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The time reversal and irreversibility in conventional quantum mechanics are compared with those of the rigged Hubert space quantum mechanics. We discuss the time evolution of Gamow and Gamow-Jordan vectors and show that the rigged Hubert space case admits a new kind of irreversibility which does not appear in the conventional case. The origin of this irreversibility can be traced back to different initial-boundary conditions for the states and observables. It is shown that this irreversibility does not contradict the experimentally tested consequences of the timereversal invariance of the conventional case but instead we have to introduce a new time reversal operator.

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Space of density matrices in quantum mechanics can be regarded as a Poisson manifold with the dynamics given by certain Lie–Poisson bracket corresponding to an infinite dimensional Lie algebra. The metric structure associated with this Lie algebra is given by a metric tensor which is not equivalent to the Cartan–Killing metric. The Lie–Poisson bracket can be written in a form involving a generalized (Lie–)Nambu bracket. This bracket can be used to generate a generalized, nonlinear and completely integrable dynamics of density matrices.

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In this contribution we review results on the kinematics of a quantum system localized on a connected configuration manifold and compatible dynamics for the quantum system including external fields and leading to non-linear Schrödinger equations for pure states.

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Quantum vortex configurations associated with representations of the group of area- or volume-preserving diffeomorphisms are obtained by geometric quantization techniques. This article reviews some of the mathematical results and physical predictions, providing a current perspective. A brief discussion of vortex creation and annihilation field operators is included.

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“It is difficult and perhaps still somewhat controversial to summarize the tenets of quantum physics.” [Haag, 1990] The basic logical structure of standard quantum mechanics and relativistic quantum field theory is outlined.

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Perturbative and non-perturbative types of approaches to quantization of ideal fluid flows are considered and compared. The results on stability of particular vortex structures obtained in the framework of the standard energy-Casimir method are reminded for the purpose of checking connection between stability and quantizability. Results on geometric quantizability derived by Goldin, Menikoff and Sharp for these structures are also reminded. The discrepancy between results of these two approaches being an evidence for non-perturbative character of quantization of ideal fluids is stressed. New non-perturbative approach exploiting ideas from Ashtekar programme of quantization of gravity is formulated. Some applications of the new approach in description of superfluid helium are briefly shown.

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A geometric approach for the formulation of the WKB expansion of multicomponent wave equations is formulated for arbitrary symplectic manifolds. The problem is reduced as much as possible to one component equations. Essential use is made of Fedosov’s star product. Obstructions against the existence of global WKB amplitudes may arise.

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Quantum electrodynamics, after more than sixty years since its discovery, still presents challenges and offers rewards to inquiring minds. This presentation describes some theoretical intricacies of this beautiful theory.

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The presence of macroscopic objects, like mirrors or cavity walls, changes the mode structure of the electromagnetic field. As a consequence, the vacuum fluctuations are also changed and they become position-dependent. This effect manifests itself in the appearance of the Casimir force, Casimir–Polder force, position-dependent energy shift and modified spontaneous emission. Also, the measurement of the electron magnetic moment is influenced by the cavity formed by the electrodes of the Penning trap.

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We investigate QED and quantized Yang–Mills theories coupled to matter fields in the framework of causal perturbation theory which goes back to Epstein and Glaser. In this approach gauge invariance is expressed by a simple commutator relation for the S-matrix. It has been proven in all orders of perturbation theory. The corresponding gauge transformations are simple transformations of the free fields only. In spite of this simplicity gauge invariance implies the usual Ward rsp. Slavnov–Taylor identities and unitarity on the physical subspace.

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There is, at the present time, a renewal of interest (M. Pawlowski, R. Raczka, Warsaw preprint SINS-IP/VIII/1995) in using massive non-Abelian vector boson theories without the unobserved Higgs particle. Such theories violate renormalizability by power counting. They are nevertheless thought to be renormalizable. A review of this renormalizability problem is made here in the light of gauge invariance.

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In these lectures we apply the method of causal perturbation theory to Yang–Mills theories with massive vector bosons. We show how the differential property of the BRS-charge leads to the introduction of scalar gauge fields. The general relationship between gauge invariance and unitarity is pointed out in detail by using Krein space techniques.

