We summarize the situation regarding non-polynomial Lagrangians: I should make the qualification that an enor mous amount of verification is needed before the problems of renormalizability are all sorted out, but one may ten tatively state: 1) All matrix elements are finite for theories where the Dyson index D is less than two. 2) For the cases when D=2 or 3, counter-terms have been explicitly written which absorb all infinities and the theories are renormalizable. 3) Mixed theories of polynomial and non-polynomial fields appear to be renormalizable provided the Dyson in dices separately and jointly fulfill renormalizability criteria. We believe that weak interactions, chiral La grangians and Yang-Mills theory fall into this class though detailed proofs have not yet been constructed. 4) It seems likely that to each order in the major coupling (and to all orders in the minor coupling}the S matrix elements, as computed by methods outlined, satisfy the necessary unitarity and analyticity requirements."

On the occasion of the 50th anniversary of the discovery of the Schrodinger equation a small symposium was organized in Vienna. It had mainly retrospective character, where after an appreciation of Schrodinger's scientific achievements the results were collected which one could extract from his equation. Of course not all the developments which originated in Schrodingers dis coveries could be included. Instead, it was attempted to present a review of the established predictions which follow directly from his equation. Despite the 50 years of its existence there are always new results of this sort being found, especially because the necessary mathe matical methods are being developed and become known to the physicists slowly only now .. I want to take the opportunity here to thank the lecturers for their efforts which they put into their excellent talks and their written versions. With their help this volume should become a useful document on the current mathematical art in the treatment of the Schrodinger equation. Finally it is my pleasant obligation to thank the Bundesministerium fUr Wissenschaft und Forschung and the Kulturamt der Gemeinde Wien for their financial support which made it possible to honor one of the great Austrian scientists."

Soluble quantum field theory models are a rare commodity. An infinite number of degrees of freedom and noncompact invariance groups have a nasty habit of ex ploding in the model-makers' face. Nevertheless, impor tant progress has recently been made in the class of superrenormalizable relativistic theories, such as a self-interacting boson in a two-dimensional space time [ 1]. These results have been obtained starting with the free field and adding the interaction in a carefully controlled way. Yet, the models successfully studied in this way do DQ~ have an infinite field strength renormalization, which, at least according to perturbation theory, should appear for realistic relativistic models in four-dimensional space time. ~2~!Y~~!9n_~g_~h~_~gg~1 The ultralocal scalar field theories discussed in these lecture notes are likewise motivated by relativistic theories but are based on a different approximatiGn. This approximation formally amounts to dropping the spatial gradient term from the Hamiltonian rather than the non linear interaction. For a self-interacting boson field in a space-time of (s+l) dimensions (s~l), the classical ultralocal model Hamiltonian reads (1-1) The quantum theory of this model is the subject of the present paper. This model differs formally from a rela tivistic theory by the term f![Z~Cl(~)]2 d~ which, it is hoped, can, in one or another way, be added as a pertur 229 bation in the quantum theory. However, that still remains a problem for the future, and we confine our remarks to . . a careful study of the "unperturbed" model (1-1).

The equations of quantum mechanics, let alone those of quantum field theory, are sufficiently complex to solve that various approximation schemes are both welcome and desirable. In applications to quantum field theory, renorm alized perturbation theory has long been the predominant calculational tool, and it has even tended to shape the entire conceptual outlook on such problems as well. How ever, the apparently irreconcilable conflict between strong interaction physics and perturbation theoretic calculations has given way in the past few years to some fairly revolut ionary (at least for field theory) modes of formulation and calculation. The fires of this revolution have been fed by exact solutions of special classical systems, [1] notably the sine-Gordon equation, and an appreciation of the relevance of classical solutions of nonlinear systems as starting points in quantum analyses [2]. The soliton solutions of the classical sine-Gordon equation exhibit extraordinary stability under scattering, but this feature is not essential for their utility in quantum studies. Consequently, broad studies have been made of nonlinear systems that exhibit localized, persistent solutions, which have also frequently been termed "solitons" (or solitary waves, or kinks) as well [31. Solitons that have their origin in con ventional symmetries and conserved currents (and thus are ~topological in origin) have been discussed [4] along with those that arise from asymptotic properties of the solution, as in the sine-Gordon model, (and thus are topo logical in origin).

