constant is irrelevant for a description of the phenomena, in our example it is zero. Let us apply this, following Bernoulli, to a simple example in hydrodynamics: a large reservoir, filled with water of density ȡ up to a level a and a small hole in the bottom, of area S (fig.5). The water spouts forth from the hole; what is its velocity v ?
Fig. 5. The reservoir is kept filled to the level a from the bottom, with outlet of area S
In one second a volume of water escapes through the hole, equal to Sv; its mass is ȡSv. That mass has velocity v, its kinetic energy is therefore:
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Frans A. Cerulus
U
1 U Sv.v 2 2
2
Sv 3 .
In the same second the same volume of water has disappeared from the tank, its level has been lowered. If the tank is large this change is imperceptible and we can assume that the water has disappeared from the height a, i.e., the total potential energy of the water in the tank (we count the height from the bottom upwards) is lowered by ȡSvga. By the law of of the conservation of energy (or, in Bernoulli’s mind, the equality of vis viva and weight lifted over the equivalent height): U Svga
2 ga
U 2
Sv 3 .
v.
(12)
i.e., the velocity of the escaping water is the same as the velocity of a mass falling over a (cf. (11)), the height to which the tank is filled. This law had been experimentally discovered by Torricelli (1608-1647).5 Bernoulli asks now: if accelerating the mass ȡSv in one second from rest (in the tank) to speed v (leaving the hole) is due to a constant force K, how large is this force? Because force = mass times acceleration we have (13) K U S v2 . The reaction force, which tends to push the tank in the opposite direction, must have the same value, by Newton’s third law that action equals reaction. Next comes a paradox: at the position of the outlet the water pressure is simply ȡga, the weight of a column of water reaching to the level of the water in the tank. But the pressure driving the jet (i.e., the force divided by the area) is U v 2 2U g a , that is, it corresponds to a column twice as high. Bernoulli mentions he discussed the point with Riccati (1676-1754), who commented,
distinguendum esse statum quietis a statu motus.6 To assume that in the tank the water is at rest is true on the whole, except in the vicinity of the outlet where some streaming must be present. But how to define this “vicinity”? Bernoulli, very cleverly, makes an artificial vicinity that is well defined. In order to solve the paradox, Bernoulli considers a tank with a spout of length l, ending in an opening of area S (fig. 6).7 As long as the outlet is closed there is a hydrostatic pressure p=ȡga at the exit D. Upon opening the tube at D the water starts to flow reaching ultimately 2 ga . its stationary value v
5
Evangelista Torricelli, De motu gravium in Opera geometrica, Florence, 1644. One ought to distinguish the state of rest from the state of motion. 7 Bernoulli takes a spout of varying section but shows that this variation does not influence the result. 6
Daniel Bernoulli and Leonhard Euler on the Jetski
81
Fig. 6. Putting a horizontal tube to the outlet allows to calculate the flow
Consider now, says our author, the increase of momentum of the water over one second, in the first instants after the hole is unplugged. It will consist of two terms:
x x
the water having left the spout in one second, which carries a momentum ȡSv2, being a cylinder of section S and length v; the water in the spout carries momentum ȡSlv and hence its increase is U S l dv . dt
The force pushing the water is therefore: K U Sv 2 U S l
dv dt
.
(14)
Bernoulli invokes now the law of conservation of energy: in one second the water leaving the spout carries away a kinetic energy 12 U S l v 3 ; the water in the spout has kinetic energy 1 2
U S l v 2 , whose change in one second is U S l v dv . dt
Those amounts are compensated by the loss of potential energy gȡSva: gU S va
USlv
dv 1 U S v3 . dt 2
or, equivalently USl
dv dt
gU S a
1 U S v2 . 2
Substituting in (14) Bernoulli obtains for the accelerating force K
gU S a
1 U S v2 , 2
(15)
which is a function of time, tending to an asymptotic value; so is v. This solves the paradox: for the steady state we saw that (cf. (12)) the second term 12 U S v 2 12 U S 2 ga and it turns out that indeed K
2 gU S a .
Bernoulli shows here that the force on a fluid in tubing has a double origin: head and flow. He uses (15) to demonstrate that the reaction force, which is equal and opposite to K,
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Frans A. Cerulus
is present as soon as the opening in D is unplugged and is growing until the steady value of the velocity v is reached. In the sequel he will concentrate on steady flow. The modern reader is tempted to go one step further: the force K on a portion of fluid in a tube is the difference of the pressure p at both ends (times the cross section), e.g., K CD
S pC p D .
