0000000000021749
AUTHOR
Florian Scheck
Muon physics — Survey
The empirical basis of the minimal standard model has been consolidated in an impressive way, over the last seventeen years, by precision experiments at the meson factories. I illustrate this by means of selected examples of muonic weak interaction processes. I then describe an extension of Yang-Mills theory, inspired by noncommutative geometry, that yields precisely the standard model but fixes and explains some of its empirical input. In particular, this new approach yields a simple geometrical interpretation of spontaneous symmetry breaking. The algebraic framework of this approach offers a natural place for the lepton and quark matter fields and for inter-family mixing.
Elementary Newtonian Mechanics
This chapter deals with the kinematics and the dynamics of a finite number of mass points that are subject to internal, and possibly external, forces, but whose motions are not further constrained by additional conditions on the coordinates. Constraints such as requiring some mass points to follow given curves in space, to keep their relative distance fixed, or the like, are introduced in Chap. 2. Unconstrained mechanical systems can be studied directly by means of Newton’s equations and do not require the introduction of new, generalized coordinates that incorporate the constraints and are dynamically independent. This is what is meant by “elementary” in the heading of this chapter — thoug…
Leptonic Generation Mixing, Noncommutative Geometry and Solar Neutrino Fluxes
Triangular mass matrices for neutrinos and their charged partners contain full information on neutrino mixing in a most concise form. Although the scheme is general and model independent, triangular matrices are typical for reducible but indecomposable representations of graded Lie algebras which, in turn, are characteristic for the standard model in noncommutative geometry. The mixing matrix responsible for neutrino oscillations is worked out analytically for two and three lepton families. The example of two families fixes the mixing angle to just about what is required by the Mikheyev-Smirnov-Wolfenstein resonance oscillation of solar neutrinos. In the case of three families we classify a…
GENERALIZED GAUGE TRANSFORMATIONS AND HIDDEN SYMMETRY IN THE STANDARD MODEL
A recently proposed, new construction of the Standard Model based on the graded Lie algebra SU (2|1) is analyzed in some depth. The essential ingredient is an algebraic superconnection which incorporates both the gauge fields and the Higgs fields and whose curvature automatically leads to a spontaneously broken realization of the theory. The mechanism of hiding the original algebraic structure is unorthodox and is due to the specific, "noncommutative" realization of SU (2|1). The model is characterized by a constant background supercurvature which is invariant under arbitrary, constant SU (2|1) gauge transformations. This background field whose effect is analogous to the action of a consta…
Stability and Chaos
In this chapter we study a larger class of dynamical systems that include but go beyond Hamiltonian systems. We are interested, on the one hand, in dissipative systems, i.e. systems that lose energy through frictional forces or into which energy is fed from exterior sources, and, on the other hand, in discrete, or discretized, systems such as those generated by studying flows by means of the Poincare mapping. The occurence of dissipation implies that the system is coupled to other, external systems, in a controllable manner. The strength of such couplings appears in the set of solutions, usually in the form of parameters. If these parameters are varied it may happen that the flow undergoes …
Electromagnetic Processes and Interactions
The electron, the muon, and their neutrinos are important tools in testing the structure of the fundamental electromagnetic and weak interactions. On the other hand, if these interactions are known, they serve as ideal probes for the internal structure of complex hadronic targets such as nucleons and nuclei. Although electroweak interactions should in fact be discussed as a whole and on the same footing, purely electromagnetic interactions play a distinctive role, for obvious experimental reasons: At low and intermediate energies the effective electromagnetic coupling is larger by many orders of magnitude than the weak couplings, so that electromagnetic processes are measurable to much high…
Quantized Fields and Their Interpretation
This chapter deals with the quantum theory of systems with an infinite number of degrees of freedom and provides elements of quantum field theory.
Measurement of the positron longitudinal polarization in muon decay
The longitudinal polarization of the ${e}^{+}$ from ${\ensuremath{\mu}}^{+}$ decay has been measured. The spin dependence of positron-electron (Bhabha) scattering and of annihilation in flight were used as the analyzing reactions. The combined statistical and systematic error was reduced by a factor of approximately 3 below that of any previous measurement. The longitudinal polarization was found to be 1.010\ifmmode\pm\else\textpm\fi{}0.064 (statistical plus systematic error) and to be consistent with the prediction of the $V\ensuremath{-}A$ interaction.
Classical Field Theory of Gravitation
The classical field theories developed in the preceding chapters all have in common that they are formulated on a flat spacetime, i.e. on a four-manifold which is a Euclidean space and which locally is decomposable into a direct product M 4 = ℝR3 ℝR of a physical space ℝR3 x of motions, and a time axis ℝRt. The first factor is the threedimensional space as it is perceived by an observer at rest while the time axis displays the (coordinate) time that he/she measures on his/her clocks. This spacetime is endowed with the Poincare group as the invariance group of physical laws and inherits the corresponding specific causality structure.