These lectures represent a condensation of a number of colloquia, seminars and discussions held at the Institute of Theoretical Physics of the University of Graz during the last years and epitomize the principal lines of research undertaken by my group. From the very beginning of my appointment at the University of Graz in 1947 I have been concerned with the task of bringing up a relatively small group of scientifically interested and open-minded co-workers and of stimulating them to sound scientific research. Since 1930 I myself have dealt with subjects of the kind treated in these lectures, to which I was introduced by my late friend and teacher TH. SEXL. But also as assistant and co-worker of E. FUES and R. THIRRING I frequently worked on these problems, constantly using new methods and lines of approach. During the last years of the war and the first ones afterwards Ihad the fortunate opportunity to receive many stimulating ideas and comments on my work from A. SOMMERFELD on the occasion of my frequent visits to Munich. Especially this last period, although partially connected with personal difficulties and troubles of many kinds stemming from the turbulence of lost-war readjustments, I consider to be one of the most valuable times in my life.

These lectures concern the properties of topological charge in gauge theories and the physical effects which have been attributed to its existence. No introduction to this subject would be adequate without a discussion of the original work of Belavin, Polyakov, Schwarz, and Tyupkin [1], of the beautiful calculation by 't Hooft [2,3], and of the occurrence of 8-vacua [4-6]. Other important topics include recent progress on solutions of the Yang-Mills equation of motion [7,8], and the problem of parity and time-reversal invariance in strong interactions [9] (axions [10,11], etc. ). In a few places, I have strayed from the conventional line and in one important case, disagreed with it. The im- portant remark concerns the connection between chirality and topological charge first pointed out by 't Hooft [2]: in the literature, the rule is repeatedly quoted with the wrong sign! If QS is the generator for Abelian chiral transformations of massless quarks with N flavours, the correct form of the rule is ssQs = - 2N {topological charge} (1. 1) where ssQS means the out eigenvalue of QS minus the in eigenvalue. The sign can be checked by consulting the standard WKB calculation [2,3], rotating to Minkowski space, and observing that the sum of right-handed chiralities of operators in a Green's function equals -ssQS. The wrong sign is an automatie consequence of a standard but incorrect derivation in which the axial charge is misidentified.

The main task of an experimental talk at a theoreticians school should probably be a tempering one. In this respect, e+e- physics may have been a bad choice. The field has so rapidly developed and dis coveries are chasing each other that much of the optimism of theory has passed over to e+e- experimentalists. A vast amount of experimental material arose from the simple reaction of e+e- annihilation. I, therefore, have to limit myself to recent results - most of them less than one year old. The paper will be organized as follows: In the first lecture (chapter I and II) I will give - a short introduction to e e machines and cross sections. In particular I will discuss the total cross section an- after a short summary on charm - concentrate on the third generation of auarks and leptons: the heavy lepton T and the T family. In my second lecture the various aspects of event topologies in the DORIS energy range will be discussed, including the T decay. In the third lecture I will then describe the new storage ring PETRA and present first results on QED checks, total cross section, jet structure, and two-photon processes."

The subject of this year's conference, as of the six previous meetings, was again elementary particle physics. One of the main topics of current interest here is the understanding of strong interactions; some of the attempts to explain them are discuBBed at length in these proceedings, including current algebra, Regge pole theory and effective Lagrangians. On the other hand, the proceBses of weak interactions, too, are not yet fully understood, especially when strong interactions intervene, as in the radiative corrections to weak decays. Several other aspects of elementary particle physics are also treated in this volume, every one of them another step towards a thorough understanding of the sub. atomic world. I want to take this opportunity to thank all the members of my staff for their assistance in organizing the meeting and for their help in preparing the manuscripts. Graz, June 1968 P. Urban Contents Urban, P. Introduction ................................................. V Nilsson, J. Discrete Symmetries in Elementary Particle Physics ............ 1 Pietschmann, H. Semi-Leptonic Weak Decay Processes .... ............... 88 Dothan, Y. Introduction to Recent Developments in Regge Pole Theory .... 126 Koller, K. On Feynman Diagrams wi h Infinite Dimensional Representations of the Lorentz Group 0 (3,1) ......................................... 161 Giirsey, F. Effective Lagrangians in Particle Physics. . . . . . . . . . . . . . . .. . . 185 . . . ."

