Electroweak Phase Transition and the Early Universe, a NATO Advanced Re- search Workshop, was held March 23-25, 1994, at the Hotel Tivoli in Sintra, Portugal. The meeting was co-sponsored by three other Lisbon-based institutions: the Fundac;ao Gulbenkian, J. N.!. C. T. (Junta Nacional para Investigac;ao Cientifica e Tecnologica) and G. T. A. E. (Grupo Teorico de Alta Energias). The workshop brought together a large number of theoretical physicists who are actively researching topics relevant to the understanding of the standard model of electroweak interactions in the early universe. We were pleased and overwhelmed by the positive, and sometimes instan- taneous response that our enterprise raised right from its inception. The old town of Sintra provided a serene and pleasant environment for the par- ticipants. Some heated and controversial discussions on many unanswered questions in the standard model took place throughout the three days of the workshop. If one consensus emerged from the meeting, it was the imperative need for non-perturbative techniques for the understanding of the electroweak phase transition.

The reader will find in this volume the Proceedings of the NATO Advanced Study Institute held in Cortina d' Ampezzo, Italy, between July 25 and August 6, 1993, under the title From Newton to Chaos: Modem Techniques for Understanding and Coping With Chaos inN-Body Dynamical Systems. This institute was the latest in a series of meetings held every three years from 1972 to 1990 in dynamical astronomy, theoretical mechanics and celestial mechanics. The proceedings from these institutes have been well-received in the international community of research workers in these disciplines. The present institute was well attended with 15 series of lectures being given by invited speakers: in addition some 40 presentations were made by the other participants. The majority of these contributions are included in these proceedings. The all-pervading influence of chaos in dynamical systems (of even a few variables) has now been universally recognised by researchers, a recognition forced on us by our ability, using powerful computer hardware and software, to tackle dynamical problems that until twenty-five years ago were intractable. Doubtless it was felt by many that these new techniques provided a break-through in celestial mechanics and its related disciplines. And so they were.

This book on the theory of three-dimensional spinors and their applications fills an important gap in the literature. It gives an introductory treatment of spinors. From the reviews: "Gathers much of what can be done with 3-D spinors in an easy-to-read, self-contained form designed for applications that will supplement many available spinor treatments. The book...should be appealing to graduate students and researchers in relativity and mathematical physics." --MATHEMATICAL REVIEWS

"General Relativity Without Calculus" offers a compact but mathematically correct introduction to the general theory of relativity, assuming only a basic knowledge of high school mathematics and physics. Targeted at first year undergraduates (and advanced high school students) who wish to learn Einstein's theory beyond popular science accounts, it covers the basics of special relativity, Minkowski space-time, non-Euclidean geometry, Newtonian gravity, the Schwarzschild solution, black holes and cosmology. The quick-paced style is balanced by over 75 exercises (including full solutions), allowing readers to test and consolidate their understanding.

The five lectures presented in this volume address very timely mathematical problems in relativity and cosmology. "Part I" is devoted to the initial value and evolution problems of the Einstein equations. Especially it deals with the Einstein-Yang-Mills-Boltzmann system, fluid models with finite or infinite conductivity, global evolution of a new (two-phase) model for gravitational collapse and the structure of maximal, asymptotically flat, vacuum solutions of the constraint equations which have the additional property of containing trapped surfaces. "Part II" focuses on geometrical-topological problems in relativity and cosmology: on the role of cosmic censorship for the global structure of the Einstein-Maxwell equations and on the mathematical structure of quantum conformal superspace.

General relativity, which lies at the heart of contemporary physics, has recently become the focus of a number of lively theoretical, experimental, and computational research programs. As a result, undergraduates have become increasingly excited to learn about the subject. A General Relativity Workbook is a textbook intended to support a one-semester upper division undergraduate course on general relativity. Through its unique workbook-based design, it enables students to develop a solid mastery of both the physics and the supporting tensor calculus by pushing (and guiding) them to work through the implications. Each chapter, which is designed to correspond to one class session, involves a short overview of the concepts without obscuring derivations or details, followed by a series of boxes that guide students through the process of working things out for themselves. This active-learning approach enables students to develop a more secure mastery of the material than more traditional approaches. More than 350 homework problems support further learning. This book more strongly emphasizes the physics than many of its competitors, and while it provides students a full grounding in the supporting mathematics (unlike certain other competitors), it introduces the mathematics gradually and in a completely physical context. Ancillaries To facilitate self-study, a complimentary Online Student Manual with Hints and Answers for Selected Problems is available online. A detailed Instructor's Manual is available to adopting professors.