Raum-Zeit-Symmetrien in der Quantenphysik
Die Transformationen in Raum und Zeit, die in der Galilei-Gruppe zusammengefasst sind, spielen in der Quantentheorie eine wichtige und – gegenuber der klassischen Mechanik – in einigen Aspekten neue Rolle. Drehungen, Translationen und die Raumspiegelung induzieren unitare Transformationen aller Elemente des Hilbert-Raums, die mit Bezug auf den physikalischen R3 und die Zeitachse Rt definiert sind. Die Umkehr der Zeitrichtung induziert eine antiunitare Transformation in H. Wenn der Hamiltonoperator H, der ein quantenmechanisches System definiert, unter Galilei-Transformationen invariant ist, dann lassen sich daraus Aussagen uber seine Eigenwerte und -funktionen ableiten, die im Experiment ge…
Rechenmethoden der Quantenmechanik
Da die Quantentheorie die Grundlage fur fast alle Gebiete der modernen Physik ist, gibt es zahlreiche und weit entwickelte Rechenmethoden zur praktischen Losung von aktuellen Problemstellungen. Diese Methoden, die storungstheoretischer oder nichtstorungstheoretischer Natur sein konnen, sind oft fur die einzelnen Gebiete spezifisch, und es wurde den Rahmen eines Lehrbuchs sprengen, wollte man sie vollstandig und in der gebotenen Ausfuhrlichkeit darstellen. Atom- und Molekulphysik zum Beispiel machen vielfachen Gebrauch von Variationsrechnungen, aber auch von den Viel-Teilchen-Methoden, die auch fur weite Bereiche der Festkorperphysik und der Kernphysik von groser Bedeutung sind. Die Elementa…
Die Maxwell-Theorie als klassische Feldtheorie
Das Hamilton’sche Extremalprinzip und die Lagrange’sche Mechanik, die darauf aufbaut, sind uberaus erfolgreich in ihrer Anwendung auf mechanische Systeme mit endlich vielen Freiheitsgraden. Das Hamilton’sche Extremalprinzip charakterisiert die physikalisch realisierbaren unter allen denkbaren Bahnen als diejenigen, die kritische Elemente des Wirkungsintegrals sind. Die Lagrangefunktion, obwohl selbst keine Observable, dient nicht nur zur rationellen Herleitung der Bewegungsgleichungen, sondern ist auch ein wichtiges Hilfsmittel, um Symmetrien der Theorie festzustellen und die zugehorigen Erhaltungsgrosen uber das Noether’sche Theorem zu konstruieren.
Exercises, Hints and Selected Solutions
1.1. Prove the formula (1.8a) in Sect. 1.3, $$\displaystyle{ \int \mathrm{d}^{n}x\; =\int _{ 0}^{+\infty }\!\!\!\mathrm{d}r\;r^{n-1}\int _{ 0}^{2\pi }\!\!\!\mathrm{d}\phi \prod _{ k=1}^{n-2}\int _{ 0}^{\pi }\!\!\!\mathrm{d}\theta _{ k}\sin ^{k}(\theta _{ k}) }$$ (1.1) by means of induction.
Maxwell Theory as a Classical FieldTheory
Hamilton’s variational principle and the Lagrangian mechanics that rests on it are exceedingly successful in their application to mechanical systems with a finite number of degrees of freedom. Hamilton’s principle characterizes the physically realizable orbits, among the set of all possible orbits, as being the critical elements of the action integral. The Lagrangian function, although not an observable on its own, is not only useful in deriving the equations of motion but is also an important tool for identifying symmetries of the theory and constructing the corresponding conserved quantities, via Noether’s theorem.
Geometric Aspects of Mechanics
In many respects, mechanics carries geometrical structures. This could be felt very clearly at various places in the first four chapters. The most important examples are the structures of the space–time continua that support the dynamics of nonrelativistic and relativistic mechanics, respectively. The formulation of Lagrangian mechanics over the space of generalized coordinates and their time derivatives, as well as of Hamilton–Jacobi canonical mechanics over the phase space, reveals strong geometrical features of these manifolds.
The Principles of Quantum Theory
This chapter develops the formal framework of quantum mechanics: the mathematical tools, generalization and abstraction of the notion of state, representation theory, and a first version of the postulates on which quantum theory rests.
