Categorical Foundation of Quantum Mechanics and String Theory

The unification of Quantum Mechanics and General Relativity remains the primary goal of Theoretical Physics, with string theory appearing as the only plausible unifying scheme. In the present work, in a search of the conceptual foundations of string theory, we analyze the relational logic developed by C. S. Peirce in the late nineteenth century. The Peircean logic has the mathematical structure of a category with the relation $R_{ij}$ among two individual terms $S_i$ and $S_j$, serving as an arrow (or morphism). We introduce a realization of the corresponding categorical algebra of compositions, which naturally gives rise to the fundamental quantum laws, thus indicating category theory as the foundation of Quantum Mechanics. The same relational algebra generates a number of group structures, among them $W_{\infty}$. The group $W_{\infty}$ is embodied and realized by the matrix models, themselves closely linked with string theory. It is suggested that relational logic and in general category theory may provide a new paradigm, within which to develop modern physical theories.Comment: To appear in International Journal of Modern Physics

Introduction to Categories and Categorical Logic

The aim of these notes is to provide a succinct, accessible introduction to some of the basic ideas of category theory and categorical logic. The notes are based on a lecture course given at Oxford over the past few years. They contain numerous exercises, and hopefully will prove useful for self-study by those seeking a first introduction to the subject, with fairly minimal prerequisites. The coverage is by no means comprehensive, but should provide a good basis for further study; a guide to further reading is included. The main prerequisite is a basic familiarity with the elements of discrete mathematics: sets, relations and functions. An Appendix contains a summary of what we will need, and it may be useful to review this first. In addition, some prior exposure to abstract algebra - vector spaces and linear maps, or groups and group homomorphisms - would be helpful.Comment: 96 page

Renormalization : A number theoretical model

We analyse the Dirichlet convolution ring of arithmetic number theoretic functions. It turns out to fail to be a Hopf algebra on the diagonal, due to the lack of complete multiplicativity of the product and coproduct. A related Hopf algebra can be established, which however overcounts the diagonal. We argue that the mechanism of renormalization in quantum field theory is modelled after the same principle. Singularities hence arise as a (now continuously indexed) overcounting on the diagonals. Renormalization is given by the map from the auxiliary Hopf algebra to the weaker multiplicative structure, called Hopf gebra, rescaling the diagonals.Comment: 15 pages, extended version of talks delivered at SLC55 Bertinoro,Sep 2005, and the Bob Delbourgo QFT Fest in Hobart, Dec 200

Breast cancer screening beliefs by practice location

BACKGROUND: This study examines variations in breast cancer screening among primary care clinicians by geographic location of clinical practice. METHODS: A cross-sectional survey design was used to examine approaches to breast cancer screening among physicians, nurse practitioners, and physician assistants involved in primary care practice. A summary index of beliefs about breast cancer screening was created by summing the total number of responses in agreement with each of four survey items; values for this summary variable ranged between zero and four. Respondents were classified into urban, rural and suburban categories based upon practise location. RESULTS: Among the 428 respondents, agreement with "correct" responses ranged from 50% to 71% for the individual survey items; overall, half agreed with three or more of the four breast cancer screening items. While no significant differences were noted by practice location, variation in responses were evident. Reported use of written breast cancer guidelines was less in both suburban (OR = 0.51) and urban areas (OR = 0.56) when compared to clinicians in rural areas. CONCLUSION: Development of an evidence-based consensus statement regarding breast cancer screening would support a single set of unambiguous guidelines for implementation in all primary care settings, thus decreasing variations in how breast cancer screening is approached across varied clinical settings

Correlating matched-filter model for analysis and optimisation of neural networks

A new formalism is described for modelling neural networks by means of which a clear physical understanding of the network behaviour can be gained. In essence, the neural net is represented by an equivalent network of matched filters which is then analysed by standard correlation techniques. The procedure is demonstrated on the synchronous Little-Hopfield network. It is shown how the ability of this network to discriminate between stored binary, bipolar codes is optimised if the stored codes are chosen to be orthogonal. However, such a choice will not often be possible and so a new neural network architecture is proposed which enables the same discrimination to be obtained for arbitrary stored codes. The most efficient convergence of the synchronous Little-Hopfield net is obtained when the neurons are connected to themselves with a weight equal to the number of stored codes. The processing gain is presented for this case. The paper goes on to show how this modelling technique can be extended to analyse the behaviour of both hard and soft neural threshold responses and a novel time-dependent threshold response is described