From there it is but a small step to the relation between pressure, velocity and head that textbooks call the Bernoulli equation; we shall meet it in (38) when discussing Euler’s essay. Bernoulli himself does not make use of the concept of internal pressure; in the Hydrodynamica his notion of pressure is linked to the concrete picture of a column of water rising in a small vertical tube applied to a vent in the flow tube. 3.2 Ship propulsion Having established the above theory, Bernoulli turns to an application: Mentem aliquando subiit, posse ea quae de vi repellente fluidorum, dum ejiciuntur, meditatus fueram, quaeque hic maximam partem exposui, utiliter applicari ad novum instituendum navigationis modum; neque enim video, quid obstet, quo minus maximae naves sine velis remisque eo modo promoveri possint, ut aquae continue in altum eleventur effluxurae per foramina in ima parte navis, faciendo ut directio aquarum effluentium versus puppem spectet. Ne quis vero opinionem hanc in ipso limine rideat, ceu nimis insulsam, e re erit nostra argumentum istud accuratius excutere & ad calculum revocare: utile enim esse potest multisque disquisitionibus geometricis est fertilissimum.8
Bernoulli remarks first that, applying the reaction force Kr (13) for propulsion, one should observe that it not only takes power to continuously refill the tank (raising water over the distance a), but that taking water on board of a moving ship supposes an additional force; indeed, that water has to be accelerated to the velocity of the ship. He confesses he had at first overlooked the effect. Suppose the ship has velocity v, says Bernoulli. The jet has velocity c and consumes therefore Sc cubic meters9 of water per second. Accelerating this to a speed v requires a force (16) Kw U S c v , pushing the ship backward. The net propulsion force is therefore: K ship K r K w U S c 2 cv
Suppose now we are given a ship, with a watertank filled to a height a and a crew who keep it filled, raising the water over a. Let the total power of the men be W watt, i.e., newton meter per second (Bernoulli uses cubic feet of water raised over one foot in one second10).
8
“It came sometime to my mind that what I had conceived about the repulsive force of fluids being ejected — and which I expounded here for the greatest part — could well be put to good use to set up a new kind of seafaring. I don’t see what could be objected to this: the largest vessels could be set in motion, without sails or oars, by waters being continuously raised to a high spot to flow out of holes in the lower part of the vessel, arranging that the flow would be directed aft. It will behove me to investigate exactly that argument and call in the calculation, that nobody would laugh from the onset at our conviction as utterly foolish: it can be useful and is most fruitful for many geometrical investigations” ([2] pp. 413-414). 9 Bernoulli speaks of course of cubic feet; in our modern paraphrase we use metric units. 10 This unit happens to coincide with the power an average man can deliver over an extended time.
Daniel Bernoulli and Leonhard Euler on the Jetski
83
The jet consumes Sc m3/s of water per second; to raise this quantity over a requires the power ȡSca watt; Bernoulli equates the two amounts:11 (17) W U gSca watt but because v
v2 2g
2 ga , i.e., a
, this means that S, the area of the outlet, has—according
to Bernoulli—to be dimensioned as S
(18)
2W
Uc
3
How fast a ship, powered by W watt, will sail depends on the resistance the sea opposes to its motion. We know since the nineteenth century this is compounded of three effects: displacement of the water, friction, and wake. Bernoulli has contributed markedly to the understanding of the first, neglects the second, and ignores the third. He assumes that the resistance is given by (1), where A is seen as roughly equal to the cross section of the ship below the waterline, but with corrections due to the form of the hull; A, in his 1753 essay [6], is presented as a parameter to be determined experimentally. When sailing at a constant speed this resistance is equal to the driving force, i.e.: (19) 1 2 2 AU v
U S c cv
1 AU v 2 2
§1 v 2W ¨¨ 2 ©c c
· ¸¸ ¹
2
and eliminating S by using (18) (20)
Bernoulli asks now what is the best choice of c (i.e., the optimal height a of the tank) to obtain the highest possible seaspeed v. At this maximum the differential dv is zero; differentiating at this point formula (20) he obtains: (21) 1 v c 2 and introducing this in (19) yields A S 4 as the most favourable outlet. Still at those optimal conditions one finds c
1
1
§ 2W · 3 ¨¨ ¸¸ ©US¹
§ 8W · 3 ¨¨ ¸¸ © AU ¹
which requires a tank filled to a height 2
a
11
2 § W ·3 ¨ ¸ g ¨© AU ¸¹
This would be correct if the ship remained still, e.g., when measuring the reaction force by a dynamometer or similar device. Once the ship has velocity v, accelerating the water to that velocity will add a term in the power balance; Bernoulli forgets here his own remark. Johann Albrecht Euler will point out the error but both his father and he omit the term (15) for the force. In addition the velocity of the ship influences the velocity of the jet. A third point lost sight of is that a ship sailing sets the sea in motion and this needs power as well; C.Truesdell made this last remark before [10] p.XXIX. Those points will be elaborated in 3.4.