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I review results from recent investigations of anomalies in fermion — Yang–Mills systems in which basic notions from noncommutative geometry (NCG) where found to appear naturally. The general theme is that derivations of anomalies from quantum field theory lead to objects which have a natural interpretation as generalization of de Rham forms to NCG, and that this allows a geometric interpretation of anomaly derivations which is useful e.g. for making these calculations efficient. This paper is intended as selfcontained introduction to this line of ideas, including a review of some basic facts about anomalies. I first explain the notions from NCG needed and then discuss several different anomaly calculations: Schwinger terms in 1+1 and 3+1 dimensional current algebras, Chern–Simons terms from effective fermion actions in arbitrary odd dimensions. I also discuss the descent equations which summarize much of the geometric structure of anomalies, and I describe that these have a natural generalization to NCG which summarize the corresponding structures on the level of quantum field theory.

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Minimally coupled classical Yang–Mills and Dirac fields in the Minkowski space-time and in spatially bounded domains are investigated. The extended phase space, defined as the space of the Cauchy data admitting solutions of the evolution equations, is identified. The structure of the gauge symmetry group, defined as the group of all gauge transformations acting in the extended phase space is analysed. In the Minkowski space-time the Lie algebra of infinitesimal gauge symmetries has an ideal giving rise to the constraints. The quotient algebra, isomorphic to the structure algebra, labels the conserved colour charges. In the case of spatially bounded domains, each set of the boundary data gives rise to an extended phase space in which the evolution is Hamiltonian. The problem of a physical interpretation of the boundary data is discussed.

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The nature of the electroweak bosonic loop corrections to which current precision experiments are sensitive is explored. The set of effective parameters \(\Delta x\), \(\Delta y\), and \(varepsilon\), which quantify SU(2) violation in an effective Lagrangian, is shown to be particularly useful for this purpose. The standard bosonic corrections are sizable only in the parameter \(\Delta y\), while \(\Delta x\) and \(varepsilon\) are sufficiently well approximated by the pure fermion-loop prediction. By analyzing the contributions to \(\Delta y\) it is shown that the bosonic loop corrections resolved by the present precision data are induced by the change in energy scale between the low-energy process muon decay and the energy scale of the LEP1 observables. If the (theoretical value of the) leptonic width of the \(W\) boson is used as input parameter instead of the Fermi constant \(G_{mu}\), no further bosonic loop corrections are necessary for compatibility between theory and experiment.

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The present status of the electromagnetic form factors of strongly interacting particles is very roughly reviewed. First, the concept of the electromagnetic form factor is introduced, then a connection of its asymptotic behaviour with the quark structure of hadrons is specified and different approaches in a theoretical prediction of form factor behaviours are mentioned. A phenomenological approach, based on a synthesis of the analyticity with an experimental fact of a creation of vector–meson in the electron–positron annihilation processes and the asymptotic behaviour of the electromagnetic form factors, is discussed in more detail.

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The model of induced nonlocal quark currents based on the hypothesis that the QCD vacuum is realized by the (anti-)self-dual homogeneous gluon field is suggested. The field produces quark confinement and chiral symmetry breaking. Nonlocal extension of the bosonization procedure of quark currents is developed, which leads to the ultraviolet finite unitary \(S\)-matrix on the space of meson states. The model has a minimal set of parameters: quark masses, vacuum field strength and the quark–gluon coupling constant. The vacuum field provides qualitative regimes in the meson spectrum: mass splitting between pseudoscalar and vector mesons, Regge trajectories, masses of heavy quarkonia and heavy-light mesons in the heavy quark limit. The masses and weak decay constants of mesons from all qualitatively different regions of the spectrum are described to within ten per cent inaccuracy.

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After reviewing basic facts about large-order behaviour of perturbation expansions in various fields of physics, I consider several alternatives to the Borel summation method and discuss their relevance to different physical situations. Then I convey news about the singularities in the Borel plane in QCD, and discuss the topical subject of the resummation of renormalon chains and its application in various QCD processes.

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A fully covariant construction of quantum field theory with antiparticles realized by negative frequency states, but positive energy density is shown to provide a qualitative explanation for the experiments on the neutral \(K\) mesons so far interpreted as symmetry violation.

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Two topics on the standard electroweak theory are discussed based on its remarkable success in precision analyses. One is a test of structure of the radiative corrections to the weak-boson masses as a further precision analysis. The other is an indirect Higgs-boson search through the radiative corrections to the various quantities measured at LEP.

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QCD Sum Rules are used to study nonperturbative behaviour of quark propagators. Instead of using operator product expansion, we use Dyson–Schwinger equations and Ward–Takahashi identity. We found that good agreement with the data is obtained for logarithmically divergent effective quark mass.

