Quantum Theory and Symmetries

The following sections are included:

• PREFACE

• CONTENTS

We prove that Haldane exclusion statistics (ES) can be included in the more general class of standard interacting Fock spaces (IFS). This result provides a standard rule to assign in a consistent way the parameters of the most general exclusion statistics and, through the results of [4], it shows that there is a canonical one–to–one correspondence between these parameters and the families of orthogonal polynomials associated to an arbitrary measure with moments of all orders. Such a correspondence has been discovered in a multiplicity of particular cases (see for example [5], [9]). Thus our result gives a rule to identify immediately the family of orthogonal polynomials associated to a given ES. Moreover it allows to extend the ES in a natural way to infinitely many species and in particular it suggests a standard way to take the thermodynamic limit of these statistics.

A fundamental discrete symmetry of quantum field theory, namely symmetry, is derived from the assumption of Lorentz invariance and positivity of the spectrum of the Hamiltonian. What happens if we assume only Lorentz invariance and symmetry? Hamiltonians having this property need not be Hermitian, but, except when is spontaneously broken, the energy levels of such Hamiltonians are all real and positive! In this talk I examine quantum mechanical and quantum field theoretic systems whose Hamiltonians are non-Hermitian but obey PCT symmetry. These systems have weird and remarkable properties, and the classical theories underlying these quantum systems also have strange behaviors. Simple examples of such Hamiltonians are H = p² + ix³ and H = p² - x⁴. Hamiltonians such as these may be thought of as complex deformations of conventional Hermitian Hamiltonians. Thus, in this talk I will be studying the analytic continuation of conventional classical mechanics and quantum mechanics into the complex plane.

Resonances and decaying particles are characterized by two numbers, either by the mass m (or resonance energy E_R) and the Breit-Wigner width Γ or by the mass m and the decay rate (inverse lifetime) 1/τ. In non-relativistic quantum mechanics, it is always assumed that Γ = 1/τ, though this is only based on an approximation (Weisskopf-Wigner). For relativistic resonances, opinions are more divided and some believe that the lineshape has a more complicated form than the Breit-Wigner, or at least that there is an energy dependent width Γ(s). This has led to disputes about the correct definition of mass and width for a relativistic resonance, in particular for the Z-boson. An answer to this problem is provided if, in analogy to Wigner's definition of stable relativistic particles, one defines relativistic resonances by irreducible representations of the Poincaré group. Since the decay of resonances is an irreversible process, one needs to generalize the unitary representations (m,j) of Wigner to semigroup representations characterized by spin j and a complex number for the invariant mass squared s_R = (m - iΓ/2)². These semigroup representations and their transformation properties under causal Poincaré transformations are presented. They provide a unique definition of mass, width and lineshape for a relativistic resonance if one accepts that the relation Γ = 1/τ holds also for relativistic resonances.

A review on the representations of symmetries in quantum field theory of local observables will be given. One part is concerned with the interplay of the locality condition in configuration space (Einstein causality) and the spectrum condition in momentum space (positive energy). Another part uses Tomita's theory of modular Hilbert algebras (Tomita-Takesaki theory) to induce symmetries for the quantum field theory.

We give a survey of recent results on counting systems, so-called beta-integers, based on Pisot-Vijayaraghavan algebraic integers β > 1 and leading to the notion of beta-lattices in ℝ^d. These beta-lattices may be equipped with an abelian group structure and reveal themselves as ideal labelling frames for discrete self-similar quasiperiodic structures with scaling invariant β.

A model unifying general relativity and quantum mechanics is proposed the fundamental symmetry of which is not that of a group but rather that of a groupoid. With this groupoid there is associated a noncommutative C*-algebra . It defines a certain noncommutative space. It is shown that both general relativity and quantum mechanics can be recovered from . Noncommutative space defined by is nonlocal. The usual concepts of space and time, as composed of points and instants, are meaningless in it. In spite of this, a generalized dynamics can be defined, but it is intrinsically probabilistic. It is demonstrated how, in the process of transition from the noncommutative regime to the commutative regime, the usual dynamics and time emerge.

In this paper an overview is given of connections between the Witten-Dijkgraaf-Verlinde-Verlinde equations and its generalization and various integrable systems. In particular its link with the n-component KP-hierarchy is treated in a fairly detailed way.