The Higgs Mechanism and Spontaneous Symmetry Breaking
As is well known all gauge bosons of a pure Yang-Mills theory are necessarily massless. This is so because any ad-hoc mass term such as $$ m_i^2 A_\mu ^{(i)} A^{(i)\mu } or \sum\limits_{ik} {M_{ik} } A_\mu ^{(i)} A^{(k)\mu } $$ is incompatible with local gauge invariance. It is saidthat W. Pauli hadd evelopednonab elian gauge theory for himself (or knew about it from the work of H. Weyl and O. Klein) before the work of C.N. Yang and R. Mills (1954) but dismissedit because he hadrealizedthat the gauge particles wouldall be massless. As there was only one massless spin-1 particle known at the time (the photon) nonabelian gauge theory was to be rejectedon physical grounds. The few facts that w…
Symmetrien und Symmetriegruppen in der Quantenphysik
Wenn man von (diskreten oder kontinuierlichen) Gruppen spricht, die Symmetrien von Quantensystemen beschreiben sollen, dann mus man zunachst feststellen, worauf die Elemente dieser Gruppen wirken. Im Falle der Galilei-Gruppe oder der Poincare-Gruppe wirkt ein Element g ∈ G auf Punkte der Raum-Zeit, wobei noch die Wahl besteht, ein gegebenes Element g als aktive oder passive Transformation aufzufassen. Die aktive Interpretation ist die richtige, wenn man zwei identische physikalische Prozesse, die in verschiedenen Bereichen der Raum-Zeit ablaufen, vergleichen und aufeinander abbilden will. Die passive Interpretation andererseits gibt die richtige Lesart, wenn ein und dasselbe physikalische G…
Supersymmetry in the standard model of electroweak interactions
Abstract Starting from the peculiar chirality pattern of weak and electromagnetic interactions, established by experiment, we show that the minimal standard model contains supersymmetry, though in a new, unconventional, realization. It appears as an action on the fields but is not an invariance of the lagrangian. This supersymmetry which is not in conflict with experiment, is seen to be the raison d'etre of the Higgs fields and provides a geometrical understanding of spontaneous symmetry breaking. It turns out that this approach which is based on the fundamental role of left- and right-chiral spinor fields in weak interactions, has many similarities to models developed in the framework of n…
Weak Interactions and the Standard Model of Strong and Electroweak Interactions
This chapter gives an introduction to the phenomenology and the theory of weak interaction processes involving leptons and hadrons. In Sects. 3.1, 3.2 we collect the most prominent and characteristic properties of weak interactions as they follow from the analysis of a set of key experiments, old and recent. The following sections 3.3–3.5 deal with the elements of non-Abelian local gauge theories in general, and with the unified theory of electroweak and strong interactions of Glashow, Salam and Weinberg (GSW), in particular.
Thermodynamics: Classical Framework
This chapter starts with a summary of the thermodynamic potentials and the relationships between them which are obtained from Legendre transformation. This is followed by an excursion to some important global properties of materials such as specific heat, expansion coefficients and others. The thermodynamic relations provide the basis for a discussion of continuous changes of state which are illustrated by the Joule-Thomson effect and the Van der Waals gas. These are models which are more realistic than the ideal gas. The discussion of Carnot cycles leads to and illustrates the second and third laws of thermodynamics. The chapter closes with a discussion of entropy as a concave function of …
Geometric Aspects of Thermodynamics
This chapter deals with mathematical aspects of thermodynamics most of which will be seen to be primarily of geometrical nature. Starting with a short excursion to differentiable manifolds we summarize the properties of functions, of vector fields and of one-forms on thermodynamic manifolds. This summary centers on exterior forms over Euclidean spaces and the corresponding differential calculus. In particular, one-forms provide useful tools for the analysis of thermodynamics. A theorem by Caratheodory is developed which is closely related to the second law of thermodynamics. The chapter closes with a discussion of systems which depend on two variables and for which there is an interesting a…
Die Maxwell’schen Gleichungen
Die empirische Basis der Elektrodynamik ist durch das Induktionsgesetz, das Gaus’sche Gesetz, das Biot-Savart’sche Gesetz sowie durch die Lorentz-Kraft und die universelle Erhaltung der elektrischen Ladung gegeben. Dies sind die Gesetzmasigkeiten, die sich in realistischen Experimenten bestatigen oder, schlimmstensfalls, widerlegen lassen. Die integrale Form der Grundgesetze enthalt ein-, zwei- oder dreidimensionale Objekte, d. h. lineare Leiter, Leiterschleifen, raumliche Ladungsverteilungen oder Ahnliches, und hangt daher immer von konkreten experimentellen Anordnungen ab. Um den Zusammenhangen zwischen scheinbar ganz unterschiedlichen Phanomenen auf den Grund zu gehen, muss man aus der i…
Fermion Fields and Their Properties
The fundamental building blocks of matter, i.e. quarks and leptons, carry spin 1/2. There are two formally different but in essence equivalent methods of describing particles with spin: The representation theory of the Poincare group, in the framework of Wigner’s classification hypothesis of particles (see e.g. [QP07], Chap. 6), and the Van der Waerden spinor calculus based on SL(2, \(\mathbb{C}\)).
Neutrino mixing and masses from long baseline and atmospheric oscillation experiments
We argue that regardless of the outcome of future Long Baseline experiments, additional information will be needed to unambiguously decide among the different scenarios of neutrino mixing. We use, for this purpose, a simple test of underground data: an asymmetry between downward and upward going events. Such an asymmetry, in which matter effects can be crucial, tests electron and muon neutrino data separately and can be compared with the theoretical prediction without relying on any simulation program.