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3.3 The trireme of Bernoulli The data for the trireme that Bernoulli uses as an example are taken from a communication by Chazelles [8] who investigated the motion of a trireme with a crew of 260 oarsmen. He found that if the ship were towed with a force equal to the weight of 72 pounds it had a speed of 2 feet/second. This allows to determine A. In metric units12 this amounts to a force of 346 newton and a speed of 0.650 m/s; consequently A = 1.638 m2. Earlier in the Hydrodynamica we are told, on the basis of experiments,13 that a man can produce 109 watt for many hours; our men represent therefore a total power of 28 340 watt. From his formulae Bernoulli computes as optimal performance of his hypothetical jetpropelled trireme:
x x x x
a = 1.18 m (height of tank); S = 0.409 m2 (area of nozzle); c = 4.81 m/s (speed of jet); v = 2.405 m/s or 8.66 km/h (seaspeed).
Each mariner has to pour Sc/260 = 7.6 liter per second in the tank, by pumping or otherwise. Bernoulli has made an estimate what speed the 260 men could give to the trireme by conventional rowing. In his later publication [6] he corrects the estimate from the Hydrodynamica, which was based on an erroneous size of the oars, and he finds the trireme could reach 7-1/5 feet/second, i.e., 8.48 km/h. In the Hydrodynamica he states that, to his mind, the jet propulsion cannot be dismissed as much inferior to the conventional oars.
Hydronamica
This article
area of outlet
M
S
height of water
A
a
velocity of jet
2A
velocity of ship
2B
v
density of water
1
ȡ
“height of resistance”
C
1 A
acceleration gravity
1
g
c
2 ga
Table 1: The notations used in the Hydrodynamica and their equivalents in the present article.
12
The translation of the old units in the metric system is tricky, because the value of feet and pound vary with the region. As our purpose is only to bring out the theoretical arguments the small differences between e.g. pied du roy and Rheinischer Fuss is unimportant. So we based our transcription on the old Paris system of units used in chapter XIII of [1]. 13 [2], p.311.
Daniel Bernoulli and Leonhard Euler on the Jetski
85
3.3.1 The trireme as a jetski. For curiosity we compute, using the formulae of section 2.3, what would have been the performance of the trireme according to those. The theoretical limit lies at a seaspeed of v = 3.259 m/s. Choosing as working point v = 3.00 we find:
x x x x x
a = 1.12 m (height of tank); S = 0.93 m2 (area of nozzle); c = 4.69 m/s (speed of jet); v = 3.00 m/s or 10.8 km/h (seaspeed); Efficiency: 0.78.
3.4 The method redeemed 3.4.1 In the frame of the ship. The formula (17), which we noted was correct in the static case, is also applicable to moving ships provided we refer all velocities to a reference frame in which the ship is at rest. This is easily visualized if we think of a boat in a fast flowing river of velocity v; its propulsion system just cancels the effect of the flow and the boat does not move with respect to the bank. The jet exerts a reaction force on the ship: (22) p j SUc 2 and requires a power (23) 1 SUc 3 2 Stopping the water that comes in requires a force and the reaction (pushing the ship backwards) is: (24) p w SUcv The sum of the two previous forces is the force on the ship (25) p s p j p w SUc c v c W jet
which cancels the force 12 AUv 2 of the flowing water on the ship (compare (2)). As the ship is at rest this force is not displaced, no work is done and no power is required. The incoming water brings kinetic energy to the ship at the rate of (26) 1 c Wwater SUcv 2 2 reducing its power requirements by the same amount. The crew has only to add the necessary work to the incoming kinetic energy of the water in order to sustain the jet.