It is shown that the local Gauss law of quantum chromodynamics on a finite lattice implies a gauge invariant, additive law giving rise to a gauge invariant Z₃-valued global charge in QCD. The total charge contained in a region of the lattice is equal to the flux through its boundary of a certain Z₃-valued, additive quantity.

The following sections are included:

• Introduction

• Quantum states

• States of composite quantum systems

• Entanglement measures and geometry of entangled states

• Manifolds of equivalent states

• Identical particles

• Outlook

• References

We consider the simplest class of Lie-algebraic deformations of space-time algebra, with the selection of κ-deformations as providing quantum deformation of relativistic framework. We recall that the κ-deformation along any direction in Minkowski space can be obtained. Some problems of the formalism of κ-deformations will be considered. We shall comment on the conformal extension of light-like κ-deformation as well as on the applications to astrophysical problems.

Tomograms introduced for the description of quantum states in terms of probability distributions are shown to be related to a standard star-product quantization with appropriate kernels. Examples of symplectic tomograms and spin tomograms are presented.

The "algorithmic cooling" of Boykin, Mor, Roychowdhury, Vatan and Vrijen (BMRVV) is a powerful cooling method for obtaining a large number of highly polarized spins in liquid nuclear-spin systems at finite temperature. The cooling is done by compressing the antropy of a string into a portion of it and then swaping that portion with rapidly thermal-relaxing bits, which transfers the antropy quickly to the environment. Algorithmic cooling could enable cooling spins to a very low temperature without cooling the environment and could resolve the scaling problem of NMR quantum computing devices. The BMRVV cooling algorithm is described in a simplified model in which all spins have identical bias (the minimum over all the different biases of the spins), and using blocks of m bits, hence the resulting cooling technique is non-optimal. We present here initial analysis of two improvements of the algorithm: working with variable-size blocks in the cooling process, and using better the fact that initial biases are not equal. As result, much less cooling steps are required in order to yield more spins which are highly polarized (with the desired bias).

We discuss the reducible representation of a particle on a torus with magnetic field, where the gauge automorphisms are unitarily implementable. Identifying these gauge automorphisms for several particles demands a soldering of this representation. The soldering procedure is unique only for integer magnetic flux. For rational magnetic flux a kind of interaction between different particles results, that becomes negligible in the thermodynamic limit.

The time behaviour of partial reflection by opaque photonic barriers was measured with microwaves. It was observed that unlike the duration of partial reflection by dielectric sheets, the measured reflection duration of barriers is independent of their length. The experimental results point to a nonlocal behaviour of evanescent modes.

The limitations of the approach based on using fields restricted to the lightfront (Lightfront Quantization or p→ ∞ Frame Approach) which drive quantum fields towards canonical and ultimately free fields are well known. Here we propose a new concept which does not suffer from this limitation. It is based on a procedure which cannot be directly formulated in terms of pointlike fields but requires "holographic" manipulations of the algebras generated by those fields. We illustrate the new concepts in the setting of factorizable d=1+1 models and show that the known fact of absence of ultraviolet problems in those models (in the presence of higher than canonical dimensions) also passes to their holographic images. In higher spacetime dimensions d>1+1 the holographic image lacks the transversal localizability; however this can be remedied by doing holography on d-2 additional lightfronts which share one lightray (Scanning by d-1 chiral conformal theories).

The following sections are included:

• Definitions

• Explanations

• Finitedimensional Algebras

• Proposition

• Corollary

• Proposition

• Subalgebras of the Weyl Algebra

The following sections are included:

• The Algebraic Concept

• Gauge theories

• References

In this work, we construct the algebra of differential forms with exterior differential d satisfying d³ = 0 on one-dimensional space. We prove that this algebra is a graded q-differential algebra where q is a cubic root of unity. Since d² ≠ 0 the algebra of differential forms is generated not only by the first order differential dx but also by the second order differential d²x of a coordinate x. We study the bimodule generated by this second order differential, and show that its structure is similar to the structure of bimodule generated by the first order differential dx in the case of the anyonic line.

Three computers, with local independent choices, genereate the EPR correlations hence violating Bell's inequality.