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We discuss two-photon and hadronic production of \(B_c\) mesons in nonrelativistic bound state approximation and to lowest order in the coupling constants \(\alpha \) and \(\alpha _s\). It is shown that in photon–photon collisions, heavy quark fragmentation is dominated by recombination of \(\bar b\) and \(c\) quarks up to the highest accessible transverse momenta. In contrast, in hadroproduction, which at high energies mainly involves gluon–gluon collisions, the fragmentation mechanism dominates at transverse momenta \(p_{\rm T} \gt m_{B_a}\), providing a simple and satisfactory approximation of the complete \(O(\alpha ^4_s)\) results in the high-\(P_{\rm T}\) regime. Contradictions in previous publications on hadroproduction of \(B_c\) mesons are clarified. We also present predictions for cross sections and differential distributions at present and future accelerators.

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Nonzero quark and gluon condensates generate nonzero value of pion mass, even in the zero limit of current quark mass. In turn, the nonzero \(\rho \) meson width is due to the nonzero pion mass. Nonzero quark and gluon condensates are also necessary conditions for permanent confinement of quarks and gluons. The notion of permanent confinement means: (i) the nonexistence of any asymptotic quark or gluon states, (ii) the nonexistence of any asymptotic continuum partonic states, and (iii) the nonexistence of any colourfull bound systems. Only colourless hadrons composed of permanently confined partons are present. These hadrons must be calculated as solutions of truly relativistic bound state equations. Masses of constituent quarks are defined in hadrons, and crucially depend on the magnitude of a space-like Wightman–Garding relative momentum. The dominating binding potential of constituents in hadrons is the QCD analog of Coulomb interaction. There is no place for any interaction between constituents which could increase indefinitely with a space-like separation of constituents. For example, a linear interaction, with singularity \((q-q')_4\) in momentum space, is ruled out on two grounds: (i) as contradicting Dyson–Schwinger equations, and (ii) as being in conflict with the cluster property of local QCD, if there is a nonzero mass gap. For QCD with quark condensate for up and down quarks the Goldstone theorem fails. If it would hold it would require massless pion for massless quarks, and the possibility of approximating the pion field by a local field. Nonperturbative QCD with permanently confined quarks says “no” to both of these claims.

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In this talk some recent developments in the overlap formulation of chiral gauge theories on a lattice is briefly reviewed. We argue that the overlap formalism correctly accounts for all the desirable features of the chiral Dirac determinant for slowly varying weak external gauge fields.

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Problems in fermion models on the lattice are discussed. A general procedure which allows to remove fermion doubling preserving gauge invariance in anomaly free chiral models on the lattice is presented. A representation of fermion determinant as a path integral of bosonic effective action is constructed.

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It is shown that there exists an intimate relationship between Bose Einstein correlations and quantum field theory. On the one hand several essential aspects of BEC cannot be understood and even formulated without second quantization. On the other hand BEC can serve as a unique tool in the investigation of modern field theory and in particular of the standard model. Some new developments on this subject related to multiparticle production and squeezed states are also discussed.

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A number of seemingly exotic phenomena seen in the highest cosmic-ray experiments are briefly discussed. We argue that they indicate existence of non-statistical fluctuations and strong correlations in the fragmentation region of multiparticle production processes unaccessible to the present accelerators.

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We present the status of experimental studies of the nucleon spin structure from the polarized deep inelastic scattering. We give an overview of results on polarized structure functions \(g_1\) and on spin distributions of valence and sea quarks. The Bjorken and the Ellis–Jaffe sum rules for the first moments of the structure functions \(g_1\) are discussed. The Bjorken sum rule is well confirmed experimentally. The Ellis–Jaffe sum rule significantly disagrees with data and its violation can be interpreted as due to negative polarization of the strange sea. The spin structure function of the proton, \(g^p_1(x)\), is positive and the neutron, \(g^n_1(x)\), is negative for \(x \lt 0.1\) and positive elsewhere. The first semi-inclusive results from CERN indicate that the difference between \(g^p_1\) and \(g^n_1\) for low \(x\) can be related to different signs of polarization of valence up and down quarks. It is found that the overall spin carried by the valence up quarks in the proton is positive and amounts to unity. The valence down quarks are polarized in the opposite direction. Spin of non-strange sea quarks is consistent with zero over the full experimentally accessible range \(0.003 \lt x \lt 0.7\).

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Selected problems in heavy quark physics are discussed. The wealth of research problems in this field of physics is stressed.

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An attempt for the formulation of a local version of the spectrum condition on globally hyperbolic spacetime is discussed. It relies on microlocal analysis.