Observing our Universe and its evolution with ever increasing sensitivity from ground-based or space-borne telescopes is posing great challenges to Fundamental Physics and Astronomy. The remnant cosmic microwave background, as beautifully measured by successive space missions COBE, WMAP, and now PLANCK, provides a unique probe of the very early stages of our Universe. The red-shift of atomic lines in distant galaxies, the dynamics of pulsars, the large scale structure of galaxies, and black holes are a few manifestations of the theory of General Relativity. Yet, today, we understand only 4% of the mass of our Universe, the rest being called dark energy and dark matter, both of unknown origin! A second family of space missions is currently emerging; rather than designing ever more re nedobservationalinstruments,physicistsandengineersseekalsotousethespaceenvironment to perform high-precision tests of the fundamental laws of physics. The technology required for such tests has become available only over the course of the last decades. Clocks of high accuracy are an example. They are based on advances in atomic and laser physics, such as cold atoms, enabling a new generation of highly sensitive quantum sensors for ground and space experiments. Two experiments in space have now tested Einstein's relativity theory: * Several decades ago, Gravity Probe A con rmed the accuracy of the gravitational red-shift ?5 according to general relativity to a level of 7x 10 [R. F. C. Vessot et al. , Test of Relativistic Gravitation with a Space-Borne Hydrogen Maser, Phys. Rev. Lett. 45, 2081-2084 (1980)].

Research on photon and electron collisions with atomic and molecular targets and their ions has seen a rapid increase in interest, both experimentally and theoretically, in recent years. This is partly because these processes provide an ideal means of investigating the dynamics of many particle systems at a fundamental level and partly because their detailed understanding is required in many other fields, particularly astrophysics, plasma physics and controlled thermonuclear fusion, laser physics, atmospheric processes, isotope separation, radiation physics and chemistry and surface science. In recent years a number of important advances have been made, both on the experimental side and on the theoretical side. On the experimental side these include absolute measurements of cross sections, experiments using coincidence techniques, the use of polarised beams and targets, the development of very high energy resolution electron beams, the use of synchrotron radiation sources and ion storage rings, the study of laser assisted atomic collisions, the interaction of super-intense lasers with atoms and molecules and the increasing number of studies using positron beams.

Dark matter in the Universe has become one of the most exciting and central fields of astrophysics, particle physics and cosmology. The lectures and talks in this book emphasize the experimental and theoretical status and perspectives of the ongoing search for dark matter, and the future potential of the field into the next millennium, stressing in particular the interplay between astro- and particle physics.

It would seem that any specialist in plasma physics studying a medium in which the interaction between particles is as distance-dependent as the inter action between stars and other gravitating masses would assert that the role of collective effects in the dynamics of gravitating systems must be decisive. However, among astronomers this point of view has been recog nized only very recently. So, comparatively recently, serious consideration has been devoted to theories of galactic spiral structure in which the dominant role is played by the orbital properties of individual stars rather than collec tive effects. In this connection we would like to draw the reader's attention to a difference in the scientific traditions of plasma physicists and astrono mers, whereby the former have explained the delay of the onset of controlled thermonuclear fusion by the "intrigues" of collective processes in the plasma, while many a generation of astronomers were calculating star motions, solar and lunar eclipses, and a number of other fine effects for many years ahead by making excellent use of only the laws of Newtonian mechanics. Therefore, for an astronomer, it is perhaps not easy to agree with the fact that the evolution of stellar systems is controlled mainly by collective effects, and the habitual methods of theoretical mechanics III astronomy must make way for the method of self-consistent fields."

This book gives a comprehensive overview of the current observational and theoretical status in the field of the local and general interstellar medium. It contains contributions presented at the IAU Colloquium No. 166. Review articles and highlight talks will serve both as an introduction to the field for the undergraduate or the non-specialist and also give a summary of the most recent developments for the expert and researcher. These articles are supplemented by a representative number of original research papers. All contributions are fully refereed and have been edited with extensive care to provide a high-standard reference book. The scientific content spans a wide range from solar system measurements of dust grains to X-ray emission from distant galaxies.