We analyse the Lorentz and Feld-Tai lemmas (or reciprocity-like theorems) having numerous applications in electrodynamics, optics, radiophysics, electronics, etc. It is demonstrated that these lemmas are valid for more general time-dependences of the electromagnetic field sources than it was suggested up to now. It is shown also that the validity of reciprocity-like theorems is intimately related to the equality of electromagnetic action and reaction: both of them are fulfilled or violated under the same conditions. Conditions are stated under which reciprocity-like theorems can be violated. The concrete example of their violation is presented.

Hierarchies of equations invariant under the Schrödinger algebras in (n+1)-dimensional spacetime are constructed for n = 1,2,3. The heat/Schrödinger equation is found as a member of the hierarchies.

It is shown that the structure of equations for the Bethe ansatz parameters is closely connected with the algebraic structure of the model. The Hamiltonians of Gaudin's models have the form of the sum of products of operators of two algebras. The original Bethe equations are connected only will the form of these algebras. But it turns out, that if one considers the concrete representations of these algebras, the structure of corresponding equations can be simplified. We will call them reduced Bethe equations. This system of Bethe equations is found for three-boson interaction in explicit form.

We present here wavelets of class Cⁿ(ℝ) living on a sequence of aperiodic discretization of ℝ, known as the Fibonacci chain, constructed via the splines functions. The construction method is based on an algorithm published by G. Bernau. Corresponding multiresolution analysis is defined and numerical examples of linear scaling functions and wavelets are shown.

A molecular algebraic model is applied to the analysis of thermodynamic properties of molecules. The local anharmonic effects are described by anharmonic bosons associated with the SU(2) algebra. The vibrational high-temperature partition function and the related thermodynamic potentials are derived in terms of the parameters of the model. Quantum analogues of anharmonic bosons, q-bosons, are used to describe anharmonic properties of molecules. A physical interpretation of a linear quantum deformation associated with the model is given.

We present a method to construct representations of quantum algebras having the structure of bicrossproduct. This method is, in some sense, the quantum counterpart of the Mackey method for Lie groups with structure of semidirect product. As an example we consider the quantum κ–Galilei algebra.

The finite oscillator in two dimensions is a system whose dynamical algebra is so(4); it has a discrete, finite configuration space whose points can be arranged following cartesian or polar coordinates. Its wavefunctions satisfy 'Schrödinger' difference equations; in the cartesian model they are Wigner d-functions involving Kravchuk polynomials, while in the radial model they are su(2) Clebsch-Gordan functions containing dual Hahn polynomials. An su(2) symmetry algebra and group can be imported from the ordinary oscillator; coherent states exist, and a covariant Wigner function on a compact phase space can be formulated.

The spectrum-generating algebra of the finite oscillator model is so(3) [or su(2)], whose generators are the position, momentum and mode number operators. The spectra thus consist of a finite number of equally-spaced values. We examine the contraction of this model to the ordinary quantum oscillator as the number and density of points increases. This is done for the algebra and for the wave functions.

Mathematical diffraction theory is concerned with the analysis of the diffraction measure of a translation bounded complex measure ω. It emerges as the Fourier transform of the autocorrelation measure of ω. The mathematically rigorous approach has produced a number of interesting results in the context of perfect and random systems, some of which are summarized here.

The su_q(2) algebra is considered as a natural dynamical symmetry algebra for some quantum optical models. This allows us to discuss spectrum and dynamical properties of these models. We use the Wigner quasiprobabilty function on the sphere to visualize quantum nonlinear dynamics.

A procedure for the systematic construction of integrable systems from symplectic realizations of a Poisson coalgebra with Casimir element is presented. Several examples of Hamiltonians with (either undeformed or 'quantum') coalgebra symmetry are given, and their Liouville integrability is discussed.

Medium modifications of the πωρ vertex are analyzed in context of the ω → π⁰γ* and ρ → πγ* decays in nuclear matter. A relativistic hadronic model with mesons, nucleons, and Δ(1232) isobars is applied. A substantial increase of the widths for the decays ω → π⁰γ* and ρ → πγ* is found for photon virtualities in the range 0.3 – 0.6GeV. This enhancement has a direct importance for the description of dilepton yields obtained in relativistic heavy-ion collisions.

