739 research outputs found
A categorical framework for the quantum harmonic oscillator
This paper describes how the structure of the state space of the quantum
harmonic oscillator can be described by an adjunction of categories, that
encodes the raising and lowering operators into a commutative comonoid. The
formulation is an entirely general one in which Hilbert spaces play no special
role. Generalised coherent states arise through the hom-set isomorphisms
defining the adjunction, and we prove that they are eigenstates of the lowering
operators. Surprisingly, generalised exponentials also emerge naturally in this
setting, and we demonstrate that coherent states are produced by the
exponential of a raising morphism acting on the zero-particle state. Finally,
we examine all of these constructions in a suitable category of Hilbert spaces,
and find that they reproduce the conventional mathematical structures.Comment: 44 pages, many figure
Nets, relations and linking diagrams
In recent work, the author and others have studied compositional algebras of
Petri nets. Here we consider mathematical aspects of the pure linking algebras
that underly them. We characterise composition of nets without places as the
composition of spans over appropriate categories of relations, and study the
underlying algebraic structures.Comment: 15 pages, Proceedings of 5th Conference on Algebra and Coalgebra in
Computer Science (CALCO), Warsaw, Poland, 3-6 September 201
Environment and classical channels in categorical quantum mechanics
We present a both simple and comprehensive graphical calculus for quantum
computing. In particular, we axiomatize the notion of an environment, which
together with the earlier introduced axiomatic notion of classical structure
enables us to define classical channels, quantum measurements and classical
control. If we moreover adjoin the earlier introduced axiomatic notion of
complementarity, we obtain sufficient structural power for constructive
representation and correctness derivation of typical quantum informatic
protocols.Comment: 26 pages, many pics; this third version has substantially more
explanations than previous ones; Journal reference is of short 14 page
version; Proceedings of the 19th EACSL Annual Conference on Computer Science
Logic (CSL), Lecture Notes in Computer Science 6247, Springer-Verlag (2010
A model structure for coloured operads in symmetric spectra
We describe a model structure for coloured operads with values in the
category of symmetric spectra (with the positive model structure), in which
fibrations and weak equivalences are defined at the level of the underlying
collections. This allows us to treat R-module spectra (where R is a cofibrant
ring spectrum) as algebras over a cofibrant spectrum-valued operad with R as
its first term. Using this model structure, we give suficient conditions for
homotopical localizations in the category of symmetric spectra to preserve
module structures.Comment: 16 page
Pre-torsors and Galois comodules over mixed distributive laws
We study comodule functors for comonads arising from mixed distributive laws.
Their Galois property is reformulated in terms of a (so-called) regular arrow
in Street's bicategory of comonads. Between categories possessing equalizers,
we introduce the notion of a regular adjunction. An equivalence is proven
between the category of pre-torsors over two regular adjunctions
and on one hand, and the category of regular comonad arrows
from some equalizer preserving comonad to on
the other. This generalizes a known relationship between pre-torsors over equal
commutative rings and Galois objects of coalgebras.Developing a bi-Galois
theory of comonads, we show that a pre-torsor over regular adjunctions
determines also a second (equalizer preserving) comonad and a
co-regular comonad arrow from to , such that the
comodule categories of and are equivalent.Comment: 34 pages LaTeX file. v2: a few typos correcte
Sums over Graphs and Integration over Discrete Groupoids
We show that sums over graphs such as appear in the theory of Feynman
diagrams can be seen as integrals over discrete groupoids. From this point of
view, basic combinatorial formulas of the theory of Feynman diagrams can be
interpreted as pull-back or push-forward formulas for integrals over suitable
groupoids.Comment: 27 pages, 4 eps figures; LaTeX2e; uses Xy-Pic. Some ambiguities
fixed, and several proofs simplifie
Calculating Colimits Compositionally
We show how finite limits and colimits can be calculated compositionally
using the algebras of spans and cospans, and give as an application a proof of
the Kleene Theorem on regular languages
A Rhamnose-Inducible System for Precise and Temporal Control of Gene Expression in Cyanobacteria
Cyanobacteria are important for fundamental studies of photosynthesis and have great biotechnological potential. In order to better study and fully exploit these organisms, the limited repertoire of genetic tools and parts must be expanded. A small number of inducible promoters have been used in cyanobacteria, allowing dynamic external control of gene expression through the addition of specific inducer molecules. However, the inducible promoters used to date suffer from various drawbacks including toxicity of inducers, leaky expression in the absence of inducer and inducer photolability, the latter being particularly relevant to cyanobacteria, which, as photoautotrophs, are grown under light. Here we introduce the rhamnose-inducible rhaBAD promoter of Escherichia coli into the model freshwater cyanobacterium Synechocystis sp. PCC 6803 and demonstrate it has superior properties to previously reported cyanobacterial inducible promoter systems, such as a non-toxic, photostable, non-metabolizable inducer, a linear response to inducer concentration and crucially no basal transcription in the absence of inducer
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