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

Completeness of dagger-categories and the complex numbers

The complex numbers are an important part of quantum theory, but are difficult to motivate from a theoretical perspective. We describe a simple formal framework for theories of physics, and show that if a theory of physics presented in this manner satisfies certain completeness properties, then it necessarily includes the complex numbers as a mathematical ingredient. Central to our approach are the techniques of category theory, and we introduce a new category-theoretical tool, called the dagger-limit, which governs the way in which systems can be combined to form larger systems. These dagger-limits can be used to characterize the dagger-functor on the category of finite-dimensional Hilbert spaces, and so can be used as an equivalent definition of the inner product. One of our main results is that in a nontrivial monoidal dagger-category with all finite dagger-limits and a simple tensor unit, the semiring of scalars embeds into an involutive field of characteristic 0 and orderable fixed field.Comment: 39 pages. Accepted for publication in the Journal of Mathematical Physic

Categorical formulation of quantum algebras

We describe how dagger-Frobenius monoids give the correct categorical description of certain kinds of finite-dimensional 'quantum algebras'. We develop the concept of an involution monoid, and use it to construct a correspondence between finite-dimensional C*-algebras and certain types of dagger-Frobenius monoids in the category of Hilbert spaces. Using this technology, we recast the spectral theorems for commutative C*-algebras and for normal operators into an explicitly categorical language, and we examine the case that the results of measurements do not form finite sets, but rather objects in a finite Boolean topos. We describe the relevance of these results for topological quantum field theory.Comment: 34 pages, to appear in Communications in Mathematical Physic

Effectiveness of the Mindfulness in Schools Programme: non-randomised controlled feasibility study

Open Access Article. Copyright ©2013 The Royal College of PsychiatristsMindfulness-based approaches for adults are effective at enhancing mental health, but few controlled trials have evaluated their effectiveness among young people

Approved Mental Health Professionals, Best Interests Assessors and People with Lived Experience - An Exploration of Professional Identities and Practice, A report prepared for Social Work England

Social Work England was established under The Children and Social Work Act 2017. It is the specialist regulator for social workers in England. Social Work England officially took over from the Health and Care Professions Council (HCPC) in December 2019. It is a non-departmental public body, operating at arm’s length from the government. Social Work England has become the professional regulator for Approved Mental Health Professionals (AMHPs) and Best Interests Assessors (BIAs). In 2020 -21, Social Work England has been developing the regulatory framework to support AMHPs and the new specialism of Approved Mental Capacity Professional (AMCP), which will succeed BIAs from April 2022. This includes the development of education & training approval standards as well as specialist standards for AMHP and AMCP practice. Social Work England commissioned this piece of work as part of a commitment to learning about the professionals in these specialisms and people’s experiences of them. The objective of this research was to undertake a study into the experiences of AMHPs and BIAs and those who have experience of their interventions. Existing research is generally inconclusive and little is known about this area

Picturing classical and quantum Bayesian inference

We introduce a graphical framework for Bayesian inference that is sufficiently general to accommodate not just the standard case but also recent proposals for a theory of quantum Bayesian inference wherein one considers density operators rather than probability distributions as representative of degrees of belief. The diagrammatic framework is stated in the graphical language of symmetric monoidal categories and of compact structures and Frobenius structures therein, in which Bayesian inversion boils down to transposition with respect to an appropriate compact structure. We characterize classical Bayesian inference in terms of a graphical property and demonstrate that our approach eliminates some purely conventional elements that appear in common representations thereof, such as whether degrees of belief are represented by probabilities or entropic quantities. We also introduce a quantum-like calculus wherein the Frobenius structure is noncommutative and show that it can accommodate Leifer's calculus of `conditional density operators'. The notion of conditional independence is also generalized to our graphical setting and we make some preliminary connections to the theory of Bayesian networks. Finally, we demonstrate how to construct a graphical Bayesian calculus within any dagger compact category.Comment: 38 pages, lots of picture