The past meetings held under the auspices of the Austrian Ministry of Education and the successes and benefits derived from these lectures in former years have encouraged us to continue this year with a survey of high energy physics. The experimental side of this field has undergone such a great advancement that the evaluation of huge masses of data called for the theoretical physicist to develop new theories and to explain facts and connections which we have not been able to clarify up to the present. In spite of the precarious situation especially in the field of Elementary Particle Theories, the theoretical physicist is about to summarize the present situation and to give further impulses for thc progress of our knowledge in nuclear physics. Thus the results of the Sixth International Meeting in Schladming and the interest shown in it fully justify our endeavours in this direction. In organizing the publication of the proceedings we shall continue on the same lines as hitherto so as to reduce delays through editing to an absolute minimum and to offer the publication at a reasonable price. I am again grateful for the assistance of the Springer-Verlag whose photo mechanical method proved so successful. Nevertheless we wish to apologize for any mistakes that might have occurred in writing the text of formulae."

The great success of the experimental research on elementary particles and their qualities during the last years suggests giving a summary of the present situation also in the theoretical description of this important branch of physics. In spite of the precarious situation in this field of theoretical physics I believe I can fully account for this choice and must see that the number of participants and the general interest justify my opinion. In organizing the proceedings it was our prime concern to reduce the delay in editing and also keep down the price. This was possihle only through the assistance of the Springer-Verlag who chose photomechanical method working quicker and cheaper. Therefore we apologize for any mistakes and errors that may occur in the text and formulae. I am very indebted to my secretary, Miss A. SCHMALDIENST and one of my assistants, Dr. H. KUHNELT, who did an the typing and correcting of the manu scripts with great patience and knowledge."

Whereas then' has been significant progress in various branches of elementary particle physics during the last decade or so, Quantum Electrodynamics seems to be in a somewhat stagnating phase. For people not involved in sophisticated questions of relativistic field theories the situation might appear as not very di(: ltinct from the time, when Schwinger, Feynman and Dyson publishl'ld their famous articles in the early 1950's. Only some arguments against the possibility of a finite Quantum Electrodynamics (1953) has caused a significant reaction and initiated a rather vehement discussion. In connection with these questions about the consistency of Quantum Electrodynamics there arises the problem of the high energy behaviour of the theory, since higher and higher energies will be available in future experiments. For these reasons I thought it quite useful to choose Quantum Electrodynamics at high energies as subject of this year's meetings and hope to meet the demands of both experimental and theoretical physi cists working in this field. Graz, October, 1965."

Weak interactions and higher symmetries are nowadays of special importance for elementary particles theory. Lately both theoretical and experimental physicists became more and more interested in the subject. Because of the complicated subject and the scarce available literature proper introductions in the subject are tiresome. The mathematical back ground such as higher Algebra and Grouptheory etc. cannot be applied immediately in all cases. The third Schladming University Courses on the above subject belong to this category. The present first supplement volume of "Acta Physica Austriaca" contains all lectures (with literature references) giving not only a review of the fundamentals but also discussing recent research work. I; incerely hope that the pUblication will find interested readers. In the last years it became customary in physics and mathematics to arrange summer schools on subjects which allow the experienced as well as the young scientiflt to get quickly acquainted with special modern fields. Discussions and private contacts make it possible for participants to touch quickly the peripheries of the subject in question, and new research work will be stimulated which otherwise could not be expected. Graz, October, 1964."

The observation of the scaling properties of the structure functions w and vw of deep inelastic electron 1 2 nucleon scattering [1]+ has been taken by many people as an indication for an approximate scale invariance of the world. It was pointed out by Wilson [2], that in many field theories it is possible to assign a dimension d to every fundamental field, which proves to be a conserved quantum number as far as the most singular term of an operator product expansion at small distances ((x-y) +a) is con- JJ cerned++. Later it was shown, at the canonical level, that in many field theories the dimension of a field seems to be a c:pod quantum number even in the terms less singular at small (x-y) , as long as they all belong to the strongest \l light cone singularity (i. e. (x-y)2+a) [3]. The assumption that this type of scale invariance on the light cone be present in the operator product ex pansion of two electromagnetic currents has provided us with a rather natural explanation of the observed scaling phenomena. We should like to mention, however, that this ex planation cannot account for the precocity with which scaling is being observed experimentally in energy regions, in which resonances still provide prominent contributions to the final states [4].