In this talk I discuss a new class of exactly solvable differential-equation eigenvalue problems -y″(x) + x^2N+2y(x) = x^N Ey(x) (N = -1,0,1,2,3,…) on the interval -∞ < x < ∞. These problems can be solved exactly for the eigenvalues E and corresponding eigenfunctions y(x). The eigenvalues are all integers and the eigenfunctions are confluent hypergeometric functions. The eigenfunctions can be rewritten as products of polynomials and functions that decay exponentially as x → ±∞. For odd N the polynomials obtained in this way are new and interesting classes of orthogonal polynomials. For example, when N = 1, the eigenfunctions are orthogonal polynomials in x³ multiplying Airy functions of x². The properties of the polynomials for all N are described.

Unification ideas lead to the consideration of a generalized Dirac equation with boson and fermion solutions. These fields naturally support an algebra and inner product which lead to vertices among them. Symmetries in the equation can be associated with flavor and gauge groups. Extended versions predict isospin and hypercharge SU(2)_L × U(1) symmetries, their vector carriers, two-flavor charged and chargeless leptons, and scalar particles. A mass term produces breaking of the symmetry to an electromagnetic U(1), a Weinberg's angle with sin²(θ_W) = .25, and respective coupling constants and .

Field equations of S² sigma model ("the A3 model") with spontaneously broken Z(2) symmetry are presented for (D + 1)-dimensional space-time. The A3 model is an extension of the sine-Gordon equation (SGE) and supports kink-like U(1) charged solitons which are generalization of neutral solitons of the SGE. Natural question arises: is the A3 model completely integrable in (1+1)-dimensional space-time? The Lorentz-invariant scalar A3 field can be viewed as promising alternative to Higgs field.

We give solutions of the quantum conformal deformations of the full Maxwell equations in terms of deformations of the plane wave.

Wavelet analysis of different patterns reveals some symmetries and singularities otherwise hidden in the pattern. Its general methods are briefly reviewed. Examples from turbulence, cavitation, Cherenkov gluon emission and quark-gluon jets structure in QCD are considered.

In this note we complement recent results on the exchange r-matrices appearing in the chiral WZNW model by providing a direct, purely finite-dimensional description of the relationship between the monodromy dependent 2-form that enters the chiral WZNW symplectic form and the exchange r-matrix that governs the corresponding Poisson brackets. We also develop the special case in which the exchange r-matrix becomes the 'canonical' solution of the classical dynamical Yang-Baxter equation on an arbitrary self-dual Lie algebra.

We briefly report on our result that, if there exists a realization of a Hopf algebra H in a H-module algebra , then their cross-product is equal to the product of itself with a subalgebra isomorphic to H and commuting with . We illustrate its application to the Euclidean quantum groups in N ≥ 3 dimensions.

The following sections are included:

• Introduction

• Discretisation of redshift differences in paired galaxies

• Redshift quantisation in the Local Group

• Redshift periodicity in the Local Supercluster

• Large-scale periodisation

• Quasar redshift quantisation

• Conclusions

• References

The aim of this paper is to introduce our idea of Holonomic Quantum Computation (Computer). Our model is based on both harmonic oscillators and non–linear quantum optics, not on spins of usual quantum computation and our method is moreover completely geometrical.

We consider Ising models defined on periodic approximants of aperiodic graphs. The model contains only a single coupling constant and no magnetic field, so the aperiodicity is entirely given by the different local environments of neighbours in the aperiodic graph. In this case, the partition function zeros in the temperature variable, also known as the Fisher zeros, can be calculated by diagonalisation of finite matrices. We present the partition function zero patterns for periodic approximants of the Penrose and the Ammann-Beenker tiling, and derive precise estimates of the critical temperatures.

We define a class of orthosymplectic superalgebras osp(m; j|2n;ω) which may be obtained from osp(m|2n) by contractions and analytic continuations in a similar way as the orthogonal and the symplectic Cayley-Klein algebras are obtained from the corresponding classical ones. Contractions of osp(1|2) and osp(3|2) are regarded as an examples.

Fine gradings of simple Lie algebras determine their distinguished bases. Certain subgroups of automorphisms yield the symmetries of these bases. Complete description of the bases and their symmetries is given for sl(2,C) and sl(3,C).

