Algebraic Quantum Field Theory

Аннотация: С помощью, специального алгоритма,принципа многоуровневой периодичности, обнаружено периодическое изменение свойств на уровне ядер химических элементов и представлен вариант периодической системы изотопов. Выполнена проверка периодического изменения свойств изотопов, а также вертикальной симметрии подгрупп, на соответствие по 10 типам экспериментальных данных: соотношения масс осколков деления; значений квадрупольного момента; магнитного момента; времени жизни радиоактивных изотопов; рассеяния нейтронов; сечения радиационного захвата тепловых нейтронов (n,γ); сечения выхода α-частиц (n,α); содержания изотопов на Земле, в Солнечной и системах других звезд; а также особенностей рудообразования и звездной эволюции. Для всех 10 случаев получены соответствия, предлагаемой периодической структуре ядра. Система сформирована в привычном 2-х мерном виде таблицы, аналогично периодической системе элементов и подразделяет массовый ряд изотопов на 8 периодов и 4 типа «ядерних» орбиталей: s я , d я , p я , f я. Представлено объяснение происхождения «магических» чисел, как совокупности заполненных зарядовых оболочек ядра. Система изотопов показывает периодическую структуру на новом уровне мироздания и открывает перспективы практического использования.

This paper examines strong operator convergence through its master equation formulation and associated variance bounds. The work develops the mathematical framework governing this convergence mode and its implications for expectation values in physical observables. Key focus areas include the master equation for all , the variance bound , and the theoretical foundations in operator theory that support these results. The paper establishes rigorous proofs for convergence properties, examines applications to quantum mechanical systems, and demonstrates the practical utility of these bounds in computational frameworks. Results indicate that strong operator convergence provides essential stability guarantees for physical observables while maintaining computational tractability in infinite-dimensional Hilbert spaces.

We extend the observer-relative formulation of black hole complementarity by integrating TomitaTakesaki modular theory into the geometric structure of causal diamonds. By treating modular time as a connection over diamond-space, we show that trajectory-dependent holonomies encode a form of "memory" of the observers path, with curvature given by the commutator of modular generators. This provides a rigorous algebraic framework for the relational encoding of information accessibility across horizons, while maintaining compatibility with algebraic quantum field theory (AQFT) and conformal symmetry. Initial computations in a 1 + 1 CFT setup reveal how bilocal entanglement kernels dynamically deform under evolving observer horizons, suggesting new avenues for exploring quantum information flow in curved spacetimes.

This paper revisits black-hole complementarity using Rovelli's relational quantum mechanics within the algebraic framework of quantum field theory in curved spacetime. For each observer, we construct a local von Neumann algebra as the inductive limit of open double-cone algebras contained strictly within their causal diamond, avoiding null-boundary ambiguities. Modeling late Hawking radiation as an entangled twomode system in the Unruh state, we prove that no single observer's algebra contains both a late exterior mode and its interior partner. This reframes complementarity as compatibility of observer-relative facts rather than a global paradox. For a static exterior observer, we explicitly compute the accessible thermal entropy and derive an operational Holevo bound on retrievable information about the interior partner. Extensions to multimode systems and modular localization are discussed.

This paper proposes an extended framework unifying Hameroff-Penrose's Orchestrated Objective Reduction (Orch-OR) with a novel Unified Coherence Field (ψ C). We derive the quantum dynamics of microtubule coherence and couple ψ C to spacetime via modified Einstein field equations. Key contributions include: • Quantum coherence dynamics in microtubules from first principles. • Modified gravitational equations incorporating consciousness field interactions. • Testable predictions for experiments spanning 2025-2035 and beyond 2040. Predictions include detectable consciousness signatures in cold atom interferometry (∆[x, p] ∼ 10-34 J•s), gravitational wave detectors (∆θ ∼ 0.1µarcsec), and EEG-Bose-Einstein condensate (BEC) correlations. The model bridges quantum biology and cosmology, addressing the Hard Problem of consciousness [1].