Klassische Feldtheorie der Gravitation
Allen bis zu diesem Punkt behandelten klassischen Feldtheorien ist gemeinsam, dass sie auf einer flachen Raumzeit formuliert sind, d. h. einer Raumzeit-Mannigfaltigkeit M, die ein Euklidischer Raum ist und die lokal in ein direktes Produkt M = ℝ3 × ℝ aus physikalischem Raum ℝ3 x der Bewegungen und einer Zeitachse ℝ t zerlegt werden kann. Der erste Anteil ist dabei der dreidimensionale Raum, wie ihn ein ruhender Beobachter wahrnimmt, wahrend die Zeitachse diejenige (Koordinaten-)Zeit darstellt, die er auf seinen Uhren misst. Dieser Raumzeit wird durch die Poincare-Gruppe --~oder im Grenzfall kleiner Geschwindigkeiten |v| ≪ c durch die Galilei-Gruppe – eine Invarianzgruppe physikalischer Gese…
Streuung von Teilchen an Potentialen
Die drei Grundtypen von Spektren selbstadjungierter Hamiltonoperatoren, das rein diskrete Spektrum mit oder ohne Entartung, das rein kontinuierliche Spektrum und das gemischte Spektrum sowie die zugehorigen Eigenfunktionen enthalten wichtige Informationen uber die physikalischen Systeme, die durch sie beschrieben werden. Aus physikalischer Sicht sind die bisherigen Ergebnisse allerdings noch weitgehend leer, solange wir nicht wissen, wie wir diese Informationen durch konkrete Messungen sichtbar machen konnen. Das statische Spektrum des Hamiltonoperators zum Beispiel, der das Wasserstoffatom beschreibt, und die raumliche Verteilung seiner stationaren Eigenfunktionen sind fur uns makroskopisc…
Teilchen mit Spin 1/2 und die Dirac-Gleichung
Um den Spin eines massiven Teilchens festzustellen, mus man in dessen momentanes Ruhesystem gehen, dort Drehungen des Bezugssystems ausfuhren und untersuchen, wie sich die Teilchenzustande transformieren. Dies war eines der wesentlichen Resultate aus Kap. 1.
Can (noncommutative) geometry accommodate leptoquarks?
We investigate the geometric interpretation of the Standard Model based on noncommutative geometry. Neglecting the $S_0$-reality symmetry one may introduce leptoquarks into the model. We give a detailed discussion of the consequences (both for the Connes-Lott and the spectral action) and compare the results with physical bounds. Our result is that in either case one contradicts the experimental results.
Scattering of Strongly Interacting Particles on Nuclei
This chapter deals with the interaction of strongly interacting particles (hadrons) with nuclei, at low and intermediate energies. The hadronic projectiles are used as probes of nuclear properties and of the hadron-nucleon interaction in the nuclear medium, and they are selected accordingly. We discuss mainly the case of pions, to some extent also kaons, because of the specific properties of the pion-(kaon)-nucleon system. Some of the methods and results can also be applied to nucleons if these are chosen to be projectiles, but we do not go into the full complexity of the spin analysis of the nucleon-nucleon system. In this sense this field is distinct from the conventional topic of nuclear…
Quantum Mechanics of Point Particles
In developing quantum mechanics of pointlike particles one is faced with a curious, almost paradoxical situation: One seeks a more general theory which takes proper account of Planck’s quantum of action \(h\) and which encompasses classical mechanics, in the limit \(h\rightarrow 0\), but for which initially one has no more than the formal framework of canonical mechanics. This is to say, slightly exaggerating, that one tries to guess a theory for the hydrogen atom and for scattering of electrons by extrapolation from the laws of celestial mechanics. That this adventure eventually is successful rests on both phenomenological and on theoretical grounds.
Space-Time Symmetries in Quantum Physics
The transformations in space and in time which belong to the Galilei group play an important role in quantum theory. In some respect and for some aspects, their role is new as compared to classical mechanics.
A New Analysis of the Tippe Top: Asymptotic States and Liapunov Stability
Asymptotic behaviour of a tippe top, under the action of gliding friction. Liapunov stability analysis of the asymptotics of states with arbitrary initial conditions.
The Mechanics of Rigid Bodies
The theory of rigid bodies is a particularly important part of general mechanics. Firstly, next to the spherically symmetric mass distributions that we studied in Sect. 1.30, the top is the simplest example of a body with finite extension. Secondly, its dynamics is a particularly beautiful model case to which one can apply the general principles of canonical mechanics and where one can study the consequences of the various space symmetries in an especially transparent manner.
Probabilities, States, Statistics
In this chapter we clarify some important notions which are relevant in a statistical theory of heat: The definitions of probability measure, and of thermodynamic states are illustrated, successively, by the classical Maxwell-Boltzmann statistics, by Fermi-Dirac statistics and by Bose-Einstein statistics. We discuss observables and their eigenvalue spectrum as well as entropy and we calculate these quantities for some examples. The chapter closes with a comparison of statistical descriptions of classical and quantum gases.
