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Quantum theory has the distinction among physical theories of currently underpinning most of modern physics, while remaining essentially mysterious, with no general agreement about the nature of its principles or the underlying reality. Recently, the rise of quantum information science has shown that thinking in operational or information-theoretic terms can be extremely enlightening, and that a fruitful direction for understanding quantum theory is to study it in the context of more general probabilistic theories. The framework for such theories will be reviewed in the Chapter Two. In Chapter Three we will study a property of quantum theory called self-duality, which is a correspondence between states and observables. In particular, we will show that self-duality follows from a computational primitive called bit symmetry, which states that every logical bit can be mapped to any other logical bit by a reversible transformation. In Chapter Four we will study a notion of probabilistic interference based on a hierarchy of interference-type experiments involving multiple slits. We characterize theories which do not exhibit interference in experiments with k slits, and give a simple operational interpretation. We also prove a connection between bit symmetric theories which possess certain natural transformations, and those which exhibit at most two-slit interference. In Chapter Five we will focus on reconstructing the algebraic structures of quantum theory. We will show that the closest cousins to standard quantum theory, namely the finite-dimensional Jordan-algebraic theories, can be characterized by three simple principles: (1) a generalized spectral decomposition, (2) a high degree of symmetry, and (3) a generalization of the von Neumann-Luders projection postulate. Finally, we also show that the absence of three-slit interference may be used as an alternative to the third principle. In Chapter Six, we focus on quantum statistical mechanics and the problem of understanding how its characteristic features can be derived from an exact treatment of the underlying quantum system. Our central assumptions are sufficiently complex dynamics encoded as a condition on the complexity of the eigenvectors of the Hamiltonian, and an information theoretic restriction on measurement resources. We show that for almost all Hamiltonian systems measurement outcome probabilities are indistinguishable from the uniform distribution.

Quantum theory has the distinction among physical theories of currently underpinning most of modern physics, while remaining essentially mysterious, with no general agreement about the nature of its principles or the underlying reality. Recently, the rise of quantum information science has shown that thinking in operational or information-theoretic terms can be extremely enlightening, and that a fruitful direction for understanding quantum theory is to study it in the context of more general probabilistic theories. The framework for such theories will be reviewed in the Chapter Two. In Chapter Three we will study a property of quantum theory called self-duality, which is a correspondence between states and observables. In particular, we will show that self-duality follows from a computational primitive called bit symmetry, which states that every logical bit can be mapped to any other logical bit by a reversible transformation. In Chapter Four we will study a notion of probabilistic interference based on a hierarchy of interference-type experiments involving multiple slits. We characterize theories which do not exhibit interference in experiments with k slits, and give a simple operational interpretation. We also prove a connection between bit symmetric theories which possess certain natural transformations, and those which exhibit at most two-slit interference. In Chapter Five we will focus on reconstructing the algebraic structures of quantum theory. We will show that the closest cousins to standard quantum theory, namely the finite-dimensional Jordan-algebraic theories, can be characterized by three simple principles: (1) a generalized spectral decomposition, (2) a high degree of symmetry, and (3) a generalization of the von Neumann-Luders projection postulate. Finally, we also show that the absence of three-slit interference may be used as an alternative to the third principle. In Chapter Six, we focus on quantum statistical mechanics and the problem of understanding how its characteristic features can be derived from an exact treatment of the underlying quantum system. Our central assumptions are sufficiently complex dynamics encoded as a condition on the complexity of the eigenvectors of the Hamiltonian, and an information theoretic restriction on measurement resources. We show that for almost all Hamiltonian systems measurement outcome probabilities are indistinguishable from the uniform distribution.