This work establishes a rigorous field-theoretic framework for consciousness-cosmos interaction through: • A non-local quantum coherence field ψC with Hubble-scale correlation length • Microtubule-mediated decoherence channels in neural systems • Three experimentally falsifiable predictions with numerical bounds • Resolution of the hard problem via ψC-induced quantum Darwinism The model requires only O(1) new parameters while unifying neural decoherence, dark energy, and quantum gravity anomalies.

We propose a unified geometric framework extending classical spacetime to a 64dimensional manifold, integrating quantum gravity effects, higher-dimensional topology, and a novel interpretation of dark matter. This model introduces a partitioned structure of extra dimensions: 10 super-spatial, 20 sub-quantum, and 30 inverse AdS dimensions. A dilatonic field geometry and anti-gravitational curvature are used to mediate unification between Einstein's field equations and quantum mechanics. A new QGG (Quark-Gluon-Graviton) plasma state is proposed to explain dark matter density and its behavior. Modified Einstein field equations incorporating quantum corrections are derived. The model includes compatibility with loop quantum gravity, Calabi-Yau manifolds, and SUSY-breaking mechanisms. Numerical methods and symbolic derivations are presented.

A link between quantization and (non-commutative) geometry is uncovered via Tomita-Takesaki modular theory for complexified ortho-symplectic spaces.
We suggest (following arXiv:1007.4094v1) that, in a covariant quantum theory, non-commutative space-time can be a-posteriori recovered from relativistic states over algebras of partial observables.
Speculative implications for cosmology will be mentioned.

The periodic system of chemical elements is represented within the framework of the weight diagram of the Lie algebra of the fourth rank of the rotation group of an eight-dimensional pseudo-Euclidean space. Spin is interpreted as the fourth generator of the Cartan subalgebra, whose two eigenvalues correspond to two three-dimensional projections of the
weight diagram containing elements of the periodic system from hydrogen to moscovium (the first projection) and from helium to oganesson (the second projection). One of the main advantages of the proposed group-theoretic construction of the periodic system is the natural inclusion of antimatter in the general scheme.

В данной статье анализируется фундаментальное отличие полей Σ, ℂ и ℍ от векторных пространств. Особое внимание уделяется критике введения операции умножения вектора на вектор («кросс-» или «буравчиковое» умножение) в этих алгебрах как условия векторизации. Показывается, что хотя в Σ, ℂ и ℍ определены билинейные или антикоммутативные произведения, они не превращают эти структуры в векторные пространства, а лишь расширяют их до алгебр с дополнительной операцией.

В статье предложена интерпретация двумерных алгебр, описанных в труде С.С. Кокарева и Д. Павлова, через геометрическую структуру устройства Сфираль. Сфираль рассматривается как физическая реализация алгебраических отношений: неассоциативность, антисимметрия и вложенность получают пространственное выражение через витки, петли и узлы. Предлагается концепция Сфирали как алгеброгеометрического механизма, в котором время порождается не внешним параметром, а внутренней логикой переходов.

The periodic system of chemical elements is represented within the framework of the weight diagram of the Lie algebra of the fourth rank of the rotation group of an eightdimensional pseudo-Euclidean space. The hydrogen realization of the Cartan subalgebra and Weyl generators of the group algebra is studied. The root structure of the subalgebras of the group algebra of a conformal group in the framework of a twofold covering is analyzed. Based on the analysis, the Cartan-Weyl basis of the group algebra is determined. The root and weight diagrams are constructed. A mass formula associated with each node of the weight diagram is introduced. Spin is interpreted as the fourth generator of the Cartan subalgebra, whose two eigenvalues correspond to two three-dimensional projections of the weight diagram containing elements of the periodic system from hydrogen to moscovium (the first projection) and from helium to oganesson (the second projection). One of the main advantages of the proposed group-theoretic construction of the periodic system is the natural inclusion of antimatter in the general scheme.