c Wcrew Wwater
c W jet
or Wcrew
c Wwater c W jet
1 SUc c 2 v 2 2
as was found in (5). The method used by Bernoulli was sound, but one term was omitted: viz. (26) for the power. The point to be made is that all velocities should be taken relative to the same frame of reference; here we chose the frame of the ship, in (2.3) and in the next section it is the frame of the sea. 3.4.2 In the frame of the sea. Bernoulli argues as if lifting the water and expelling it in the jet were a two-step process: first taking the water on board, say in an auxiliary tank B at sea level, and then lifting it to level a in the tank A, from which it spouts forth. If we follow this
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Frans A. Cerulus
argument in the frame of the sea we can distinguish the following forces and power consuming processes: Jet. Accelerating the mass Sȡc from tank B, with velocity ív to the jet with velocity cív in one second, we need a force Sȡc 2, the reaction force on the ship is íSȡc 2. The power expended is (27) · 1 §1 c W jet SUc 2 ¨ c v ¸ SUc 3 SUc 2 v ¹ 2 ©2 Water aboard. Accelerating water from the sea to tank B, i.e., from zero to ív needs a force whose reaction on the ship is Sȡcv. The power needed is (28) 1 Wwater SUcv 2 2 Ship. From those two effects the ship experiences a force (29) SUc c v which balances the reaction force of the sea; this is what we found in (2). Maintaining the velocity ív requires the power Wship SUc c v v ;
(30)
this is what was found in (4). Total power. The sum of those three terms gives the total power: 1 c Wwater Wship Wtotal W jet SUc c 2 v 2 2 as was found in (5) and in the previous subsection.
4 Bernoulli and jet power after 1738 4.1 Cogitations and letters There is much evidence in his correspondence that he kept thinking about those problems. In a letter, presumably from 1738, he proposes to his friend Gabriel Cramer (1704-1752), professor of mathematics at the university of Geneva, to perform experiments on the lake; fig. 7 is a partial copy from his draft. He imagines a pump in a small boat, sucking in water through a number of holes in the side; those are covered with valves when the piston is pushed down and the water is expelled through a single opening towards the rear of the boat, well under the waterline. He projects to mount a set of four pumps, two to each side; the four pistons fixed to a single arm, such that the downstrokes of one pair are simultaneous with the upstrokes of the other. He mentions this project in a letter to Euler (May 24, 1738). (In those years there was an intense correspondence between Leonhard Euler, Johann Bernoulli and Daniel Bernoulli on hydrodynamical questions.) Euler tried—without success—to have similar experiments performed in St Petersburg. Apparently Cramer had difficulties organizing the project, but two years later a younger colleague from Geneva takes over: Jean Jallabert (1712-1768), and on July 6, 1740 Bernoulli repeats his proposal to him.
87
Daniel Bernoulli and Leonhard Euler on the Jetski
4.2 Experiments on the lake of Geneva Jallabert reports some time later (only an undated draft is conserved) on a simplified experiment where only one pump was used. It proved disappointing because the man working the pump reached a speed of about two feet per second, much less than he could achieve by rowing. Other experiments, done by a Mr Rivaz14 on the lake were a little better, but stayed still well under the normal rowing speed. After 1742 Bernoulli seems not to have followed up the project. Those experiments differ of course from the idea in the Hydrodynamica because the jet is not produced into the air but in the surrounding water. One could remark that this induces a current at the rear of the boat which, by Bernoulli’s equation, will induce a pressure sucking the boat back. At any rate, the interaction of the jet with the water will not allow to use the simple mechanics that worked for the jet in air, and there is nothing astonishing in those negative experiments. 4.3 Suppléer à l’action du vent In 1752 the Paris Académie des Sciences put up for the 1753 competition the following question: La manière la plus avantageuse de suppléer à l’action du vent sur les vaisseaux soit en y appliquant les rames soit en employant quelque autre moyen que ce puisse être.