Techn. Univ., Habil.-Schr.--Clausthal, 2000. / Literaturverz. S. [256] - 262.

viii, 125 p. A print copy of this thesis is available through the UO Libraries. Search the library catalog for the location and call number. / Finite p -groups are studied using bilinear methods which lead to using nonassociative rings. There are three main results, two which apply only to p -groups and the third which applies to all groups. First, for finite p -groups P of class 2 and exponent p the following are invariants of fully refined central decompositions of P : the number of members in the decomposition, the multiset of orders of the members, and the multiset of orders of their centers. Unlike for direct product decompositions, Aut P is not always transitive on the set of fully refined central decompositions, and the number of orbits can in fact be any positive integer. The proofs use the standard semi-simple and radical structure of Jordan algebras. These algebras also produce useful criteria for a p -group to be centrally indecomposable. In the second result, an algorithm is given to find a fully refined central decomposition of a finite p -group of class 2. The number of algebraic operations used by the algorithm is bounded by a polynomial in the log of the size of the group. The algorithm uses a Las Vegas probabilistic algorithm to compute the structure of a finite ring and the Las Vegas MeatAxe is also used. However, when p is small, the probabilistic methods can be replaced by deterministic polynomial-time algorithms. The final result is a polynomial time algorithm which, given a group of permutations, matrices, or a polycyclic presentation; returns a Remak decomposition of the group: a fully refined direct decomposition. The method uses group varieties to reduce to the case of p -groups of class 2. Bilinear and ring theory methods are employed there to complete the process. / Adviser: William M. Kantor

Orientador: Plamen Emilov Kochloukov / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-17T03:42:15Z (GMT). No. of bitstreams: 1 Silva_DiogoDinizPereiradaSilvae_D.pdf: 1168055 bytes, checksum: 49c676076235e3eef6f8a27594f092f7 (MD5) Previous issue date: 2010 / Resumo: Neste trabalho apresentamos um estudo sobre identidades polinomiais graduadas em álgebras não associativas. Mais precisamente estudamos as identidades polinomiais graduadas da álgebra de Lie das matrizes de ordem 2 com traço zero com as três graduações naturais, a Z2-graduação, a Z2 _ Z2-graduação e a Z-graduação, neste caso conseguimos uma nova demonstração baseada em métodos elementares dos resultados de [27] que não se baseia em resultados da Teoria de Invariantes, estes resultados foram publicados em [30]. Estudamos também as identidades graduadas da álgebra de Jordan das matrizes simétricas de ordem 2, neste caso obtivemos bases para as identidades graduadas dessa álgebra de Jordan em todas as possíveis graduações, obtivemos também bases para as identidades fracas para os pares (Bn; Jn) e (B; J), onde Bn e B denotam as álgebras de Jordan de uma forma bilinear simétrica não degenerada nos espaços vetoriais Vn e V respectivamente, onde Vn tem dimensão n e V tem dimensão 1, esses resultados estão no artigo [29], aceito para publicação / Abstract: In this thesis we study graded identities in non associative algebras. Namely we study graded polynomial identities for the Lie algebra of the 2_2 matrices with trace zero with it's three natural gradings, the Z2-grading, the Z2_Z2-grading and the Z-grading, in this case we obtained a new proof of the results of [27] that doesn't involve use of Invariant Theory, this results were published in [30]. We also studied the graded identities of the Jordan algebra of the symmetric matrices of order two, we obtained basis for the graded identities of this Jordan algebra in all possible gradings, we also obtained basis for the weak identities of the pairs (Bn; Jn) and (B; J), where Bn and B are the Jordan algebras of a symmetric bilinear form in a the vector spaces Vn and V respectively, where Vn has dimension n and V has countable dimension, this results are in the article [29], accepted for publication / Doutorado / Álgebra Não-Comutativa / Doutor em Matemática