Advances in micro-technology of the last years have made it possible to carry optics textbooks experiments over to atomic and molecular beams, such as diffraction by a double slit or transmission grating. The usual wave-optical approach gives a good qualitative description of these experiments. However, small deviations therefrom and sophisticated quantum mechanics yield new surprising insights on the size of particles and on their interaction with surfaces.

A new non-standard quantum deformation of so(4,2), the conformal algebra of the (3 + 1)-dimensional Minkowskian spacetime, is explicitly constructed by starting from the non-standard (or Jordanian) classical r matrix of sl(2,ℝ). Since the deformation parameter has the dimension of a time, this quantum so(4, 2) algebra is called 'time-type'. The corresponding Hopf structure, universal R matrix and differential-difference realization are presented. The latter allows us to show, in a natural way, that this quantum algebra is the symmetry algebra of a time discretization of the (3 + 1)-dimensional wave equation on a uniform lattice. The space-type deformation counterpart is also pointed out.

The aim of this paper is to study q-Laplace operator and the space of q-harmonic polynomials on the quantum complex vector space generated by q-commuting elements z_i, w_i, i = 1, 2, …, n, on which the quantum group GL_q(n) acts.

Lie algebra contractions from o(n+1) to e(n) are used to construct the asymptotic limit of the interbasis expansions between two general subgroup chains O(n + 1) ⊃ O(n_α+n_β)⊗O(n_γ) and O(n+1) ⊃ O(n_α)⊗O(n_γ+n_β), where n+1 = n_α+n_γ+n_β. It provides the asymptotic formula of Racah coefficients for SU(2) and SU(1,1) groups.

The interrelation between certain quadratic algebras occurring in quantized enveloping algebras on the one hand side, and Poisson structures and deformation quantization on the other, is discussed. It is shown that there are two different methods of constructing ⋆-products available. Further implications in the direction of quantized wave- and Dirac-operators are investigated.

We show that the only requirement of general covariance determines essentially the quantum operators associated with a classical quantisable function and the Schrödinger operator. Our framework is the covariant quantum mechanics of a scalar quantum particle in a curved spacetime which is fibred over absolute time and equipped with given spacelike metric, gravitational field and electromagnetic field. In particular, in the flat case, we recover essentially the standard operators.

We recall first the exceptional propagation properties of classical Born-Infeld theory discovered by G. Boillat in 1970 (³). Then we display some ancient and more recent static solutions of solitonic type, and discuss their properties

It has recently been shown that quasicrystalline structures of interest in physics (namely the 5 and 10-fold, 8-fold and 12-fold structures) are subsets of specific quasiperiodic point sets, named beta-lattices. Beta-lattices are eligible labelling frames for quasiperiodic sets. They are based on discrete sequences related to the irrational number β > 1 as lattices are based on integers. In this article we show how it is possible to embed a 12-fold quasiperiodic pattern observed in the hydrodynamic Faraday instability in an adapted beta-lattice.

We study the quantum dynamics of the SU(2) quasiprobability distribution ("Wigner function") for the simple nonlinear Hamiltonian (finite analog of the Kerr medium, ). The quasiclassical approximation for the Wigner function and corresponding evolution of mean values are considered and compared with the exact and classical solutions.

Just hundred years ago in 1900 D.Hilbert formulated 23 problems which, in his opinion, the mathematicians of XX century would have to solve. Among them the sixth problem pointed to necessity to state the mathematical formulation of the axioms of physics. As particular case of this problem Hilbert considered the possibility of the physical axioms construction according to the model of the axioms of geometry. Thus the sixth Hilbert's problem contained the problem of physics geometrization. For all XX century long this problem formed the strategies of scientific researces in theoretical physics and in some new mathematical topics, especially in geometry. Appearence of the special and general relativity as well as the geometrical gauge field theory can be regarded as consequent stages in the sixth Hilbert's problem solution. According to these physical theories the corresponding new geometries appeared: Minkowskian 4D-geometry (for SR), Riemannian 4D-geometry and its Cartan's formulation by tetrads (for GR), and after all, the fibre bundle space geometry which is the base of the geometrical gauge field theory. The gauge field theory is successful in explanation of phenomena of particle physics and gravity. Now the problem of today consists in application of the geometrical gauge field theory for relativistic nuclear physics.

