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

A Constructive Mathematic approach for Natural Language formal grammars

A mathematical description of natural language grammars has been proposed first by Leibniz. After the definition given by Frege of unsaturated expression and the foundation of a logical grammar by Husserl, the application of logic to treat natural language grammars in a computational way raised the interest of linguists, for example applying Lambek's categorial calculus. In recent years, the most consolidated formal grammars (e.g., Minimalism, HPSG, TAG, CCG, Dependency Grammars) began to show an interest in giving a strong psychological interpretation to the formalism and hence to natural language data on which they are applied. Nevertheless, no one seems to have paid much attention to cognitive linguistics, a branch of linguistics that actively uses concepts and results from cognitive sciences. Apparently unrelated, the study of computational concepts and formalisms has developed in pair with constructive formal systems, especially in the branch of logic called proof theory, see, e.g., the Curry-Howard isomorphism and the typed functional languages. In this paper, we want to bridge these worlds and thus present our natural language formalism, called Adpositional Grammars (AdGrams), that is founded over both cognitive linguistics and constructive mathematics

Quantum field theory on affine bundles

We develop a general framework for the quantization of bosonic and fermionic field theories on affine bundles over arbitrary globally hyperbolic spacetimes. All concepts and results are formulated using the language of category theory, which allows us to prove that these models satisfy the principle of general local covariance. Our analysis is a preparatory step towards a full-fledged quantization scheme for the Maxwell field, which emphasises the affine bundle structure of the bundle of principal U(1)-connections. As a by-product, our construction provides a new class of exactly tractable locally covariant quantum field theories, which are a mild generalization of the linear ones. We also show the existence of a functorial assignment of linear quantum field theories to affine ones. The identification of suitable algebra homomorphisms enables us to induce whole families of physical states (satisfying the microlocal spectrum condition) for affine quantum field theories by pulling back quasi-free Hadamard states of the underlying linear theories.Comment: 34 pages, no figures; v2: 35 pages, compatible with version to be published in Annales Henri Poincar

The Minimal Levels of Abstraction in the History of Modern Computing

From the advent of general-purpose, Turing-complete machines, the relation between operators, programmers, and users with computers can be seen in terms of interconnected informational organisms (inforgs) henceforth analysed with the method of levels of abstraction (LoAs), risen within the Philosophy of Informa- tion (PI). In this paper, the epistemological levellism proposed by L. Floridi in the PI to deal with LoAs will be formalised in constructive terms using category the- ory, so that information itself is treated as structure-preserving functions instead of Cartesian products. The milestones in the history of modern computing are then analysed via constructive levellism to show how the growth of system complexity lead to more and more information hiding

Cheeger-Simons differential characters with compact support and Pontryagin duality

By adapting the Cheeger-Simons approach to differential cohomology, we establish a notion of differential cohomology with compact support. We show that it is functorial with respect to open embeddings and that it fits into a natural diagram of exact sequences which compare it to compactly supported singular cohomology and differential forms with compact support, in full analogy to ordinary differential cohomology. We prove an excision theorem for differential cohomology using a suitable relative version. Furthermore, we use our model to give an independent proof of Pontryagin duality for differential cohomology recovering a result of [Harvey, Lawson, Zweck - Amer. J. Math. 125 (2003) 791]: On any oriented manifold, ordinary differential cohomology is isomorphic to the smooth Pontryagin dual of compactly supported differential cohomology. For manifolds of finite-type, a similar result is obtained interchanging ordinary with compactly supported differential cohomology.Comment: 33 pages, no figures - v3: Final version to be published in Communications in Analysis and Geometr