Functorial Semantics for Petri Nets under the Individual Token Philosophy

Although the algebraic semantics of place/transition Petri nets under the collective token philosophy has been fully explained in terms of (strictly) symmetric (strict) monoidal categories, the analogous construction under the individual token philosophy is not completely satisfactory because it lacks universality and also functoriality. We introduce the notion of pre-net to recover these aspects, obtaining a fully satisfactory categorical treatment centered on the notion of adjunction. This allows us to present a purely logical description of net behaviours under the individual token philosophy in terms of theories and theory morphisms in partial membership equational logic, yielding a complete match with the theory developed by the authors for the collective token view of net

Distances between States and between Predicates

This paper gives a systematic account of various metrics on probability distributions (states) and on predicates. These metrics are described in a uniform manner using the validity relation between states and predicates. The standard adjunction between convex sets (of states) and effect modules (of predicates) is restricted to convex complete metric spaces and directed complete effect modules. This adjunction is used in two state-and-effect triangles, for classical (discrete) probability and for quantum probability

Computability of Operators on Continuous and Discrete Time Streams

A stream is a sequence of data indexed by time. The behaviour of natural and artificial systems can be modelled bystreams and stream transformations. There are two distinct types of data stream: streams based on continuous time and streamsbased on discrete time. Having investigated case studies of both kinds separately, we have begun to combine their study in aunified theory of stream transformers, specified by equations. Using only the standard mathematical techniques of topology, wehave proved continuity properties of stream transformers. Here, in this sequel, we analyse their computability. We use the theoryof computable functions on algebras to design two distinct methods for defining computability on continuous and discrete timestreams of data from a complete metric space. One is based on low-level concrete representations, specifically enumerations, andthe other is based on high-level programming, specifically ‘while’ programs, over abstract data types. We analyse when thesemethods are equivalent. We demonstrate the use of the methods by showing the computability of an analog computing system.We discuss the idea that continuity and computability are important for models of physical systems to be “well-posed”

Models of free quantum field theories on curved backgrounds

Free quantum field theories on curved backgrounds are discussed via three explicit examples: the real scalar field, the Dirac field and the Proca field. The first step consists of outlining the main properties of globally hyperbolic spacetimes, that is the class of manifolds on which the classical dynamics of all physically relevant free fields can be written in terms of a Cauchy problem. The set of all smooth solutions of the latter encompasses the dynamically allowed configurations which are used to identify via a suitable pairing a collection of classical observables. As a last step we use such collection to construct a $*$-algebra which encodes the information on the dynamics and on the canonical commutation or anti-commutation relations depending whether the underlying field is a Fermion or a Boson.Comment: 41 page

Two-dimensional models as testing ground for principles and concepts of local quantum physics

In the past two-dimensional models of QFT have served as theoretical laboratories for testing new concepts under mathematically controllable condition. In more recent times low-dimensional models (e.g. chiral models, factorizing models) often have been treated by special recipes in a way which sometimes led to a loss of unity of QFT. In the present work I try to counteract this apartheid tendency by reviewing past results within the setting of the general principles of QFT. To this I add two new ideas: (1) a modular interpretation of the chiral model Diff(S)-covariance with a close connection to the recently formulated local covariance principle for QFT in curved spacetime and (2) a derivation of the chiral model temperature duality from a suitable operator formulation of the angular Wick rotation (in analogy to the Nelson-Symanzik duality in the Ostertwalder-Schrader setting) for rational chiral theories. The SL(2,Z) modular Verlinde relation is a special case of this thermal duality and (within the family of rational models) the matrix S appearing in the thermal duality relation becomes identified with the statistics character matrix S. The relevant angular Euclideanization'' is done in the setting of the Tomita-Takesaki modular formalism of operator algebras. I find it appropriate to dedicate this work to the memory of J. A. Swieca with whom I shared the interest in two-dimensional models as a testing ground for QFT for more than one decade. This is a significantly extended version of an ``Encyclopedia of Mathematical Physics'' contribution hep-th/0502125.Comment: 55 pages, removal of some typos in section