The prize was won by Daniel Bernoulli [6] and Euler [3] obtained an accessit,15 the academy judging it was the next best contribution [7, p.36]. The particular merit of Bernoulli lies in his systematic use of the concepts of power and efficiency in the discussion of the various means of propulsion. He points out that e.g., in a rowing boat only part of the power expended by the rowers goes into the effort needed to displace the water, with unavoidable subsequent eddies and waves, to make room for the advancing boat. This is given by the product of the force (1) and the velocity, i.e., W advance
1 AU v 3 2
(31)
This quantity is related to the mechanics of bodies moving through fluids and there is no hope to improve on it. The blades of the oars are moved through the water, with a velocity that can be calculated, set the water in motion and absorb power according to the same law. On the other hand punting a boat does not and is therefore a much more efficient way of propulsion. Bernoulli thoroughly analyzes the conventional oars in galleys (i.e., triremes) and proposes a number of alternatives of his own invention. He adds a short note on the possibilities of steam as a substitute for human musclepower. On every occasion he stresses the power aspect of the problem, and he has in general a very modern view of the sources of energy (he says “sources of work”) their interchangeability and the general problem of their efficient use. Probably this made the judges of the Académie prefer Bernoulli to Euler, whose arguments are centred on maximizing the thrust of the propulsion system. 14
Mr Rivaz was a watchmaker, mechanic and amateur scientist from “St Gingoux” (actually StGingolph), near Vevey, on the lake of Geneva. He invented a clock that worked without being wound up. 15 The information in [4] p. IL on the prize is mistaken.
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Fig. 7. A page from the letter of Bernoulli to Gabriel Cramer with his instructions for the experiments on hydro-propulsion. The pump should be three feet high and one foot wide. The intake is on the side, with valves which are not shown. The piston is moved between L and C. Four pumps are planned, linked by a yoke pivoting in O; the men push (or pull) in M and N. Courtesy of the Bernoulli Edition
89
Daniel Bernoulli and Leonhard Euler on the Jetski
The inventions of Euler, paddle wheels and a genuine propeller, were effectively developed and used in the industrial age; those of Bernoulli never left his drawing board, but that the Académie could not foresee. On the subject of jet propulsion Bernoulli was very brief in his memoir: … si M.Bernouilli (sic) avoit examiné la chose suivant nos principes, il auroit jugé comme moi, qu’il faut attendre moins d’effets du travail des hommes qui pompent que de ceux qui rament; car en adoptant toutes les proportions qu’il trouve les plus avantageuses, je remarque que tous les effets inutiles sont au moins trois fois aussi grands que l’effet utile, de sorte que l’effet utile n’est que le 1/4 de l’effet entier…16
We are amused at the way the author of the Hydrodynamica criticizes himself under the cloak of anonymity imposed by the Paris competition. But we are also puzzled, because we can only guess how he arrived at this conclusion, which is opposed to what he wrote five years earlier. One hypothesis is the following. In the Hydrodynamica he thought mainly of forces. But in the memoir of 1753 he has the efficiency in mind; a plausible way of reasoning suggests itself:
x
the power for the jet is (unconsciously taken in the ship’s frame, cf. (23)) 1 SU cc 3 2
cc W jet
x
power for the intake is (unconsciously taken in the sea’s frame, cf. (28)) 1 SU cv 2 2
cc W water
x
(32)
propelling power (cf. 30)
(33)
SU c c v v .
(34) The last formula follows immediately from his correct formula (19) for the force by multiplying force and velocity. Making the ratio of Wship to the sum of the three terms gives W ship
17
Wship Wsum
vc 2 1 2 vc vc 2 1 vc
Substituting the alleged optimum ratio vc 12 from the Hydrodynamica, one obtains an efficiency of 2/7=0.286, close to the announced 25%. Of course, this is only an hypothesis; another one was mentioned in [7, p.71].
5 Euler’s De promotione navium sine vi venti 5.1 An original memoir and a masquerading translation On the subject of propulsion by reaction, Euler, in his memoir of 1753 [3], was much more complete than Bernoulli in the competing essay. Both essays were published by the 16 “…had Mr Bernouilli examined the thing according to our principles, he would have estimated as I do that one should expect less effect from the work of the men who pump than from the men who row; because, adopting all the proportions he finds most favorable, I remark that all the superfluous effects are at least three times larger than the useful effect; hence the useful effect is only the 1/4 part of the whole effect…” [7, p. 213]. 17
In Bernoulli’s notation: 2 MA 2 M AB .