Quantisierung von Feldern und ihre Interpretation
Dieses Kapitel behandelt die Quantentheorie von Systemen mit unendlich vielen Freiheitsgraden und stellt somit die Grundlagen fur die Quantenfeldtheorie bereit. Unterwirft man ein klassisches Feld wie beispielsweise das reelle Skalarfeld, die Maxwellschen Felder oder die dynamischen Variablen eines kontinuierlichen Systems der Mechanik den Regeln der Quantentheorie, so entstehen aus diesen Feldoperatoren, die Quanten dieses Feldes erzeugen oder vernichten konnen und die gleichzeitig die Kinematik und die Spineigenschaften dieser Quanten beschreiben. Damit wird es moglich, die Streutheorie auf solche Prozesse zu erweitern, bei denen Quanten oder Teilchen wirklich erzeugt oder vernichtet werd…
Parity-violating asymmetry in elastic electron-nucleus scattering due to weak neutral currents
Parity-violating asymmetries are calculated for elastic scattering of electrons on nuclei of arbitrary spin and isospin. Possible dependences on nuclear-model input are shown to be weak. Elastic electron scattering at intermediate energies may become an important tool in studying the structure of hadronic weak neutral currents.
Scattering of Particles by Potentials
The three prototypes of spectra of self-adjoint operators, the discrete spectrum, with or without degeneracy, the continuous spectrum, and the mixed spectrum, as well as the corresponding wave functions, contain important information about the physical systems that they describe
Mixed Phases, Phase Transitions, Stability of Matter
Phase mixtures and phase transitions are two major themes of thermodynamics. A third one, related to the former, is the stability of macroscopic matter around us. Mixed phases can be analyzed and illustrated in a nice geometric way. Phase transitions are dealt with from the point of view of classical thermodynamics as well as in the framework of models of statistical mechanics.
Triangular mass matrices of quarks and Cabibbo-Kobayashi-Maskawa mixing
Every nonsingular fermion mass matrix, by an appropriate unitary transformation of right-chiral fields, is equivalent to a triangular matrix. Using the freedom in choosing bases of right-chiral fields in the minimal standard model, reduction to triangular form reduces the well-known ambiguities in reconstructing a mass matrix to trivial phase redefinitions. Furthermore, diagonalization of the quark mass sectors can be shifted to one charge sector only, without loosing the concise and economic triangular form. The corresponding effective triangular mass matrix is reconstructed, up to trivial phases, from the moduli of the CKM matrix elements, and vice versa, in a unique way. A new formula fo…
Beyond the Minimal Standard Model
The GSW theory is a great step forward in our understanding of electroweak interactions because it allows the well-known extremely successful theory of quantized electrodynamics and the theory of the weak CC and NC interactions to be cast into one unified, renormalizable local gauge theory. Renormalizability, in particular, is a very desirable property of the theory because it makes covariant perturbation theory a reasonable and well-defined approximation method for calculating physical quantities beyond the lowest order diagrams. Nevertheless, this model, very likely, is not the corner stone of a final theory of weak and electromagnetic interactions. It contains very many parameters which …
Scattering Matrix and Observables in Scattering and Decays
As an interlude in the analysis of canonical field quantization, this section describes important concepts of scattering theory for Lorentz covariant quantum field theories that will be needed for the calculation of observables such as scattering cross sections and decay probabilities.
Anomalies from nonfree action of the gauge group
Abstract The question whether new anomalies appear, connected with the finite dimensional part of the gauge group isomorphic to the structure group of the theory, is investigated in a systematical way. The stability groups from the stratification of the gauge group on the space of connections lead to anomalies. The detection of these anomalies within the equivariant approach pursued here is extremely simple. For a large class of theories it is shown that no new anomalies appear.
Elementare Newtonsche Mechanik
Dieses erste Kapitel befasst sich mit der Kinematik und Dynamik von endlich vielen Massenpunkten, die zwar inneren und eventuell auch auseren Kraften unterworfen sein mogen, deren Bewegung aber nicht durch zusatzliche Bedingungen (wie die Vorgabe von starren Abstanden, von Kurven, entlang derer einzelne Massenpunkte gleiten sollen, Begrenzungsflachen und dergleichen) eingeschrankt sind. Dies bedeutet, dass man solche mechanischen Systeme direkt mit den Newton’schen Gleichungen angehen kann und noch nicht gezwungen ist, zunachst die dynamisch wirklich unabhangigen, verallgemeinerten Koordinaten aufzusuchen, bevor man die Bewegungen selbst studieren kann. Hierauf bezieht sich die Bezeichnung …
Quantenmechanik eines Punktteilchens
Beim Aufbau der unrelativistischen Quantenmechanik eines Teilchens befindet man sich in einer merkwurdigen, fast paradoxen Ausgangslage: Man sucht eine allgemeinere Theorie, die der Existenz des Planck’schen Wirkungsquantums h Rechnung tragt und die im Grenzfall h → 0 die klassische Mechanik einschliest, man hat als formalen Rahmen aber zunachst nicht mehr als die kanonische Mechanik zur Verfugung. Etwas uberspitzt ausgedruckt heist das, dass man durch Extrapolation aus den Gesetzen der Himmelsmechanik eine Theorie fur das Wasserstoffatom und fur die Streuung von Elektronen erraten will. Dass dieses Abenteuer letztlich gelingt, hat sowohl phanomenologische als auch theoretische Grunde.