Cette thèse concerne quatre thèmes apparemment différents, mais en fait intimement liés : problèmes variationnels sur les algèbres de Jordan euclidiennes, problèmes de complémentarité sur l’espace des matrices symétriques, analyse angulaire entre deux cônes convexes fermés et analyse du chemin central en programmation conique symétrique.Dans la première partie de ce travail, le concept de “commutation au sens opérationnel” dans les algèbres de Jordan euclidiennes est étudié en fournissant un principe de commutation pour problèmes variationnels avec données spectrales.Dans la deuxième partie, nous abordons l’analyse et la résolution numérique d’une large classe de problèmes de complémentarité sur l’espace des matrices symétriques. Les conditions de complémentarité sont exprimées en termes de l’ordre de Loewner ou, plus généralement, en termes d’un cône du type Loewnerien.La troisième partie de ce travail est une tentative de construction d’une théorie générale des angles critiques pour une paire de cônes convexes fermés. L’analyse angulaire pour une paire de cônes spécialement structurés est également considérée. Par-exemple, nous travaillons avec des sous-espaces linéaires, des cônes polyédriques, des cônes de révolution, des cônes “topheavy” et des cônes de matrices.La dernière partie de ce travail étudie la convergence et le comportement asymptotique du chemin central en programmation conique symétrique. Ceci est fait en utilisant des techniques propres aux algèbres de Jordan. / This thesis deals with four different but interrelated topics: variational problems on Euclidean Jordan algebras, complementarity problems on the space of symmetric matrices, angular analysis between two closed convex cones and the central path for symmetric cone linear programming.In the first part of this work we study the concept of “operator commutation” in Euclidean Jordan algebras by providing a commutation principle for variational problems involving spectral data.Our main concern of the second part is the analysis and numerical resolution of a broad class of complementarity problems on spaces of symmetric matrices. The complementarity conditions are expressed in terms of the Loewner ordering or, more generally, with respect to a dual pair of Loewnerian cones.The third part of this work is an attempt to build a general theory of critical angles for a pair of closed convex cones. The angular analysis for a pair of specially structured cones is also covered. For instance, we work with linear subspaces, polyhedral cones, revolution cones, topheavy cones and cones of matrices.The last part of this work focuses on the convergence and the limiting behavior of the central path in symmetric cone linear programming. This is done by using Jordan-algebra techniques.

In this thesis, we defind a new invariant of quadratic Lie algebras and quadratic Lie superalgebras and give a complete study and classification of singular quadratic Lie algebras and singular quadratic Lie superalgebras, i.e. those for which the invariant does not vanish. The classification is related to adjoint orbits of Lie algebras o(m) and sp(2n). Also, we give an isomorphic characterization of 2-step nilpotent quadratic Lie algebras and quasi-singular quadratic Lie superalgebras for the purpose of completeness. We study pseudo-Euclidean Jordan algebras obtained as double extensions of a quadratic vector space by a one-dimensional algebra and 2-step nilpotent pseudo-Euclidean Jordan algebras, in the same manner as it was done for singular quadratic Lie algebras and 2-step nilpotent quadratic Lie algebras. Finally, we focus on the case of a symmetric Novikov algebra and study it up to dimension 7.

Dans cette thèse, nous définissons un nouvel invariant des algèbres de Lie quadratiques et des superalgèbres de Lie quadratiques et donnons une étude et classification complète des algèbres de Lie quadratiques singulières et des superalgèbres de Lie quadratiques singulières, i.e. celles pour lesquelles l’invariant n’est pas nul. La classification est en relation avec les orbites adjointes des algèbres de Lie o(m) et sp(2n). Aussi, nous donnons une caractérisation isomorphe des algèbres de Lie quadratiques 2-nilpotentes et des superalgèbres de Lie quadratiques quasi-singulières pour le but d’exhaustivité. Nous étudions les algèbres de Jordan pseudoeuclidiennes qui sont obtenues des extensions doubles d’un espace vectoriel quadratique par une algèbre d’une dimension et les algèbres de Jordan pseudo-euclidienne 2-nilpotentes, de la même manière que cela a été fait pour les algèbres de Lie quadratiques singulières et des algèbres de Lie quadratiques 2-nilpotentes. Enfin, nous nous concentrons sur le cas d’une algèbre de Novikov symétrique et l’étudions à dimension 7. / In this thesis, we defind a new invariant of quadratic Lie algebras and quadratic Lie superalgebras and give a complete study and classification of singular quadratic Lie algebras and singular quadratic Lie superalgebras, i.e. those for which the invariant does not vanish. The classification is related to adjoint orbits of Lie algebras o(m) and sp(2n). Also, we give an isomorphic characterization of 2-step nilpotent quadratic Lie algebras and quasi-singular quadratic Lie superalgebras for the purpose of completeness. We study pseudo-Euclidean Jordan algebras obtained as double extensions of a quadratic vector space by a one-dimensional algebra and 2-step nilpotent pseudo-Euclidean Jordan algebras, in the same manner as it was done for singular quadratic Lie algebras and 2-step nilpotent quadratic Lie algebras. Finally, we focus on the case of a symmetric Novikov algebra and study it up to dimension 7.