This book addresses graduate students in the first place and is meant as a modern compendium to the existing texts on black hole astrophysics. The authors present in pedagogically written articles our present knowledge on black holes covering mathematical models including numerical aspects and physics and astronomical observations as well. In addition, in their write-up of a panel discussion the participants of the school address the existence of black holes consenting that it has by now been verified with certainty.

This book offers a presentation of the special theory of relativity that is mathematically rigorous and yet spells out in considerable detail the physical significance of the mathematics. It treats, in addition to the usual menu of topics one is accustomed to finding in introductions to special relativity, a wide variety of results of more contemporary origin. These include Zeeman s characterization of the causal automorphisms of Minkowski spacetime, the Penrose theorem on the apparent shape of a relativistically moving sphere, a detailed introduction to the theory of spinors, a Petrov-type classification of electromagnetic fields in both tensor and spinor form, a topology for Minkowski spacetime whose homeomorphism group is essentially the Lorentz group, and a careful discussion of Dirac s famous Scissors Problem and its relation to the notion of a two-valued representation of the Lorentz group. This second edition includes a new chapter on the de Sitter universe which is intended to serve two purposes. The first is to provide a gentle prologue to the steps one must take to move beyond special relativity and adapt to the presence of gravitational fields that cannot be considered negligible. The second is to understand some of the basic features of a model of the empty universe that differs markedly from Minkowski spacetime, but may be recommended by recent astronomical observations suggesting that the expansion of our own universe is accelerating rather than slowing down. The treatment presumes only a knowledge of linear algebra in the first three chapters, a bit of real analysis in the fourth and, in two appendices, some elementary point-set topology.

The first edition of the book received the 1993 CHOICE award for Outstanding Academic Title.

Reviews of first edition:

a valuable contribution to the pedagogical literature which will be enjoyed by all who delight in precise mathematics and physics. (American Mathematical Society, 1993)

Where many physics texts explain physical phenomena by means of mathematical models, here a rigorous and detailed mathematical development is accompanied by precise physical interpretations. (CHOICE, 1993)

his talent in choosing the most significant results and ordering them within the book can t be denied. The reading of the book is, really, a pleasure. (Dutch Mathematical Society, 1993)

"

The discovery of the ?rst case of superluminal radio jets in our galaxy in 1994 from the bright and peculiar X-ray source GRS 1915+105 has opened the way to a major shift in the direction of studies of stellar-mass accreting binaries. The past decade has seen an impressive increase in multi-wavelength studies. It is now known that all black hole binaries in our galaxy are radio sources and most likely their radio emission originates from a powerful jet. In addition to the spectacular events related to the ejection of superluminal jets, steady jets are known from many systems. Compared with their supermassive cousins, the nuclei of active galaxies, stellar-mass X-ray binaries have the advantage of varying on time scales accessible within a human life (sometimes even much shorter than a second). This has led to the ?rst detailed studies of the relation between accretion and ejection. It is even possible that, excluding their "soft" periods, the majority of the power in gal- tic sources lies in the jets and not in the accretion ?ows. This means that until a few years ago we were struggling with a physical problem, accretion onto compact objects, without considering one of the most important components of the system. Models that associate part of the high-energy emission and even the fast aperiodic variability to the jet itself are now being proposed and jets can no longer be ignored.

Most astronomers and physicists now believe that the matter content of the Universe is dominated by dark matter: hypothetical particles which interact with normal matter primarily through the force of gravity. Though invisible to current direct detection methods, dark matter can explain a variety of astronomical observations. This book describes how this theory has developed over the past 75 years, and why it is now a central feature of extragalactic astronomy and cosmology. Current attempts to directly detect dark matter locally are discussed, together with the implications for particle physics. The author comments on the sociology of these developments, demonstrating how and why scientists work and interact. Modified Newtonian Dynamics (MOND), the leading alternative to this theory, is also presented. This fascinating overview will interest cosmologists, astronomers and particle physicists. Mathematics is kept to a minimum, so the book can be understood by non-specialists.