Stabilität und Chaos
In diesem Kapitel studieren wir eine grosere Klasse von dynamischen Systemen, die uber die Hamiltonschen Systeme hinausgehen. Dabei sind einerseits Systeme mit Dissipation besonders interessant, bei denen Energie durch Reibung verlorengeht und bei denen Energie aus auseren Quellen eingespeist wird, andererseits diskrete oder diskretisierte Systeme, wie sie auf naturliche Weise beim Studium von Flussen vermittels der Poincareabbildung auftreten. Dissipation bedeutet immer, das das dynamische System an andere Systeme in einer kontrollierbaren Weise gekoppelt ist. Die Starke solcher Kopplungen erscheint in der betrachteten Dynamik in Form von Parametern, von denen die Losungsscharen abhangen. …
Symmetries and Symmetry Groups in Quantum Physics
When one talks about discrete or continuous groups which are to describe symmetries of quantum systems, one must first identify the objects on which the elements of these groups are acting.
Mechanik des starren Körpers
Die Theorie des starren Korpers ist ein besonders wichtiges Teilgebiet der allgemeinen Mechanik: Zum einen ist der Kreisel nachst den kugelsymmetrischen Massenverteilungen des Abschn. 1.29 das einfachste Beispiel eines ausgedehnten Korpers. Zum zweiten stellt die Dynamik des starren Korpers einen besonders schonen Modellfall dar, an dem man die allgemeinen Prinzipien der kanonischen Mechanik ausprobieren und die Folgerungen aus den jeweiligen raumlichen Symmetrien besonders anschaulich studieren kann. Zum dritten stellen die Bewegungsgleichungen des Kreisels, die Eulerschen Gleichungen, ein interessantes Beispiel fur nichtlineare Dynamik dar. (Damit ist gemeint, das diese Gleichungen nicht …
Tau neutrinos from muon storage rings
Charged tau leptons emerging in a long baseline experiment with a muon storage ring and a far-away detector will positively establish neutrino oscillations. We study the conversion of $\nu_\mu$ ($\bar{\nu}_\mu$) and of $\bar{\nu}_e$ ($\nu_e$) to $\nu_\tau$ or $\bar{\nu}_\tau$ for neutrinos from a 20 GeV muon storage ring, within the strong mixing scheme and on the basis of the squared mass differences which are compatible with all reported neutrino anomalies, including the LSND data. In contrast to other solutions which ignore the Los Alamos anomaly, we find charged tau production rates which should be measurable in a realistic set up. As a consequence, determining the complete mass spectru…
Basic Notions of the Theory of Heat
This chapter summarizes some basic notions of thermodynamics and defines the empirical variables which are needed for the description of thermodynamic systems in equilibrium. Empirical temperature and several scales used to measure temperature are defined. The so-called “zeroth law of thermodynamics” is formulated which says that systems which are in mutual equilibrium have the same temperature. Thermodynamic ensembles corresponding to different macroscopic boundary conditions are introduced and are illustrated by simple models such as the ideal gas. Also, entropy appears on the scene for a first time, both in its statistical and its thermodynamical interpretation. Gibb’s fundamental form i…
Symmetrien und Kovarianz der Maxwell’schen Gleichungen
Schon bei einer festen Aufteilung der vier dimensionalen Raumzeit in den Raum,in dem Experimente ausgefuhrt werden,und in die Laborzeit zeigen die Maxwell’schen Felder ein interessantes Transformations-verhalten unter kontinuierlichen und diskreten Transformationen.Ihre volle Symmetriestruktur entfaltet sich aber erst wirklich, wenn man die Wirkungder Lorentz-Gruppe auf die Maxwell’schen Gleichungen studiert.Ihre Kovarianz unter dieser Gruppe wird besonders anschaulich am Beispiel der elektromagnetischen Felder einer gleichformig bewegten Punktladung. Die Reformulierung der Maxwell-Theorie in der Sprache der auseren Formen uber dem R4 wirft einerseits Licht auf einige ihrer Eigenschaften,di…
Commutators of second-class axial currents with normal weak currents and consequences for particle decays
Second-class weak axial currents are studied in the framework of normal weak and electromagnetic currents. Equal time commutators between normal and abnormal axial currents and the isoscalar electromagnetic current are postulated and their consequences are worked out. A number of predictions for masses, coupling constants and decay properties are derived and are compared to available data.
Only three flavours
Abstract It is shown that it is possible to account for all the experimental indications for neutrino oscillations with just three flavours. In particular we suggest that the atmospheric neutrino anomaly and the LSND result can be explained by the same mass difference and mixing. Possible implications and future test of the resulting mass and mixing pattern are given.
Geometrische Aspekte der Mechanik
Die Mechanik tragt in vielerlei Hinsicht geometrische Zuge, die an verschiedenen Stellen in den ersten vier Kapiteln deutlich hervorgetreten sind. Die Struktur des Raum-Zeitkontinuums in der nichtrelativistischen und der speziell-relativistischen Mechanik, in welches die Dynamik eingebettet ist, ist ein erstes und wichtiges Beispiel. Besonders aber die Formulierung der Lagrangeschen Mechanik sowie der kanonischen Hamilton-Jacobischen Mechanik auf dem Raum der verallgemeinerten Koordinaten bzw. dem Phasenraum bringt starke geometrische Zuge dieser Mannigfaltigkeiten zutage. (Man denke z.B. an die symplektische Struktur des Phasenraums und den Liouvilleschen Satz.) Die geometrische Natur der …
Structure of the space of reducible connections for Yang-Mills theories
Abstract The geometrical structure of the gauge equivalence classes of reducible connections are investigated. The general procedure to determine the set of orbit types (strata) generated by the action of the gauge group on the space of gauge potentials is given. In the so obtained classification, a stratum, containing generically certain reducible connections, corresponds to a class of isomorphic subbundles given by an orbit of the structure and gauge group. The structure of every stratum is completely clarified. A nonmain stratum can be understood in terms of the main stratum corresponding to a stratification at the level of a subbundle.