We give a non-perturbative construction of the fermionic projector in Minkowski space coupled to a time-dependent external potential which is smooth and decays faster than quadratically for large times. The weak and strong mass oscillation properties are proven. We show that the integral kernel of the fermionic projector is of Hadamard form, provided that the time integral of the spatial supnorm of the potential satisfies a suitable bound. This gives rise to an algebraic quantum field theory of Dirac fields in an external potential with a distinguished pure quasi-free Hadamard state. Contents 1. Introduction 2. Preliminaries 2.1. Dirac Green's Functions and the Time Evolution Operator 2.2.

A common misunderstanding, one of which we need to disabuse ourselves, is that quantum theory, while possessing astounding predictive power, actually explains the phenomena it describes. It does not. Quantum theory offers mathematical descriptions of measurable phenomena with great facility and accuracy, but it provides absolutely no understanding of why any particular quantum outcome is observed. The concepts of description, prediction, and explanation are conceptually distinct, and we must always keep this fact in mind. Mathematical descriptions, if they are accurate, tell us what mathematical relationships hold among phenomena, but not why they hold. Empirical predictions, if they are correct, tell us what we will or might observe under certain experimental conditions, but not necessarily why these things will happen. It is the province of genuine explanations to tell us how things actually
work—that is, why such descriptions hold and why such predictions are true. The failure to appreciate these differences has given rise to a lot of confusion about what the impressive edifice of modern physical theory has, and has not, achieved. Quantum theory is long on the what, both mathematically and observationally, but almost completely silent on the how and the why. What is even more interesting is that, in some sense, this state of affairs seems to be a necessary consequence of the empirical adequacy of quantum descriptions. One of the most noteworthy achievements of quantum theory, I dare suggest, is the accurate prediction of phenomena that, on pain of experimental contradiction, have no physical explanation. This is perhaps a startling way to state the matter, but no less true because of it. It is such phenomena, and arguments concerning their significance, that will occupy us in this essay.

We investigate the notion of subsystem in the framework of spectral triple as a generalized notion of noncommutative submanifold. In the case of manifolds, we consider several conditions on Dirac operators which turn embedded submanifolds into isometric submanifolds. We then suggest a definition of spectral subtriple based on the notion of submanifold algebra and the already existing notions of Riemannian, isometric, and totally geodesic morphisms. We have shown that our definitions work at least in some relevant almost commutative examples.

- A deep link between quantization and (non-commutative) geometry is uncovered via a careful usage of modular theory.
- It is premature to assume that space-time geometry in quantum gravity becomes relevant only at the emergent macroscopic level:
* At the covariant quantum level, there is still some viable option for a-posteriori derived space-time (non-commutative) geometries, induced by states (see arXiv:1007.4094v1).
* The ideas involved are strongly motivated by algebraic quantum field theory, suitably freed from its old reliance on background manifolds.
* The mathematical techniques rely on an interplay between Tomita-Takesaki modular theory and derived categorical non-commutative geometry that seems to hint at further deep modifications of the notions of space and geometry, waiting to be further explored.

We take a look at the “forbidden route” backward from the Wilsonian/Feynmanian paradigm in (effective) quantum field theory and emergent geometry in quantum gravity.
Our rear-ward exploration of the forgotten origins of quantum (field) theory will reveal crucial missing ingredients in order to recover a relation between “quantum” and “geometry”.

We review the status and current prospects of Modular Algebraic Quantum Gravity arXiv:1007.4094v1).
- It is premature to assume that space-time geometry in QG becomes relevant only at the emergent macroscopic level.
- At the covariant quantum level, there is still a viable option for a-posteriori derived space-time geometries induced by states: state-induced generalized geometry might exist also in QG.
- The ideas involved are strongly motivated by algebraic quantum field theory, suitably freed from its old reliance on background manifolds.
- The mathematical techniques rely on an interplay between
* Tomita-Takesaki modular theory and
* derived categorical non-commutative geometry
that seems to hint at further deep modifications of the
notions of space and geometry, waiting to be uncovered.