This book is devoted to one of the central problems of contemporary thinking, for which c.P. Snow in 1959 coined the phrase of the "Two Cultures". In this concept, human endeavour is directed on one side to the (forward-looking) sci ences (mathematics, physics, chemistry, biology, etc.) and on the other side to the (backward-looking) humanities (including psychology, linguistics, sociology, etc.). In this dichotomy Snow saw no possibility of unification. On the other hand the urge towards self-consistency and harmony in the mental and spiritual lives of both man and society as a whole is clearly one of the major forces of creativity, both scientific and artistic. This force aims at the unification of the "Two Cultures" in order to build an integrated self-consistent system for our intellectual life. Some attempts in this direction have been made before, and will be described in this book. It is our aim to contribute to the achievement of an integrated mental life on the basis of information theory. In order to construct our model, we examine the laws of information theory, leading us to the deduction of the main laws inherent in both "cultures". Thus, we consider the evolution of both non-living and living matter, human behaviour, the phenomenon of language, the sphere of aesthetics, etc. We hope that our work will be useful both for researchers (who are trying to derive different integral theories) and for various other "consumers" of scientific knowledge (meaning broad circles of intellectuals).

The present book grew out of lecture notes prepared for a "Cours du troisieme cycle de la Suisse Romande", 1983 in Lausanne. The original notes are considerably extended and brought up to date. In fact the book offers at many instances completely new derivations. Half-way between textbook and research monograph we believe it to be useful for students in elementary particle physics as well as for research workers in the realm of supersymmetry. In writing the book we looked back not only on ten years of super- symmetry but also on ten years of our own life and work. We realize how deeply we are indebted to many friends and colleagues. Some shared our efforts, some helped and encouraged us, some provided the facili- ties to work. Their list comprises at least C. Becchi, S. Bedding, P. Breitenlohner, T. E. Clark, S. Ferrara, R. Gatto, M. Jacob, W. Lang, J. H. Lowenstein, D. Maison, H. Nicolai, J. Prentki, A. Rouet, H. Ruegg, M. Schweda, R. Stora, J. Wess, W. Zimmermann, B. Zumino. During the last ten years we had the privilege to work at CERN (Geneva), Departement de Physique Theorique (University of Geneva), Institut fUr Theoretische Physik (University of Karlsruhe) and at the Max-Planck-Institut fUr Physik und Astrophysik (Munich) for which we are most grateful. Grate- fully acknowledged is also the support we received by "the Swiss National Science Foundation" (O. P. ), the "Deutsche Forschungsgemeinschaft" (Heisenberg-Fellowship; K. S. ).