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Frans A. Cerulus
Académie after an outrageously long time: 1771 for Euler. In the meantime he had moved back to St Petersburg from Berlin (1766) and his first son, Johann Albrecht (1734-1800), had also become a mathematician, with some help from his father. In particular Johann Albrecht published under his own name an article in French (subtitled “translated from Latin”)[9], which refers loosely to the discoveries of his father but is on the whole a translation of the latter’s memoir of 1753. We remark en passant that by this memoir Leonhard Euler could claim to be the inventor of the ship’s propeller. 5.2 Derivation of the basic equations The memoir is in two parts. The first has the title (in Latin) “On the forces that arise from percussion of the water” and is concerned with oars, paddle wheels and a propeller. The second part works out the idea of the jet motor published in Bernoulli’s Hydrodynamica. This is never mentioned in the text for the prize of 1753, but in the translation of Johann Albrecht that reference is gallantly acknowledged. Euler starts the second part by deriving the results of the Hydrodynamica according to the theory of hydrodynamics he had just completed. Clifford Truesdell has admirably sketched the arduous path to this discovery, from the point-mechanics of Newton to the field description of continuous fluids; see [10, pp.XLI-XLVI]. In the memoir of 1753 Euler presents the judges of the Paris competition with the application of that theory to the simplest case: water flowing in a tube, a one-dimensional problem. Euler was convinced that deriving the results from the Newtonian principles of mechanics was a better starting base than the equivalence between vis viva and height, as used by Daniel Bernoulli. The editor of those works of Euler, Walter Habicht, has given a modern paraphrase of the argument in [4], pp. IL-LX; we shall sketch very briefly the gist of the reasoning.
Fig. 8. The derivation of the hydrodynamic equation; the inlet of the tube is at the origin; the independent variable—besides the time—is the distance measured along the tube, which has length L
One considers a curved tube in a vertical plane whose central line is given by the vector & function x s , where s is the distance from the origin along the curve (fig.8).
Daniel Bernoulli and Leonhard Euler on the Jetski
91
& We denote the velocity field of the water in the tube by v t,s and the unit vector along & the tangent by h s . The area of the lower exit is S and the ratio of the area of the cross section at s to the exit area is 1/Ȝ(s ). The acceleration of a particle of fluid contains two terms: & & & dv t , s wv wv , v dt ws wt
a term due to the rate of change at the given point in space and a term due to change because the particle is taken along by the flow; this is the essential idea of the theory; because of the second term the theory is non-linear. Newton’s laws relate the acceleration to the forces. The forces on the liquid are those exerted by the curving walls, by the weight and by the gradient of pressure. Because the water is & incompressible the speed at a given distance s , i.e., v t,s is simply given by the exit speed
v (t) and the ratio of the cross sections at s and at the exit, which we called Ȝ (s) and the equation allows to relate the pressure at the origin, p(t), with the exit velocity v (t) (L is the length of the tube, a the difference in height between origin and end). One obtains the equation: (35) dv t L 1 2 ª º p t U « ga ³ O s ds v t 1 O 0 » . dt
¬
0
2
¼
Fig. 9. A simple case: a tube of uniform section with a piston applying a pressure p at the upper end and at the lower end a nozzle of area S
If we consider the simplest case: a piston exerting a constant pressure p at the origin, a tube with a constant cross section and a nozzle of area S at the end (fig. 9) we obtain: dv 1 O2 2 S v dt 2
p
U
ga
Taking the water initially at rest this can be solved (it is a solvable type of Riccati equation) v t
2 p U ga
U 1 O2
ª 1 tanh « « 2S ¬
º §p · 2¨¨ ga ¸¸ 1 O 2 t t 0 » . » ©U ¹ ¼
and we see that the exit velocity tends to a stationary value
(36)
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Frans A. Cerulus
v
2 p U ga 2
(37)
,
U 1 O but that the rate at which the steady flow is reached is inversely proportional to the area of the nozzle; this is a point to be kept in mind in the applications.
For the stationary regime the equation of Euler goes over in p s U gy s
1 U v 2 s constant 2
(38)
(y(s) is the height, counting downwards from the origin, v(s) = Ȝ(s)v is the stationary velocity at the distance s). This is exactly the so-called equation of Bernoulli in full generality, derived from Newton’s principles of mechanics. The flow in the tube will tend to force the tube to the outside of the bend: this is the reaction force. For the steady flow this is simply & & * (39) K U Sv 2 h L O 0 h 0 and for ship propulsion only the horizontal component is effective. By (37) or, more generally, by (38) v is related to the input in the hydraulic system.
5.3 Euler’s system 1 We adopt again the notations of the earlier sections: c is the speed of the jet, v the speed of the ship, W the power of the crew. The first system Euler considers is the one Bernoulli proposed in the Hydrodynamica: a large and broad tank at the rear of a ship, filled to a level a, discharges water through a narrow slit of area S at the bottom of the tank; see fig.10.
