121 research outputs found
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 among two
individual terms and , 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
. The group 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
Relational lattices via duality
The natural join and the inner union combine in different ways tables of a
relational database. Tropashko [18] observed that these two operations are the
meet and join in a class of lattices-called the relational lattices- and
proposed lattice theory as an alternative algebraic approach to databases.
Aiming at query optimization, Litak et al. [12] initiated the study of the
equational theory of these lattices. We carry on with this project, making use
of the duality theory developed in [16]. The contributions of this paper are as
follows. Let A be a set of column's names and D be a set of cell values; we
characterize the dual space of the relational lattice R(D, A) by means of a
generalized ultrametric space, whose elements are the functions from A to D,
with the P (A)-valued distance being the Hamming one but lifted to subsets of
A. We use the dual space to present an equational axiomatization of these
lattices that reflects the combinatorial properties of these generalized
ultrametric spaces: symmetry and pairwise completeness. Finally, we argue that
these equations correspond to combinatorial properties of the dual spaces of
lattices, in a technical sense analogous of correspondence theory in modal
logic. In particular, this leads to an exact characterization of the finite
lattices satisfying these equations.Comment: Coalgebraic Methods in Computer Science 2016, Apr 2016, Eindhoven,
Netherland
Higher Structures in M-Theory
The key open problem of string theory remains its non-perturbative completion
to M-theory. A decisive hint to its inner workings comes from numerous
appearances of higher structures in the limits of M-theory that are already
understood, such as higher degree flux fields and their dualities, or the
higher algebraic structures governing closed string field theory. These are all
controlled by the higher homotopy theory of derived categories, generalised
cohomology theories, and -algebras. This is the introductory chapter
to the proceedings of the LMS/EPSRC Durham Symposium on Higher Structures in
M-Theory. We first review higher structures as well as their motivation in
string theory and beyond. Then we list the contributions in this volume,
putting them into context.Comment: 22 pages, Introductory Article to Proceedings of LMS/EPSRC Durham
Symposium Higher Structures in M-Theory, August 2018, references update
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
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
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
Positive definite metric spaces
Magnitude is a numerical invariant of finite metric spaces, recently
introduced by T. Leinster, which is analogous in precise senses to the
cardinality of finite sets or the Euler characteristic of topological spaces.
It has been extended to infinite metric spaces in several a priori distinct
ways. This paper develops the theory of a class of metric spaces, positive
definite metric spaces, for which magnitude is more tractable than in general.
Positive definiteness is a generalization of the classical property of negative
type for a metric space, which is known to hold for many interesting classes of
spaces. It is proved that all the proposed definitions of magnitude coincide
for compact positive definite metric spaces and further results are proved
about the behavior of magnitude as a function of such spaces. Finally, some
facts about the magnitude of compact subsets of l_p^n for p \le 2 are proved,
generalizing results of Leinster for p=1,2, using properties of these spaces
which are somewhat stronger than positive definiteness.Comment: v5: Corrected some misstatements in the last few paragraphs. Updated
reference
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
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 , 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 when it is a field. This gives a new definition
for \'etale \pi_1(\spec(R)) in terms of the category of -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 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
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