InMay1988,theFirstInternationalConferenceonDissociativeRecombination:Theory, ExperimentandApplicationswasheldatChateauLakeLouise,Alberta,Canada. Thismeeting gaveaconsiderableimpetustothissubject,whichisofparticularinterestforalargevarietyof fieldsincludinginterstellarclouds,planetaryatmospheres,gaslasers,plasmaprocessing,ion sourcesandthermonuclearplasmas. Sincethen,indeed,severalcollaborationswereinitiated betweenexperimentalistsworkingwithdifferenttechniques,betweentheoreticiansdealingwith molecularstructureononehandanddynamicsontheotherhandandalsobetween experimentalistsandtheoreticians. Duringthelastfouryears,therefore,alargenumberof studieswerecarriedoutandraisedanewsetofquestions. Moreparticularly,theresultsthat wereobtainedconcerningcontroversialspeciessuchasH!andHeir,castingsomedoubton theverymechanismsbywhichdissociativerecombinationproceeds. The Second International Conference on Dissociative Recombination: Theory, ExperimentandApplications heldat"I'AbbayedeSaintJacutdelaMer",Brittany,France, May3-8,1992,camethereforeattherighttimetosurveythecurrentstateofthesubject. The symposiumbroughttogetherleadinginvestigatorsinthefieldsofmolecularionrecombination research,atomicandmoleculartheoryandexperiment,plasmasphysics,astrochemistryand aeronomy. Speakerspresentedtalksreviewingtheirownworkandthesewerefollowedby livelydiscussionsessions. Freetimeperiodsallowedparticipantstodiscoveranenchanting peninsulaofNorthBrittanywhilepursuingstimulatingscientificdiscussions. Thepapersinthis volumearebasedonthesetalksandfurtherdiscussions,withtheexceptionofcontributions fromT. AmanoandB. M. McLaughlinwhowereunabletoattend. Alistofparticipantsas wellasasouvenirgroupphotoisgivenattheendofthebook. WeareindebtedtoNATOforitsfinancialsupportwhichcontributedtothegreat successofthissecondmeeting. TheUniversityofRennesIandtheBalzerscompanyarealso acknowledgedfortheiradditionalsupportManythankstothewholestaffoftheabbeyforits kindnessandeffortsinprovidinguswithapropitiousenvironmentforsuchaworkshop. We arealsogratefultoL. Caubetforhervaluableadministrativeassistanceduringthepreparation ofthemeeting. Finallyaspecialmentionmustbegiventoallthesessionchairpersonsfortheir skillinorchestratingthediscussions. Lookingforwardtoattendingthethirdmeeting. TheEditors Bertrand R. Rowe J. Bria/l A. Mitchell Andre Callosa DepartementdePhysique DepartmentofPhysics DepartementdePhysique AtomiqueetMoleculaire TheUniversityofWesternOntario AtomiqueetMoleculaire UniversitedeRennesI London,Canada,N6A3K7 UniversitedeRennesI CampusdeBeaulieu CampusdeBeaulieu 35042RennesCedex, 35042RennesCedex, France France v CONTENTS ORALCONTRIBUTIONS PolyatomicIonDissociativeRecombination . 1 D. R. Bates RecentDevelopmentsandPerspectivesintheTreatmentofDissociative RecombinationandRelatedProcesses...11 A. Giusti-Suzor, I. F Schneider, and 0. Dulieu CharacteristicsofSuperexcitedStatesofMoleculesandMQDTStudiesofNO+ DissociativeRecombination...25 H. Sun, K. Nakashima, and H. Nakamura CalculationsforAr +Xe*andArXe+ +e 35 A. P. Hickman, DL Huestis, and R. P. Saxon Electron-IonContinuum-ContinuumMixinginDissociativeRecombination 47 s. L. Guberman ATheoreticalStudyoftheHCO+andHCS+ElectronicDissociativeRecombinations. . 59 D. Talbi, and Y Ellinger DissociativeRecombinationofCH;:SomeBasicInformationfromElectronic 2 StructureCalculations. 67 WP. Kraemer TheoreticalProblemsintheDissociativeRecombinationofH~ +e...75 H. Takagi RecentMergedBeamsInvestigationsofHydrogenMolecularIonRecombination...87 J B. A. Mitchell, F B. Yousif, P. Van der Donk, and T. J Morgan vii FlowingAfterglowStudiesofElectron-IonRecombinationusingLangmuirProbesand OpticalSpectroscopy...99 NG. Adams RecentFlowingAfterglowMeasurements . 113 B. R. Rowe InfraredSpectroscopicStudiesoftheDissociativeRecombinationProcessesofH...127 3 T. Amano RecombinationofClusterIons . 135 R. Johnsen PredissociationofExcitedStatesofH . . 145 3 H. Helm AStudyofHe e,3I,;)BoundandContinuumStates...155 2 CJ Gillan, B. M McLaughlin. and P. G. Burke ElectronCollisionInducedExcitationsandDissociationofHeH+usingthe R-MatrixMethod . . 163 BK Sarpal, J Tennyson, and L. A. Morgan AssociativeIonisationofHydrogen:ExperimentswithFastMergedBeams...173 F Brouillard, andX Urbain TheoryoftheAsssociativeIonisationReactionbetweenTwoLaser-Excited . 187 SodiumAtoms 0. Du!ieu, A. Giusti-Suzor, andF Masnou-Seeuws ResonantTheoryofDissociativeAttachment...195 J. J. Fabrikant MicroscopicandMacroscopicTheoriesofTermolecularRecombinationbetween AtomicIons...205 MR. Flannery DissociativeRecombinationinPlanetaryIonospheres . . 219 JL. Fox ChemistryofSupernova1987a . 243 A. Dalgarno DissociativeRecombinationinInterstellarClouds . .