The uncertainty relations for a quantum particle on a circle are introduced minimized by the corresponding coherent states. The sqeezed states for the quantum mechanics on a circle and the related generalized uncertainty relations are identified as well.

We consider the dual picture of the Yang-Mills theory at large distances. The dual Higgs model is reformulated in terms of two-point Wightman functions with the equations of motion involving higher derivatives.

The main goals of the programme of studying CP violation in B decays are briefly presented. It is stressed that reliable ways of handling final-state interactions have to be found if these goals are to be achieved. On the example of decays it is shown that rescattering may strongly affect phases of decay amplitudes. The need to understand SU(3) breaking effects is also pointed out.

We consider supersymmetry algebras in arbitrary spacetime dimension and signature. Minimal and maximal superalgebras are given for single and extended supersymmetry. It is seen that the supersymmetric extensions are uniquely determined by the properties of the spinor representation, which depend on the dimension D mod 8 and the signature |ρ| mod 8 of spacetime.

Some traditional and new types of transforms of analytic signal used in information processing including recently developed method of noncommutative tomography are discussed. The analogy of analytic signal to wave functions and vectors in Hilbert space of states in quantum mechanics is used to present different integral transforms as different obvious relations among the vectors and basis vectors in the linear space. The approach developed permits to construct new relations between wavelet transforms, quasidistibutions and tomograms of analytic signal.

We describe three spectral triples related to the Kronecker foliation and analyze the related differential calculi Ω_D.

The realization of the two-dimensional Poincare algebra in terms of the noncommutative differential calculus on the commutative algebra of functions is considered. Corresponding algebra of functions is commutative and is defined by the spectrum of unitary irreducible representations of the two-dimensional De Sitter group. Gauge invariance principle consistent with this quantum geometry is considered

The following sections are included:

• Introduction

• The relativistic three body problem with spin

• The eigenstates of our problem

• The system of three quarks and the spectra for non-strange barions

• Conclusion

• References

The following sections are included:

• Introduction

• S0₀(p+1,q), and their Lie algebras

• Representations of from representations of SO₀(p+2,q) and SO₀(p+1,q+1)

• References

A slight generalization of the C*-algebraic approach to quantum theory is proposed. It is based on the well-known notion of covariance system, also called C*-dynamical system. New definitions of state and of representation of a covariance system are introduced. The concepts are clarified with standard quantum mechanics as an example.

We consider a super generalization of the Duffin-Kemmer-Petiau algebra. It is shown that it is related to the Lie superalgebra osp(N, M).

A description of the quantum algebra U_q[sl(n + 1)] via a new set of generators, called deformed Jacobson generators, is given. It provides an alternative to the canonical description of U_q[sl(n + 1)] in terms of Chevalley generators. The Jacobson generators satisfy trilinear commutation relations and define U_q[sl(n + 1)] as a deformed Lie triple system. Fock representations are constructed and the action of the Jacobson generators on the Fock basis is written down. Finally, Dyson and Holstein-Primakoff realizations are given.

We construct coherent states using sequences of combinatorial numbers such as various binomial and trinomial numbers, and Bell and Catalan numbers. We show that these states satisfy the condition of the resolution of unity in a natural way. In each case the positive weight functions are given as solutions of associated Stieltjes or Hausdorff moment problems, where the moments are the combinatorial numbers.

Two new types of coherent states associated with the C_λ-extended oscillator, where C_λ is the cyclic group of order λ, are introduced. They satisfy a unity resolution relation in the C_λ-extended oscillator Fock space (or in some subspace thereof) and give rise to Bargmann representations of the latter, wherein the generators of the C_λ-extended oscillator algebra are realized as differential-operator-valued matrices.

Classical thermodynamics of a linear chain with a local asymmetric double sinh-Gordon potential is studied in a continuum limit. Transfer integral method converts the problem of finding the partition function into the pseudo Schrödinger equation with temperature dependent mass and unchanged potential. Due to its hidden symmetry this double well potential belongs to the class of quasi exactly solvable quantum mechanical models. In consequence, classical partition function and other thermodynamic properties of corresponding (1+1)-dimensional systems are known exactly at several temperatures.