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We discuss the choice of the Lagrangian in the Poincaré gauge theory of gravity. Drawing analogies to earlier de Sitter gauge models, we point out the possibility of deriving the Einstein–Cartan Lagrangian without cosmological term from a modified quadratic curvature invariant of topological type.

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Supergravity theories admit a large variety of extended-object solutions that are characterized by the saturation of a Bogomol’ny bound, with a consequent partial preservation of unbroken supersymmetry. We present a scheme for the classification of such solutions into families related by dimensional reduction and oxidation, each headed by a maximal non-isotropically-oxidizable, or “stainless” solution.

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I review different methods used in description of the high-energy processes in quantum gravity. As a first I discuss the result obtained within the eikonal approximation. Next I describe the derivation of the effective action for the quantum gravity in the multi-Regge kinematics.

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According to Newton, the ultimate building blocks of matter are hard frictionless spheres. This conjecture is here analyzed under different assumptions, which are: 1. The ultimate objects of matter are frictionless positive and negative Planck mass particles obeying nonrelativistic Newtonian mechanics. 2. The Planck mass particles interact with the Planck force \(c^4G\) (\(c\) velocity of light, \(G\) Newton’s constant) locally within a Planck length \(r_{\rm p}\), with the positive Planck mass particles exerting a repulsive and the negative Planck mass particles an attractive force, whereby particles of equal sign are accelerated away from each other and those of opposite sign — as a particle-hole interaction — accelerated towards each other. 3. Space if filled with an equal number of positive and negative Planck mass particles, whereby each Planck length volume \(r^3p\) occupies one Planck mass particle. Making these three assumptions we derive: 1. Nonrelativistic quantum mechanics as an approximation with departures from this approximation suppressed by the Planck length. 2. Lorentz invariance as a dynamic symmetry for energies small compared to the Planck energy. 3. The operator field equation for the previously proposed Planck ether model of a unified theory of elementary particles. In contrast to theories in which the ultimate objects are strings at the Planck scale, the alternative theory proposed here does not need a higher dimensional space, but rather can be formulated in \(3+1\) dimensions.

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A new method of deriving equations of motion from field equations is proposed. It is applied to classical electrodynamics. As a result, we obtain a new, perfectly gauge-invariant, second order Lagrangian for the motion of classical, charged test particles.

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Symplectic formalism for classical fields with intention to apply it to geometric quantization in field theory is presented.

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The internal space-time symmetries of relativistic particles are dictated by Wigner’s little groups. The \(O(3)\)-like little group for a massive particle at rest and the \(E(2)\)-like little group of a massless particle are two different manifestations of the same covariant little group. Likewise, the quark model and parton pictures are two different manifestations of the one covariant entity.

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In this paper usefulness of Kaluza–Klein-like description of charge carriers in some condensed matter systems is suggested. Application for description of polarons and bipolarons in some synthetic metals is proposed. Connection between this approach and the more standard, one-dimensional one, is shown. The multidimensional description is relativistic, applies multidimensional Dirac or Klein–Gordon equation. The one-dimensional description applies Schrödinger or Dirac equation with appropriate periodic potential. Physical consequences for the “multidimensional” mechanism of conductivity in synthetic metals are discussed and comparison with experiments is given.

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Given an algebra, a finite projective right module and a differential algebra over this algebra, a graded Lie algebra with derivation is constructed. It is shown that the algebraic structure of the Mainz–Marseille approach to the standard model may be obtained making use of this general construction in a special case. Thereby, a rigorous mathematical link between Connes’ noncommutative geometry and the Mainz–Marseille approach is established.

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Using the method of averaging we construct two perturbation algorithms analogous to the classical constructions. One of them reproduces the known Rayleigh-Schrödinger perturbation theory (PT) in quantum mechanics but with new closed form expressions. The other (Kolmogorov’s PT) yields a new PT where the resulting expansion is in terms of functions of the perturbation parameter.

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These pedagogical lectures contain some motivation for the study of quantum groups; a definition of “quasitriangular Hopf algebra” with explanations of all the concepts required to build it up; descriptions of quantised universal enveloping algebras and the quantum double; and an account of quantised function algebras and the action of quantum groups on quantum spaces.

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The existence of the theory of ‘twisted cotangent bundles’ (symplectic groupoids) allows to study classical mechanical systems which are generalized in the sense that their configurations form a Poisson manifold. It is natural to study from this point of view first such systems which arise in the context of some basic physical symmetry (space-time, rotations, etc.). We review results obtained so far in this direction.