Introduction to Lattice Theory

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An introduction to lattice gauge theory and spin systems John B. Kogut Rev. Abstract Authors References.

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Abstract This article is an interdisciplinary review of lattice gauge theory and spin systems. Issue Vol. Authorization Required. Log In.

Rev. Mod. Phys. 51, () - An introduction to lattice gauge theory and spin systems

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Beams Phys. Here, we show that q-observable functions can be interpreted as generalised quantile functions for quantum observables interpreted as random variables. More generally, when L is a complete meet-semilattice, we show that L-valued cumulative distribution functions and the corresponding L-quantile functions form a Galois connection.

Catalog Record: Introduction to lattice theory | HathiTrust Digital Library

Moreover, using some constructions from the topos approach to quantum theory, we show that there is a joint sample space for all quantum observables, despite no-go results such as the Kochen-Specker theorem. Observables of a quantum system, described by self-adjoint operators in a von Neumann algebra or affiliated with it in the unbounded case, form a conditionally complete lattice when equipped with the spectral order.

Using this Using this order-theoretic structure, we develop a new perspective on quantum observables. In this first paper of two , we show that self-adjoint operators affiliated with a von Neumann algebra can equivalently be described as certain real-valued functions on the projection lattice of the algebra, which we call q-observable functions. Bounded self-adjoint operators correspond to q-observable functions with compact image on non-zero projections. These functions, originally defined in a similar form by de Groote, are most naturally seen as adjoints in the categorical sense of spectral families.

demo-new.nplan.io/tersia-el-ao-nada-es-lo.php We show how they relate to the daseinisation mapping from the topos approach to quantum theory. Moreover, the q-observable functions form a conditionally complete lattice which is shown to be order-isomorphic to the lattice of self-adjoint operators with respect to the spectral order. In a subsequent paper, we will give an interpretation of q-observable functions in terms of quantum probability theory, and using results from the topos approach to quantum theory, we will provide a joint sample space for all quantum observables.

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Abelian subalgebras and the Jordan structure of a von Neumann algebra. The converse also holds. Daseinisation and the Liberation of Quantum Theory. This paper is the second in a series whose goal is to develop a fundamentally new way of constructing theories of physics. The motivation comes from a desire to address certain deep issues that arise when contemplating quantum theories of The motivation comes from a desire to address certain deep issues that arise when contemplating quantum theories of space and time.

Our basic contention is that constructing a theory of physics is equivalent to finding a representation in a topos of a certain formal language that is attached to the system. Classical physics arises when the topos is the category of sets. Other types of theory employ a different topos. In this paper, we study in depth the topos representation of the propositional language, PL S , for the case of quantum theory.

In doing so, we make a direct link with, and clarify, the earlier work on applying topos theory to quantum physics. In the second part of the paper we change gear with the introduction of the more sophisticated local language L S. From this point forward, throughout the rest of the series of papers, our attention will be devoted almost entirely to this language.

These are objects in the topos that play the role of states: a necessary development as the spectral presheaf has no global elements, and hence there are no microstates in the sense of classical physics. Truth objects therefore play a crucial role in our formalism. Reconstructing an atomic orthomodular lattice from the poset of its Boolean sublattices. We show that an atomic orthomodular lattice L can be reconstructed up to isomorphism from the poset B L of Boolean subalgebras of L. A motivation comes from quantum theory and the so-called topos approach, where one considers the poset On Lattice Coding for the Gaussian Channel.

The definition and basic properties of lattice codes for the Gaussian channel are first recalled. Then, an estimation of the code rate and bounds on the error probability are presented. Lattice codes are shown to exit which reach the Lattice codes are shown to exit which reach the channel capacity.

Separate coding and shape gains are introduced. Finally, the asymptotic behaviour of lattice codes is examined and we are led to conjecture that achieving capacity is a covering - rather than packing - problem. On J, M, m -extensions of Boolean algebras. The class K of all J, M, m -extensions of a Boolean algebra A can be partially ordered and always contains a maximum and a minimal element, with respect to this partial ordering.

However, it need not contain a smallest element.

Should K Should K contain a smallest element, then K has the structure of a complete lattice. Necessary and sufficient conditions under which K does contain a smallest element are derived. A Boolean algebra A is constructed for each cardinal m such that the class of all m-extensions of A does not contain a smallest element.

Crack propagation induced by thermal shocks in structured media. This paper describes the propagation of an edge crack in a structured thermoelastic solid. A rapid change of temperature, represented by a time-periodic series of high-gradient temperature pulses, is applied at the boundary of the A rapid change of temperature, represented by a time-periodic series of high-gradient temperature pulses, is applied at the boundary of the structured solid. A lattice approximation is employed in the model analysis discussed here. In order to describe the crack advance through the lattice a failure criterion is imposed, whereby the links break as soon as they attain a critical elastic elongation.

The elongations of the links are produced both by a variation in temperature and by elastic waves generated at the boundary due to thermal shocks, as well as waves created by the propagating crack through the breakage of the elastic ligaments. The analysis is compared to the quasi-static and dynamic models of thermal striping in thermally loaded solids containing edge cracks. The emphasis is on the effect of the structure on the crack trapping. The nonlinear simulations presented in this paper show that the average speed of crack propagation can be estimated from the analysis of the dispersion properties of waves initiated by the crack.

Temperature and inertia contributions to crack propagation are also investigated. It is found that inertia amplifies the elongations of the links, and thus influences the crack advance through the structured solid. Natasha Movchan. An algebraic study of the notion of independence of frames. Families of frames can be given several algebraic Families of frames can be given several algebraic interpretations in terms of semi-modular lattices, matroids, and geometric lattices.

Each of those structures are endowed with a particular extended independence relation, which we prove to be distinct even though related to independence of frames.

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