The International Conference, Orbis Scientiae 1996, focused on the topics: The Neutrino Mass, Light Cone Quantization, Monopole Condensation, Dark Matter, and Gravitational Waves which we have adopted as the title of these proceedings. Was there any exciting news at the conference? Maybe, it depends on who answers the question. There was an almost unanimous agreement on the overall success of the conference as was evidenced by the fact that in the after-dinner remarks by one of us (BNK) the suggestion of organizing the conference on a biannual basis was presented but not accepted: the participants wanted the continuation of the tradition to convene annually. We shall, of course, comply. The expected observation of gravitational waves will constitute the most exciting vindication of Einstein's general relativity. This subject is attracting the attention of the experimentalists and theorists alike. We hope that by the first decade of the third millennium or earlier, gravitational waves will be detected, opening the way for a search for gravitons somewhere in the universe, presumably through the observations in the CMBR. The theoretical basis of the graviton search will take us to quantum gravity and eventually to the modification of general relativity to include the Planck scale behavior of gravity -at energies 19 of the order of 10 Ge V.

Cosmology in Scalar-Tensor Gravity covers all aspects of cosmology in scalar-tensor theories of gravity. Considerable progress has been made in this exciting area of physics and this book is the first to provide a critical overview of the research. Among the topics treated are:
-Scalar-tensor gravity and its limit to general relativity,
-Effective energy-momentum tensors and conformal frames,
-Gravitational waves in scalar-tensor cosmology,
-Specific scalar-tensor theories,
-Exact cosmological solutions and cosmological perturbations,
-Scalar-tensor scenarios of the early universe and inflation,
-Scalar-tensor models of quintessence in the present universe and their far-reaching consequences for the ultimate fate of the cosmos.

Because of recent progress in the development of quasistationary toroidal mag- netic confinement systems, especially tokamaks, these systems are at the center of research on controlled thermonuclear fusion. Tokamaks were proposed and first built at the Kurchatov Institute of Atomic Energy. In the 1960s the basic features of plasma behavior in toroidal magnetic confinement systems were investigated in ex- periments on the first tokamaks and the possibility of obtaining effective confine- ment in them was demonstrated. The successes of this first stage led to a rapid ex- pansion in tokamak research around the world. The development of a thermonu- clear power reactor based on the tokamak is now actively under way. During the earliest phase of research on tokamaks, it was already clear that the ohmic heating used in them was not sufficient to obtain the temperatures needed for initiation of a self-sustaining thermonuclear reaction. At the beginning of the 1970s, therefore, a search was begun for methods of heating which could supple- ment ohmic heating. The best of these auxiliary heating techniques are neutral beam injection, various methods based on the collisionless absorption of rf (radio fre- quency) waves, and adiabatic compression of the plasma by a rising magnetic field.

This monograph presents in detail the reduction method for studying the unification of fundamental actions. The mathematical (differential geometrical) methods make extensive use of Lie Groups and the concept of homogeneous spaces. The main topic of the book is the dimensional reduction of pure Yang-Mills theories. A rather complete analysis of the structure of the scalar field potential is given and a general procedure for solving the equations of spontaneous compactification within Einstein-Yang-Mills systems is presented. The authors also discuss gravity and theories with fermions included and they review attempts to construct realistic models. The book presents the basic ideas and the calculations in detail and should be of interest to researchers and graduate students in mathematical physics.

Covering all aspects of gravitation in a contemporary style, this advanced textbook is ideal for graduate students and researchers in all areas of theoretical physics. The 'Foundation' section develops the formalism in six chapters, and uses it in the next four chapters to discuss four key applications - spherical spacetimes, black holes, gravitational waves and cosmology. The six chapters in the 'Frontier' section describe cosmological perturbation theory, quantum fields in curved spacetime, and the Hamiltonian structure of general relativity, among several other advanced topics, some of which are covered in-depth for the first time in a textbook. The modular structure of the book allows different sections to be combined to suit a variety of courses. Over 200 exercises are included to test and develop the reader's understanding. There are also over 30 projects, which help readers make the transition from the book to their own original research.

The book provides readers with an understanding of the mutual conditioning of spacetime and interactions and matter. The spacetime manifold will be looked at to be a reservoir for the parametrization of operation Lie groups or subgroup classes of Lie groups. With basic operation groups or Lie algebras, all physical structures can be interpreted in terms of corresponding realizations or representations. Physical properties are related eigenvalues or invariants. As an explicit example of operational spacetime is proposed, called electroweak spacetime, parametrizing the classes of the internal hypercharge - isospin group in the general linear group in two complex dimensions, i.e., the Lorentz cover group, extended by the casual (dilation) and phase group. Its representations and invariants will be investigated with the aim to connect them, qualitatively and numerically, with the properties of interactions and particles as arising in the representations of its tangent Minkowski spaces.