A Coding Theoretic Study on MLL proof nets

Coding theory is very useful for real world applications. A notable example is digital television. Basically, coding theory is to study a way of detecting and/or correcting data that may be true or false. Moreover coding theory is an area of mathematics, in which there is an interplay between many branches of mathematics, e.g., abstract algebra, combinatorics, discrete geometry, information theory, etc. In this paper we propose a novel approach for analyzing proof nets of Multiplicative Linear Logic (MLL) by coding theory. We define families of proof structures and introduce a metric space for each family. In each family, 1. an MLL proof net is a true code element; 2. a proof structure that is not an MLL proof net is a false (or corrupted) code element. The definition of our metrics reflects the duality of the multiplicative connectives elegantly. In this paper we show that in the framework one error-detecting is possible but one error-correcting not. Our proof of the impossibility of one error-correcting is interesting in the sense that a proof theoretical property is proved using a graph theoretical argument. In addition, we show that affine logic and MLL + MIX are not appropriate for this framework. That explains why MLL is better than such similar logics.Comment: minor modification

About Dynamical Systems Appearing in the Microscopic Traffic Modeling

Motivated by microscopic traffic modeling, we analyze dynamical systems which have a piecewise linear concave dynamics not necessarily monotonic. We introduce a deterministic Petri net extension where edges may have negative weights. The dynamics of these Petri nets are well-defined and may be described by a generalized matrix with a submatrix in the standard algebra with possibly negative entries, and another submatrix in the minplus algebra. When the dynamics is additively homogeneous, a generalized additive eigenvalue may be introduced, and the ergodic theory may be used to define a growth rate under additional technical assumptions. In the traffic example of two roads with one junction, we compute explicitly the eigenvalue and we show, by numerical simulations, that these two quantities (the additive eigenvalue and the growth rate) are not equal, but are close to each other. With this result, we are able to extend the well-studied notion of fundamental traffic diagram (the average flow as a function of the car density on a road) to the case of two roads with one junction and give a very simple analytic approximation of this diagram where four phases appear with clear traffic interpretations. Simulations show that the fundamental diagram shape obtained is also valid for systems with many junctions. To simulate these systems, we have to compute their dynamics, which are not quite simple. For building them in a modular way, we introduce generalized parallel, series and feedback compositions of piecewise linear concave dynamics.Comment: PDF 38 page

Notions of Infinity in Quantum Physics

In this article we will review some notions of infiniteness that appear in Hilbert space operators and operator algebras. These include proper infiniteness, Murray von Neumann's classification into type I and type III factors and the class of F{/o} lner C*-algebras that capture some aspects of amenability. We will also mention how these notions reappear in the description of certain mathematical aspects of quantum mechanics, quantum field theory and the theory of superselection sectors. We also show that the algebra of the canonical anti-commutation relations (CAR-algebra) is in the class of F{/o} lner C*-algebras.Comment: 11 page

Algebraic Structures in Euclidean and Minkowskian Two-Dimensional Conformal Field Theory

We review how modular categories, and commutative and non-commutative Frobenius algebras arise in rational conformal field theory. For Euclidean CFT we use an approach based on sewing of surfaces, and in the Minkowskian case we describe CFT by a net of operator algebras.Comment: 21 pages, contribution to proceedings for "Non-commutative Structures in Mathematics and Physics" (Brussels, July 2008

From vertex operator algebras to conformal nets and back

We consider unitary simple vertex operator algebras whose vertex operators satisfy certain energy bounds and a strong form of locality and call them strongly local. We present a general procedure which associates to every strongly local vertex operator algebra V a conformal net A_V acting on the Hilbert space completion of V and prove that the isomorphism class of A_V does not depend on the choice of the scalar product on V. We show that the class of strongly local vertex operator algebras is closed under taking tensor products and unitary subalgebras and that, for every strongly local vertex operator algebra V, the map W\mapsto A_W gives a one-to-one correspondence between the unitary subalgebras W of V and the covariant subnets of A_V. Many known examples of vertex operator algebras such as the unitary Virasoro vertex operator algebras, the unitary affine Lie algebras vertex operator algebras, the known c=1 unitary vertex operator algebras, the moonshine vertex operator algebra, together with their coset and orbifold subalgebras, turn out to be strongly local. We give various applications of our results. In particular we show that the even shorter Moonshine vertex operator algebra is strongly local and that the automorphism group of the corresponding conformal net is the Baby Monster group. We prove that a construction of Fredenhagen and J\"{o}rss gives back the strongly local vertex operator algebra V from the conformal net A_V and give conditions on a conformal net A implying that A= A_V for some strongly local vertex operator algebra V.Comment: Minor correction

Is Quantum Field Theory ontologically interpretable? On localization, particles and fields in relativistic Quantum Theory

In this paper, I provide a formal set of assumptions and give a natural criterion for a quantum field theory to admit particles. I construct a na\"ive approach to localization for a free bosonic quantum field theory and show how this localization scheme, as a consequence of the Reeh-Schlieder theorem, fails to satisfy this criterion. I then examine the Newton-Wigner concept of localization and show that it fails to obey strong microcausality and thus is subject to a more general version of the Reeh-Schlieder theorem. I review approaches to quantum field theoretic explanations of particle detection events and explain how particles can be regarded as emergent phenomena of a relativistic field theory. In particular, I show that effective localization of Hilbert space vectors is equivalent to an approximate locality of observable algebras.Comment: 33 page