Unlike the standard quantum field theory (QFT) in this review paper the positivity condition is alternatively overlooked. Consequently indefinite metric quantization and the presence of negative norm states as the auxiliary unphysical states has been included. The elimination of the negative norm states in the standard QFT, results in the appearance of the following anomalies: (1) infrared and ultraviolet divergence and (2) breaking of the de Sitter covariance for the minimally coupled scalar field and the breaking of the de Sitter covariance for linear quantum gravity. Inclusion of negative norm states automatically removes the divergence and symmetry breaking. Consequently the QFT is automatically renormalized and the covariance of the QFT in de Sitter space is conserved. The compatibility of this method with the results of some existing calculation is discussed in the last section.

A method for determining the orbit types, as well as their partial ordering, of the action of the group of gauge transformations on the space of connections for gauge theories with gauge group SU(n) in space-time dimension d ≤ 4 is presented. The method enables one to determine explicitly the corresponding stratification of the gauge orbit space. As an application, we discuss a criterion that characterizes kinematical nodes for physical states in 2+1-dimensional Chern-Simons-Yang-Mills theory proposed by Asorey et al..

Group-theoretical aspects of control of quantum systems are presented and a concise summary of some important results on controllability of quantum systems is provided.

Bifurcation is a common phenomenon in nonlinear dynamics and a first step into the domain of chaos. In quantum mechanics, which is supposedly a linear theory, nonlinear equations also occur. So the dynamics of position and momentum uncertainties can be described by a nonlinear differential equation of Riccati-type. Despite the nonlinearity of this equation, no bifurcation occurs, as long as the conservative standard problems that can be solved analytically are considered. However, the situation changes drastically if the influence of a dissipative environment is taken into account in an effective way (via a nonlinear Schrödinger equation). The introduction of an additional parameter - the friction coefficient - leads to bifurcation in the dynamics of the quantum fluctuations. The consequences, e.g. for the wave-packet solutions of the damped free motion, will be discussed.

In spite of widespread opinion it is possible to construct a spectral measure covariant with respect to the time translations. This spectral mesure corresponds to the time operator. However, its physical interpretation is not clear.

Discussed are classical and quantum rigorously solvable one-dimensional lattice models. The approach is based on Hamiltonian systems on the groups and .

The q-boson realization of the U_q(2,1) irreducible representations belonging to the positive discrete series is constructed. The complementary relation between U_q(2,1) and SU_q(2) algebras is used to find the connection of Weyl coefficients for SU_q(2,1) with Racah coefficients for SU_q(2).

The most general isotropic quantum deformation of Minkowski space is constructed. It depends on two deformation parameters. The four-dimensional differential calculi on the quantum Minkowski space are obtained and classified. Quantum Poincaré group acting covariantly on two-parameter quantum Minkowski space is constructed. The action of this group on the space is braided and the group itself is quantized braided group. The antipode in the group algebra is constructed with help of a pair of quantum metric tensors.

In the case of at root of unity q-deformed analogues are proposed for the generator of the maximal compact subalgebra, J, and for the raising and lowering operators. We prove an algebraic identity which implies that J has similar properties as in the nondeformed case.

A method is elaborated of constructing a time-dependent periodic Hamiltonian for which a system of Schrödinger equations admits analytic solutions. Time-independent soluble problems are transformed into time-dependent ones by a set of unitary time-dependent transformations and a proper choice of initial states. A new class of 2 × 2 periodic time-dependent Hamiltonians with cyclic solutions is constructed in a closed analytic form. As an application of the method, the expectation value of Hamiltonian, total, dynamical, and geometric phases are derived in terms of the obtained solutions.

Diffusion systems are algebraic structures underlying simple exclusion models, that is stochastic processes on linear lattices modelling diffusive dynamics. Representations for two representative types of Diffusion systems are considered with the aim to emphasize the structural differences in the representation theory of different families of Diffusion systems.

The following sections are included:

• Introduction

• The Planck Aether Hypothesis

• The Origin of Quantum Mechanics

• Heisenberg-Type Operator Field Equation

• Lorentz Invariance and Quantum Mechanics

• References

Many non-Hermitian –symmetric Hamiltonians support the real spectrum. We offer an explanation, viz., the Hermiticity of their Feshbach "effective" reduction. This point of view is new and enables us to propose an immediate generalization of symmetry. A super-integrable model in two dimensions is chosen for illustration.