We informally describe a circle of still conjectural ideas (that have been the central motivation for my research) on the intrinsic reconstruction of quantum space-time from a state on a category of partial observables, via modular covariance. Among the technical mathematical tools that will be mentioned: a) Connes non-commutative (spectral) geometries, b) Tomita-Takesaki modular theory, c) categories and non-commutative spaceoids.

Diffeologies, introduced by J.-M.Souriau, are a vast generalization
of smooth manifolds that, from the point of view of category
theory, is much better behaved.
Not much is known on the possible ways to describe such spaces in
the language of spectral geometry and it is the purpose of this
quite preliminary talk to explore some of the problems involved in
such study.
We will also see how categories of spectral triples might be of help
in these matters and more.

We prove that a certain bialgebroid introduced recently by Kadison is isomorphic to a bialgebroid introduced earlier by Connes and Moscovici. At the level of total algebras, the isomorphism is a consequence of the general fact that an L-R-smash product over a Hopf algebra is isomorphic to a diagonal crossed product.

Recently, it has been shown that the quantum equilibrium distribution in the original Bohm's model is unstable and so it isn't a tenable physical theory [Proc. R. Soc. A 470 20140288 (2014)]. In this paper we show that a natural modification of the Bohm's quantum force leads to a stable quantum equilibrium without any change to statistical predictions of this model in equilibrium state. Moreover, it is shown that an initial non-equilibrium phase space distribution relaxes to equilibrium distribution via a coarse grained H-theorem. But before end of the relaxation process, for example in the early universe, this modified quantum force can lead to the new predictions beyond standard quantum mechanics.

A new non-Archimedean approach to interacted quantum fields is presented. In proposed approach, a field operator , no longer a standard tempered operator-valued distribution, but a non-classical operator-valued function. We prove using this novel approach that the quantum field theory with Hamiltonian exists and that the corresponding *algebra of bounded observables satisfies all the Haag-Kastler axioms except Lorentz covariance. We prove that the quantum field theory model is Lorentz covariant. CONTENT CHAPTER I.

The root structure of the subalgebras of the group algebra of a conformal group in the framework of a twofold covering is analyzed. Based on the analysis, the Cartan–Weyl basis of the group algebra is determined. The root and weight diagrams are constructed. A mass formula associated with each node of the weight diagram is introduced.

The Higgs mechanism is invoked to explain how gauge bosons can be massive while Yang-Mills theory describes only massless gauge fields. Central to it is the notion of spontaneous symmetry breaking (SSB), applied to the SU(2) × U(1) gauge symmetry of the electroweak theory. However, over the past two decades, philosophers of physics have challenged the standard narrative of the Higgs mechanism as an instance of gauge symmetry breaking. They have pointed out the apparent contradiction between the status of gauge symmetries as mathematical redundancies and the account of mass generation in the Higgs mechanism by means of gauge symmetry breaking. In addition, they have pointed to Elitzur's theorem, a result from lattice gauge theory forbidding local gauge symmetry breaking. This has led philosophers to the conclusion that there cannot be any SSB in the Higgs mechanism, an idea supported by the dressing field method of gauge symmetry reduction. In this thesis we mitigate this conclusion by showing that global gauge symmetries, i.e. transformations independent of spacetime, are not mere mathematical redundancies but carry direct empirical significance. This can be seen from constrained Hamiltonian analysis by the fact that the Gauss constraint in Yang-Mills theory only generates gauge transformations which asymptotically become the identity. The classical Higgs mechanism can indeed be reformulated as a breaking of only this global gauge symmetry. We subsequently extend this result to quantum field theory by considering SSB in algebraic quantum field theory (AQFT). The Abelian U(1) Higgs mechanism can be shown to be an instance of SSB in the algebraic sense and we discuss the extent to which this can be generalised to the non-Abelian case. Finally we discuss the implications of our results for the interpretation of the electroweak phase transition and the analogy between the Higgs mechanism and superconductivity.