Fig. 10. The Euler version of the Bernoulli proposal: water flows aft from a large tank at the rear of the ship
This slit is at a distance i above sea level. This last point is an improvement, because Bernoulli takes i = 0; Euler points out it should be high enough so that it stays at all times— pitching of the ship and high waves notwithstanding—above the sea surface. He takes into account the power required to accelerate the water from rest to the speed v of the ship, an
Daniel Bernoulli and Leonhard Euler on the Jetski
93
effect pointed out by Bernoulli, but forgotten at a certain point;18 he comes up with the formula v 6 iv 4
W2
12 A 2
for the seaspeed v. Euler applies this to the kind of naval vessel he considered in the first part of his memoir, viz. a ship having an effective cross section of 100 square feet, set into motion by 100 men, each man producing a power of 85 Watt.19 A first choice of parameters yields:
x x x x
i = 0.44 feet = 0.14 m (exit slit above sea level) a = 0.587 feet = 0.19 m (height of tank) S = 12.5 feet squared = 1.28 m2 (area of exit slit) v = 3 feet/second = 0.96 m/s (seaspeed) (= 3.46 km/h))
This seems utterly impractical: i is much too small, the low depth of the tank demands a slit less than 1/4 foot wide, requiring a tank 50 feet wide, rather wider than the ship considered. Making i larger—always with a crew of 100—takes us to an example where it turns out, writes Euler, that the speed obtained is much smaller than could be obtained by the same crew if they applied their forces to, for example, paddle wheels. At this point Johann Albrecht departs from the Latin text of his father and considers only a small boat, manned by four and having a resistance of only 10 square feet, but his conclusion is similar. Having written this, he goes—as his father did not—into a detailed comparison of “his” results with those of Bernoulli, by using his own formulae to compute the performance of Bernoulli’s trireme. He points out the omission of the power needed to take the water aboard and finds, all calculations done, the trireme should proceed at 6 feet/second (6.92 km/h). Johann Albrecht adds that if the selfsame trireme were fitted with paddlewheels as described in the first part of his memoir it would make 9 feet/second (10.5 km/h), i.e., 50% more than with jet propulsion. 5.4 Euler’s system 2 Doing away with a tank on deck, Euler proposes to obtain the pressure driving the jet by a couple of pumps which would alternately suck up the water and expel it. The proposal is clear from the figure he published (fig.11). Positioned at the rear of the ship the intake is at the bottom of the vessel, in B (as in a jetski), the nozzle is at F. A piston T is pushed or pulled by the men, grabbing bars connected to P and not shown. The bar VC pivots around C and the dimensions of that lever are chosen to afford the optimal stroke and speed of the operators. The valves m and n direct the flow. The pump cylinder is horizontal on purpose: the inflow as well as the
18
But he adds up (32) and (33), taking the first in the ship’s frame (as in (23)), the second in the sea’s frame (as in (28)), which we saw is inconsistent. The term (34) is omitted. Curiously enough he forgets the effect of accelerating the water on the total reaction force (the term (16)) which Bernoulli did include. This was already remarked by G.K. Mikhailov [2, p.77]. 19 This number is calculated from the data given by Euler, based on the Rhenan foot; see [4], p.IL.
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Frans A. Cerulus
outflow is bent at A and will give a reaction force towards the outside of the bend, that is they will push the ship forward.
Fig. 11. The pump devised by Euler; the jet issues from F
De promotione navium
This article
area of nozzle
ff
S
waterhead
a
a
velocity of jet
a
velocity of ship
c
speed head
c
v2 2g
density of water
2
ȡ
c
2 ga
v
resistance
kkc
acceleration gravity
1/2
g
fall in 1 second
l=1/4
g/2
power of crew
2nA bl
W
1 2
AU v 2
Table 2: The notations used in Euler’s memoir and their equivalents in the present article
We take over the results and notations of Walter Habicht, the editor of the volume II/20 of Euler’s works. Let H be the force on the ship, h the height of the piston over the sea level, a the difference in height between the piston and the nozzle, Ȝ the ratio of the area’s of nozzle to piston; then, in optimal circumstances,
Daniel Bernoulli and Leonhard Euler on the Jetski
H
1 2O2 1 2O
2 g a h
(40)
W 1 2
.
W.
2 g h O2 a h For a very low nozzle (a=h) and Ȝ