SU(2|1) symmetry, algebraic superconnections and a generalized theory of electroweak interactions
We discuss an extension of the standard model of electroweak interactions which incorporates the usual gauge fields and the Higgs fields in one generalized Yang-Mills field (or superconnection). It is shown that both this Yang-Mills field and the corresponding field strength (supercurvature) take their values in the real, graded Lie algebra SU(2|1). The lagrangian as obtained from this superconnection yields the standard model with interesting predictions for masses and couplings. The primordial, larger symmetry is realized as a hidden symmetry. The odd part survives in relations between couplings and masses, while the even part is broken to U(1)em, as usual, though in “reverse order” as co…
Maxwell’s Equations
The empirical basis of electrodynamics is defined by Faraday’s law of induction, by Gauss’ law, by the law of Biot and Savart and by the Lorentz force and the principle of universal conservation of electric charge. These laws can be tested – confirmed or falsified – in realistic experiments. The integral form of the laws deals with physical objects that are one-dimensional, two-dimensional, or three-dimensional, that is to say, objects such as linear wires, conducting loops, spatial charge distributions, etc. Thus, the integral form depends, to some extent, on the concrete experimental set-up. To unravel the relationships between seemingly different phenomena, one must switch from the integ…
CP Violation with Three Oscillating Neutrino Flavours
We explore the prospects of observing leptonic CP violation in a neutrino factory in the context of a scenario with three strongly oscillating neutrinos able to account for the solar, the atmospheric and the LSND results. We address also the problems related with the fake asymmetries induced by the experimental device and by the presence of matter.
Radiative muon capture on 12C and 16O
Abstract We study exclusive radiative muon capture on 12 C(g.s.) leading to 12 B(g.s.), and on 16 O(g.s.) leading to 16 N(0 − , 120 keV). Our investigation, which is based on impulse approximation, puts special emphasis on elementary particle aspects of radiative capture, such as tests of magnitude and momentum transfer dependence of the induced pseudoscalar, and possible contributions of axial currents of second kind (weak electricity). We calculate several correlation observables which are sensitive to the nucleonic currents but depend very little on the structure of the nuclear initial and final states. In particular, it is shown that in capture on 12 C the polarization and alignment of …
Particles with Spin 1/2 and the Dirac Equation
In order to identify the spin of a massive particle one must go to its rest system, perform rotations of the frame of reference, and study the transformation behaviour of one-particle states. This prescription was one of the essential results of Chap. 6. Furthermore, the spin \(1/2\) (electrons, protons, other fermions) is described by the fundamental representation of the group \(SU(2)\). The eigenstates of the observables \(\mathbf{{s}}^2\) and \(s_3\) transform by the \(D\)-matrix \(\mathbf{D }^{(1/2)}(\mathbf R )\) which is a two-valued function on \(\mathbb{R }^3\).
Streumatrix und Observable in Streuung und Zerfällen
An dieser Stelle unterbrechen wir die Entwicklung der kanonischen Quantisierung freier Felder, um einige wichtige Begriffe der Streutheorie fur Lorentz-kovariante Quantenfeldtheorien mit Wechselwirkung aufzustellen, die wir fur die Berechnung von Observablen — Streuquerschnitten und Zerfallswahrscheinlichkeiten — benotigen. Der damit zur Verfugung gestellte Rahmen baut auf allgemeinen, physikalisch plausiblen Voraussetzungen auf und ist sehr allgemein, infolgedessen aber auch vergleichsweise abstrakt. Dies ist der Grund, warum ich zunachst zur nichtrelativistischen Streutheorie zuruckkehre, die der Leserin, dem Leser schon bekannt ist. Diese Streutheorie wird so umgeschrieben und formalisie…
Einfache Anwendungen der Maxwell-Theorie
Aus der ungeheuren Fulle von elektromagnetischen und optischen Phanomenen, die durch die Maxwell’schen Gleichungen erfolgreich beschrieben werden, greifen wir hier einige wenige charakteristische Beispiele auf. Dabei handelt es sich in diesem Band ausschlieslich um Anwendungen, auf die die klassische, nichtquantisierte Version der Theorie anwendbar ist. Der Bereich der semiklassischen Wechselwirkung von (Quanten-)Materie mit dem (klassischen) Strahlungsfeld ebenso wie die volle quantenfeldtheoretische Behandlung der Maxwell-Theorie wird in Band 4 behandelt.