The structure of the group algebra of a conformal group (the group underlying the
group-theoretic description of the periodic system of chemical elements) is considered within the framework of a twofold covering. The hydrogen realization of the Cartan subalgebra and Weyl generators of the group algebra is studied.

After reviewing the theory of "causal fermion systems" (CFS theory) and the "Events, Trees, and Histories Approach" to quantum theory (ETH approach), we compare some of the mathematical structures underlying these two general frameworks and discuss similarities and differences. For causal fermion systems, we introduce future algebras based on causal relations inherent to a causal fermion system. These algebras are analogous to the algebras previously introduced in the ETH approach. We then show that the spacetime points of a causal fermion system have properties similar to those of "events", as defined in the ETH approach. Our discussion is underpinned by a survey of results on causal fermion systems describing Minkowski space that show that an operator representing a spacetime point commutes with the algebra in its causal future, up to tiny corrections that depend on a regularization length.

Recently, examples of an index theory for KMS states of circle actions were discovered, [9, 13]. We show that these examples are not isolated. Rather there is a general framework in which we use KMS states for circle actions on a C *-algebra A to construct Kasparov modules and semifinite spectral triples. By using a residue construction analogous to that used in the semifinite local index formula we associate to these triples a twisted cyclic cocycle on a dense subalgebra of A. This cocycle pairs with the equivariant KK-theory of the mapping cone algebra for the inclusion of the fixed point algebra of the circle action in A. The pairing is expressed in terms of spectral flow between a pair of unbounded self adjoint operators that are Fredholm in the semifinite sense. A novel aspect of our work is the discovery of an eta cocycle that forms a part of our twisted residue cocycle. To illustrate our theorems we observe firstly that they incorporate the results in [9, 13] as special cases. Next we use the Araki-Woods III λ representations of the Fermion algebra to show that there are examples which are not Cuntz-Krieger systems.

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Present day physics rests on two main pillars: General relativity and quantum field theory. We discuss the deep and at the same time problematic interplay between these two theories. Based on an argument by Doplicher, Fredenhagen and Roberts, we propose a possible universality property for noncommutative quantum field theory in the sense that any theory of quantum gravity should involve quantum field theories on noncommutative space-times as a special limit. We propose a mathematical framework to investigate such a universality property and start the discussion of its mathematical properties. The question of its connection to string theory could be a starting point for a new perspective on string theory.

Pimsner introduced the C *-algebra O X generated by a Hilbert bimodule X over a C *-algebra A. We look for additional conditions that X should satisfy in order to study the simplicity and, more generally, the ideal structure of O X when X is finite projective. We introduce two conditions: "(I)-freeness" and "(II)-freeness", stronger than the former, in analogy with [J. Cuntz, W.

The questions of interpretation of the algebraic formulation of a quantum theory with a binary structure are considered. The possibility of constructing a quantum theory without using any classical analogies and visual images and mechanical models associated with these analogies is discussed. It is shown that the inconsistency of the quark model, as well as the Bohr model in the theory of the atom, is a consequence of the introduction of classical space-time representations to the subatomic (hadron) level.

We propose an alternative representation for linear quantum gravity. It is based on the use of a structure that bears some resemblance to the Abelian loop representation used in electromagnetism but with the difference that the space of extended object on which waves functions take values has a structure of commutative monoide instead of Abelian group. The generator of duality of the theory is realized in this representation and a geometrical interpretation is discussed.