Quantum Picturalism

The quantum mechanical formalism doesn't support our intuition, nor does it elucidate the key concepts that govern the behaviour of the entities that are subject to the laws of quantum physics. The arrays of complex numbers are kin to the arrays of 0s and 1s of the early days of computer programming practice. In this review we present steps towards a diagrammatic `high-level' alternative for the Hilbert space formalism, one which appeals to our intuition. It allows for intuitive reasoning about interacting quantum systems, and trivialises many otherwise involved and tedious computations. It clearly exposes limitations such as the no-cloning theorem, and phenomena such as quantum teleportation. As a logic, it supports `automation'. It allows for a wider variety of underlying theories, and can be easily modified, having the potential to provide the required step-stone towards a deeper conceptual understanding of quantum theory, as well as its unification with other physical theories. Specific applications discussed here are purely diagrammatic proofs of several quantum computational schemes, as well as an analysis of the structural origin of quantum non-locality. The underlying mathematical foundation of this high-level diagrammatic formalism relies on so-called monoidal categories, a product of a fairly recent development in mathematics. These monoidal categories do not only provide a natural foundation for physical theories, but also for proof theory, logic, programming languages, biology, cooking, ... The challenge is to discover the necessary additional pieces of structure that allow us to predict genuine quantum phenomena.Comment: Commissioned paper for Contemporary Physics, 31 pages, 84 pictures, some colo

Topos theory and `neo-realist' quantum theory

Topos theory, a branch of category theory, has been proposed as mathematical basis for the formulation of physical theories. In this article, we give a brief introduction to this approach, emphasising the logical aspects. Each topos serves as a `mathematical universe' with an internal logic, which is used to assign truth-values to all propositions about a physical system. We show in detail how this works for (algebraic) quantum theory.Comment: 22 pages, no figures; contribution for Proceedings of workshop "Recent Developments in Quantum Field Theory", MPI MIS Leipzig, July 200

Bisimilarity congruences for open terms and term graphs via tile logic

The definition of sos formats ensuring that bisimilarity on closed terms is a congruence has received much attention in the last two decades. For dealing with open terms, the congruence is usually lifted from closed terms by instantiating the free variables in all possible ways; the only alternatives considered in the literature are Larsen and Xinxin’s context systems and Rensink’s conditional transition systems. We propose an approach based on tile logic, where closed and open terms are managed uniformly, and study the ‘bisimilarity as congruence’ property for several tile formats, accomplishing different concepts of open system

The fundamental pro-groupoid of an affine 2-scheme

A natural question in the theory of Tannakian categories is: What if you don't remember \Forget? Working over an arbitrary commutative ring $R$, we prove that an answer to this question is given by the functor represented by the \'etale fundamental groupoid \pi_1(\spec(R)), i.e.\ the separable absolute Galois group of $R$ when it is a field. This gives a new definition for \'etale \pi_1(\spec(R)) in terms of the category of $R$-modules rather than the category of \'etale covers. More generally, we introduce a new notion of "commutative 2-ring" that includes both Grothendieck topoi and symmetric monoidal categories of modules, and define a notion of $\pi_1$ for the corresponding "affine 2-schemes." These results help to simplify and clarify some of the peculiarities of the \'etale fundamental group. For example, \'etale fundamental groups are not "true" groups but only profinite groups, and one cannot hope to recover more: the "Tannakian" functor represented by the \'etale fundamental group of a scheme preserves finite products but not all products.Comment: 46 pages + bibliography. Diagrams drawn in Tik

Topos Theory and Consistent Histories: The Internal Logic of the Set of all Consistent Sets

A major problem in the consistent-histories approach to quantum theory is contending with the potentially large number of consistent sets of history propositions. One possibility is to find a scheme in which a unique set is selected in some way. However, in this paper we consider the alternative approach in which all consistent sets are kept, leading to a type of `many world-views' picture of the quantum theory. It is shown that a natural way of handling this situation is to employ the theory of varying sets (presheafs) on the space \B of all Boolean subalgebras of the orthoalgebra \UP of history propositions. This approach automatically includes the feature whereby probabilistic predictions are meaningful only in the context of a consistent set of history propositions. More strikingly, it leads to a picture in which the `truth values', or `semantic values' of such contextual predictions are not just two-valued (\ie true and false) but instead lie in a larger logical algebra---a Heyting algebra---whose structure is determined by the space \B of Boolean subalgebras of \UP.Comment: 28 pages, LaTe