Spatial variations in lead isotopes, Tasman Element, eastern Australia

Lead isotope data from ore deposits and mineral occurrences in the Tasman Element of eastern Australia have been used to construct isotopic maps of this region. These maps exhibit systematic patterns in parameters derived from isotope ratios. The parameters include μ (238U/204Pb), as calculated using the Cumming and Richards (1975) lead evolution model, and the difference between true age of mineralisation and the Cumming and Richards lead isotope model age of mineralisation (Δt). Variations in μ coincide with boundaries at the orogen, subprovince and zone scales. The boundary between the Lachlan and New England orogens is accompanied by a decrease in μ, and within the Lachlan Orogen, the Central Subprovince is characterised by μ that is significantly higher than in the adjacent Eastern and Western subprovinces. Within the Eastern Subprovince, the Cu-Au-rich Macquarie Arc is characterised by significantly lower μ relative to adjacent rocks. The Macquarie Arc is also characterised by very high Δt (generally above 200 Myr). Other regions characterised by very high Δt include western Tasmania, the southeastern New England Orogen, and the Hodgkinson Province in northern Queensland. These anomalies are within a broad pattern of decreasing Δt from east to west, with Paleozoic deposits within or adjacent to Proterozoic crust characterised by Δt values of 50 Myr or below. The patterns in Δt are interpreted to reflect the presence of the two major tectonic components involved in the Paleozoic Tasman margin in Australia (cf., Münker, 2000): subducting proto-Pacific crust (Δt >150 Myr), and Proterozoic Australia crust (Δt < 50 Myr) on the over-riding plate. Proterozoic Australia crustal sources are interpreted to dominate the western parts of the Tasman Element and Proterozoic crust further to the west, whereas Pacific crustal sources are inferred to characterise western Tasmania and much of the eastern part of the Tasman Element. Contrasts in Δt between the Cambrian Mount Read Volcanics in western Tasmania and similar aged rocks in western Victoria and New South Wales make direct tectonic correlation between these rocks problematic

Causal categories: relativistically interacting processes

A symmetric monoidal category naturally arises as the mathematical structure that organizes physical systems, processes, and composition thereof, both sequentially and in parallel. This structure admits a purely graphical calculus. This paper is concerned with the encoding of a fixed causal structure within a symmetric monoidal category: causal dependencies will correspond to topological connectedness in the graphical language. We show that correlations, either classical or quantum, force terminality of the tensor unit. We also show that well-definedness of the concept of a global state forces the monoidal product to be only partially defined, which in turn results in a relativistic covariance theorem. Except for these assumptions, at no stage do we assume anything more than purely compositional symmetric-monoidal categorical structure. We cast these two structural results in terms of a mathematical entity, which we call a `causal category'. We provide methods of constructing causal categories, and we study the consequences of these methods for the general framework of categorical quantum mechanics.Comment: 43 pages, lots of figure

Elesclomol restores mitochondrial function in genetic models of copper deficiency

© The Author(s), 2018. This article is distributed under the terms of the Creative Commons Attribution License. The definitive version was published in Proceedings of the National Academy of Sciences of the United States of America 115 (2018): 8161-8166, doi:10.1073/pnas.1806296115.Copper is an essential cofactor of cytochrome c oxidase (CcO), the terminal enzyme of the mitochondrial respiratory chain. Inherited loss-of-function mutations in several genes encoding proteins required for copper delivery to CcO result in diminished CcO activity and severe pathologic conditions in affected infants. Copper supplementation restores CcO function in patient cells with mutations in two of these genes, COA6 and SCO2, suggesting a potential therapeutic approach. However, direct copper supplementation has not been therapeutically effective in human patients, underscoring the need to identify highly efficient copper transporting pharmacological agents. By using a candidate-based approach, we identified an investigational anticancer drug, elesclomol (ES), that rescues respiratory defects of COA6-deficient yeast cells by increasing mitochondrial copper content and restoring CcO activity. ES also rescues respiratory defects in other yeast mutants of copper metabolism, suggesting a broader applicability. Low nanomolar concentrations of ES reinstate copper-containing subunits of CcO in a zebrafish model of copper deficiency and in a series of copper-deficient mammalian cells, including those derived from a patient with SCO2 mutations. These findings reveal that ES can restore intracellular copper homeostasis by mimicking the function of missing transporters and chaperones of copper, and may have potential in treating human disorders of copper metabolism.This work was supported by National Institutes of Health Awards R01GM111672 (to V.M.G.), R01 DK110195 (to B.-E.K.), and DK 44464 (to J.D.G.); Welch Foundation Grant A-1810 (to V.M.G.); and Canadian Institutes of Health Research Operating Grant MOP 133562 (to S.C.L.)

Mobile Criminals, Immobile Crime: The Efficiency of Decentralized Crime Deterrence

In this paper we examine a class of local crimes that involve perfectly mobile criminals, and perfectly immobile criminal opportunities. We focus on local non-rival crime deterrence that is more efficient against criminals pursuing domestic crimes than criminals pursuing crimes elsewhere. In a standard case of sincerely delegated politicians and zero transfers to other districts, we show that centralized deterrence unambiguously dominates the decentralized deterrence. With strategic delegation and voluntary in-kind transfers, the tradeoff is exactly the opposite: Decentralization achieves the social optimum, whereas cooperative centralization overprovides for enforcement. This is robust to various cost-sharing modes. We also examine the effects of the growing interdependence of districts, stemming from criminals' increasing opportunities to strategically displace. Contrary to the supposition in Oates's decentralization theorem, increasing interdependence makes centralization less desirable