We propose a profound consequence of symmetry towards the axiomatic derivation of Hilbert space quantum theory. Specifically, we show that the symmetry of information gain in minimal error state discrimination induces a non-trivial proviso on the state space structure of a physical theory. The symmetry considered here puts a restriction on the means of optimal guessing of a system's state from that of an ensemble. We coin the term information symmetry (IS) since it constrains the way optimal information gain occurs in the act of measurement. It is found that IS rules out a broad class of generalized probabilistic theories while being perfectly compatible with physical theories such as quantum and classical mechanics.

Symmetry shares an entwined history with the structure of physical theory. We propose a consequence of symmetry towards the axiomatic derivation of Hilbert space quantum theory. We introduce the notion of information symmetry (IS) and show that it constraints the state-space structure in any physical theory. To this end we study the minimal error binary state discrimination problem in the framework of generalized probabilistic theories. A theory is said to satisfy IS if the probability of incorrectly identifying each of two randomly prepared states is same for both the states. It is found that this simple principle rules out several classes of theories while being perfectly compatible with quantum theory.

We present a numerical approximation scheme for the Tomita-Takesaki modular operator of local subalgebras in linear quantum fields, working at one-particle level. This is applied to the local subspaces for double cones in the vacuum sector of a massive scalar free field in (1 + 1)and (3 + 1)-dimensional Minkowski spacetime, using a discretization of time-0 data in position space. In the case of a wedge region, one component of the modular generator is well known to be a mass-independent multiplication operator; our results strongly suggest that for the double cone, the corresponding component is still at least close to a multiplication operator, but that it is dependent on mass and angular momentum. Communicated by Karl-Henning Rehren.

We analyze the algebraic, topological, and order properties of /*-algebras: complex unital topological *-algebras for which S^/*^/=0 implies #f=0 (Je/), JcJV any finite subset. We consider the ergodic properties of states on an I*-algebra with a distinguished group of automorphisms. Particular attention is given to I*-algebras of the form £=Sf® n £ where E is a nuclear LF-space. When E=^(JR 4) (0(K»)©$(R 3) respectively) then E has applications to relativistic quantum field theory (the canonical anticommutation or commutation relations, respectively).

We consider the positive energy representations of the algebra of quasilocal observables for the free massless Majorana field described in preceding papers [1,2]. We show that by an appropriate choice of the (partially) occupied one particle modes we can find irreducible, type II^ or IΠ A representations in this class which are unitarily equivalent to the vacuum representation when restricted to any forward light cone and disjoint from it when restricted to any backward light cone, or conversely. We give an elementary explicit proof of local normality of each representation in the above class. 2 otherwise. (Actually all properly infinite, injective von Neumann algebras with separable predual can appear in positive energy representations of this model [2].) Here we discuss these representations in connection with the notion of charge Research supported by Ministero della Pubblica Istruzione and C.N.R.-G.N.A.F.A.

This paper is dedicated to a detailed analysis and computation of quantum states of causal fermion systems. The mathematical core is to analyze integrals over the unitary group asymptotically for a large dimension of the group, for various integrands with a specific scaling behavior in this dimension. It is shown that, in a well-defined limiting case, the localized refined pre-state is positive and allows for the description of general entangled states.

We give a functional analytic construction of the fermionic projector on a globally hyperbolic Lorentzian manifold of finite lifetime. The integral kernel of the fermionic projector is represented by a two-point distribution on the manifold. By introducing an ultraviolet regularization, we get to the framework of causal fermion systems. The connection to the "negative-energy solutions" of the Dirac equation and to the WKB approximation is explained and quantified by a detailed analysis of closed Friedmann-Robertson-Walker universes. Contents 1. Introduction 1 2. Preliminaries 4 3. Functional Analytic Construction of the Fermionic Projector 6 3.1. The Space-Time Inner Product as a Dual Pairing 6 3.2. Space-Times of Finite Lifetime 8 3.3. The Fermionic Signature Operator and the Fermionic Projector 9 3.4. Explicit Formulas in a Foliation 11 3.5. Representation as a Distribution 12 4. Connection to the Framework of Causal Fermion Systems 14 5. Example: A Closed Friedmann-Robertson-Walker Universe 17 5.1. Computation of S WKB and P WKB 19 5.2. Estimates of U − U WKB and S − S WKB 21 5.3. An Estimate of P − P WKB 24 6. Discussion of Examples with a Piecewise Constant Scale Function 26 References 28