Elements of Quantum Electrodynamics and Weak Interactions
Quantum field theory in its application to electroweak and strong interactions has two rather different facets: A pragmatic, empirical one, and an algebraic, systematical one. The pragmatic approach consists in a set of rules and formal calculational procedures which are extremely successful in their application to concrete physical processes, but rest on mathematically shaky ground. The mathematically rigorous approach, in turn, is technically difficult and not very useful, from a practical point of view, for reaching results which can be compared with phenomenology. Generally speaking, quantum field theory quickly becomes rather technical if one wants to understand it in some depth, and g…
Sweeping the Space of Admissible Quark Mass Matrices
We propose a new and efficient method of reconstructing quark mass matrices from their eigenvalues and a complete set of mixing observables. By a combination of the principle of NNI (nearest neighbour interaction) bases which are known to cover the general case, and of the polar decomposition theorem that allows to convert arbitrary nonsingular matrices to triangular form, we achieve a parameterization where the remaining freedom is reduced to one complex parameter. While this parameter runs through the domain bounded by a circle with radius R determined by the up-quark masses around the origin in the complex plane one sweeps the space of all mass matrices compatible with the given set of d…
Elemente der Quantenelektrodynamik und der Schwachen Wechselwirkung
Die Quantenfeldtheorie in ihrer Anwendung auf die elektroschwachen oder starken Wechselwirkungen hat zwei ganz unterschiedliche Gesichter, ein pragmatisch-empirisches und ein algebraisch-systematisches. Der pragmatische Zugang besteht in einer Reihe von Regeln und formalen Rechenvorschriften, die in der Anwendung auf konkrete physikalische Prozesse sehr erfolgreich sind, aber mathematisch auf einer schlingernden Basis stehen. Der mathematisch strenge, formale Zugang andererseits ist schwierig und fuhrt erst uber einen langen und muhsamen Weg zu Resultaten, die mit der Phanomenologie verglichen werden konnen. Uberhaupt ist Quantenfeldtheorie, wenn man sie in einiger Tiefe verstehen will, seh…
Symmetries and Covariance of the Maxwell Equations
Already within a given, fixed division of four-dimensional spacetime into the space where experiments are performed, and the laboratory time variable, Maxwell’s equations show interesting transformation properties under continuous and discrete space-time transformations. However, only the action of the whole Lorentz group on them reveals their full symmetry structure. A good example that illustrates the covariance of Maxwell’s equations is provided by the electromagnetic fields of a point charge uniformly moving along a straight line.
Die Prinzipien der kanonischen Mechanik
Dies ist ein zentrales Stuck der allgemeinen Mechanik, in dem man an einigen, zunachst recht kunstlich anmutenden Beispielen lernt, sich von dem engen Rahmen der Newtonschen Mechanik fur Bahnkoordinaten im dreidimensionalen Raum ein wenig zu losen, zugunsten einer allgemeineren Formulierung von mechanischen Systemen, die einer wesentlich groseren Klasse angehoren. Das ist der erste Schritt der Abstraktion, weg von Wurfparabeln, Satellitenbahnen, schiefen Ebenen und schlagenden Pendeluhren; er fuhrt auf eine neue Ebene der Beschreibung, die sich in der Physik weit uber die Mechanik hinaus als tragfahig erweist. Man lernt, zunachst uber die „Rauberleiter“ des d’Alembertschen Prinzips, die Lag…
Oscillations, neutrino masses and scales of new physics
We show that all the available experimental information involving neutrinos can be accounted for within the framework of already existing models where neutrinos have zero mass at tree level, but obtain a small Dirac mass by radiative corrections.
Die Prinzipien der Quantentheorie
Dieses Kapitel befasst sich mit dem formalen Rahmen der Quantenmechanik: ihren mathematischen Hilfsmitteln, der Verallgemeinerung und Abstraktion des Zustandsbegriffs, der Darstellungstheorie und einer ersten Fassung der Postulate, auf denen die Quantentheorie beruht.
Simple Applications of MaxwellTheory
In this chapter we select some characteristic examples from the great wealth of electromagnetic and optical phenomena which are described successfully by Maxwell’s equations. These case studies are restricted to the classical, non quantized version of the theory. The field of semi-classical interactions of quantum matter and classical radiation field, as well as the full quantum field theoretic treatment of Maxwell theory is described in many monographs or textbooks, such as, e.g., [QP].
Applications of Quantum Mechanics
Quantum mechanics provides the basis for most fields of modern physics and there are many well advanced methods of practical solution of specific and topical problems
The Principles of Canonical Mechanics
Canonical mechanics is a central part of general mechanics, where one goes beyond the somewhat narrow framework of Newtonian mechanics with position coordinates in the three-dimensional space, towards a more general formulation of mechanical systems belonging to a much larger class. This is the first step of abstraction, leaving behind ballistics, satellite orbits, inclined planes, and pendulum-clocks; it leads to a new kind of description that turns out to be useful in areas of physics far beyond mechanics. Through d’Alembert’s principle we discover the concept of the Lagrangian function and the framework of Lagrangian mechanics that is built onto it. Lagrangian functions are particularly …