Causal variational principles, which are the analytic core of the physical theory of causal fermion systems, are found to have an underlying Hamiltonian structure, giving a formulation of the dynamics in terms of physical fields in space-time. After generalizing causal variational principles to a class of lower semi-continuous Lagrangians on a smooth, possibly non-compact manifold, the corresponding Euler-Lagrange equations are derived. In the first part, it is shown under additional smoothness assumptions that the space of solutions of the Euler-Lagrange equations has the structure of a symplectic Fréchet manifold. The symplectic form is constructed as a surface layer integral which is shown to be invariant under the time evolution. In the second part, the results and methods are extended to the non-smooth setting. The physical fields correspond to variations of the universal measure described infinitesimally by one-jets. Evaluating the Euler-Lagrange equations weakly, we derive linearized field equations for these jets. In the final part, our constructions and results are illustrated in a detailed example on R 1,1 × S 1 where a local minimizer is given by a measure supported on a two-dimensional lattice.

We consider a structure of the K-Hilbert space, where K ≃ R is a field of real numbers, K ≃ C is a field of complex numbers, K ≃ H is a quaternion algebra, within the framework of division rings of Clifford algebras. The K-Hilbert space is generated by the Gelfand-Naimark-Segal construction, while the generating C *-algebra consists of the energy operator H and the generators of the group SU(2, 2) attached to H. The cyclic vectors of the K-Hilbert space corresponding to the tensor products of quaternionic algebras define the pure separable states of the operator algebra. Depending on the division ring K, all states of the operator algebra are divided into three classes: 1) charged states with K ≃ C; 2) neutral states with K ≃ H; 3) truly neutral states with K ≃ R. For pure separable states that define the fermionic and bosonic states of the energy spectrum, the fusion, doubling (complexification) and annihilation operations are determined.

Обсуждается проблема построения эффективного алгоритма обращения целочисленной матрицы. Один из способов вычисления обратной матрицы опирается на предварительное вычисление матрицы Смита. Известен вероятностный алгоритм вычисления матрицы Смита с кубической зависимостью числа бит-операций от размеров матрицы. Предлагается некоторое детерминистское продолжением этого подхода для вычисления обратной матрицы. Ключевые слова: форма Смита; символьные вычисления; обращение матриц

Due to an example indicated to us in September 2009 we have to add one more restriction to the suppositions on the imprimitivity bimodules treated in Proposition 4.1, Theorem 5.1, Theorem 6.2 and Proposition 6.3. In the situation when the Banach-Saks property holds for the imprimitivity bimodule we can describe all possible additional examples violating the newly invented supposition. So the classification of Hilbert C *-modules with the Banach-Saks property is complete. Beyond that, there is still an open problem for a certain class of imprimitivity bimodules with the weak or uniform weak Banach-Saks property which might violate the additional condition.

It is shown that uncertainty relations, as well as coherent and squeezed states, are structural properties of stochastic processes with Fokker-Planck dynamics. The quantum mechanical coherent and squeezed states are explicitly constructed via Nelson stochastic quantization. The method is applied to derive new minimum uncertainty states in time-dependent oscillator potentials.

It is shown that uncertainty relations, as well as coherent and squeezed states, are structural properties of stochastic processes with Fokker-Planck dynamics. The quantum mechanical coherent and squeezed states are explicitly constructed via Nelson stochastic quantization. The method is applied to derive new minimum uncertainty states in time-dependent oscillator potentials.