Quantising on a category

We review the problem of finding a general framework within which one can construct quantum theories of non-standard models for space, or space-time. The starting point is the observation that entities of this type can typically be regarded as objects in a category whose arrows are structure-preserving maps. This motivates investigating the general problem of quantising a system whose `configuration space' (or history-theory analogue) is the set of objects \Ob\Q in a category \Q. We develop a scheme based on constructing an analogue of the group that is used in the canonical quantisation of a system whose configuration space is a manifold $Q\simeq G/H$, where $G$ and $H$ are Lie groups. In particular, we choose as the analogue of $G$ the monoid of `arrow fields' on \Q. Physically, this means that an arrow between two objects in the category is viewed as an analogue of momentum. After finding the `category quantisation monoid', we show how suitable representations can be constructed using a bundle (or, more precisely, presheaf) of Hilbert spaces over \Ob\Q. For the example of a category of finite sets, we construct an explicit representation structure of this type.Comment: To appear in a volume dedicated to the memory of James Cushin

A Topos Perspective on State-Vector Reduction

A preliminary investigation is made of possible applications in quantum theory of the topos formed by the collection of all $M$-sets, where $M$ is a monoid. Earlier results on topos aspects of quantum theory can be rederived in this way. However, the formalism also suggests a new way of constructing a `neo-realist' interpretation of quantum theory in which the truth values of propositions are determined by the actions of the monoid of strings of finite projection operators. By these means, a novel topos perspective is gained on the concept of state-vector reduction

Histories quantisation of parameterised systems: I. Development of a general algorithm

We develop a new algorithm for the quantisation of systems with first-class constraints. Our approach lies within the (History Projection Operator) continuous-time histories quantisation programme. In particular, the Hamiltonian treatment (either classical or quantum) of parameterised systems is characterised by the loss of the notion of time in the space of true degrees of freedom (i.e. the `problem of time'). The novel temporal structure of the HPO theory (two laws of time transformation that distinguish between the temporal logical structure and the dynamics) persists after the imposition of the constraints, hence the problem of time does not arise. We expound the algorithm for both the classical and quantum cases and apply it to simple models.Comment: 34 pages, Late

Spatiotemporal evolution of radio wave pump-induced ionospheric phenomena near the fourth electron gyroharmonic

On 12 November 2001, the European Incoherent Scatter (EISCAT) high-frequency (HF) radio wave transmitter facility, operating in O-mode at 5.423 MHz with 550 MW effective radiated power, produced artificial optical rings which appeared immediately at transmitter turn-on and collapsed into blobs after ∼60 s while descending in altitude. A similar descent in altitude was observed in the EISCAT ultra high frequency (UHF) ion line enhancements. Likewise, the stimulated electromagnetic emission (SEE) spectra changed as the pump frequency approached the fourth electron gyroharmonic due to pump-induced variations in electron concentration. Optical recordings were made from Skibotn at 630.0 and 557.7 nm and from Ramfjord in white light. The altitude of the initial optical ring and steady state blob has been estimated by triangulation. The evolution in altitude of the optical emissions, ion line enhancements, and SEE spectra all show a similar morphology but are generally not at exactly the same height. Typically, the optical height is close to and a few kilometers below that of the radar backscatter but sometimes above it, both of which are above the SEE generation altitude. There is evidence that upper hybrid (UH) waves, which propagate perpendicular to the magnetic field line, and Langmuir (L) waves, which propagate parallel to the magnetic field line, act simultaneously to accelerate electrons even in the steady state

Entropy of Classical Histories

We consider a number of proposals for the entropy of sets of classical coarse-grained histories based on the procedures of Jaynes, and prove a series of inequalities relating these measures. We then examine these as a function of the coarse-graining for various classical systems, and show explicitly that the entropy is minimized by the finest-grained description of a set of histories. We propose an extension of the second law of thermodynamics to the entropy of histories. We briefly discuss the implications for decoherent or consistent history formulations of quantum mechanics.Comment: 35 pages RevTeX 3.0 + 5 figures (postscript). Minor corrections and typos. To appear in Physical Review

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

General relativity histories theory I: The spacetime character of the canonical description

The problem of time in canonical quantum gravity is related to the fact that the canonical description is based on the prior choice of a spacelike foliation, hence making a reference to a spacetime metric. However, the metric is expected to be a dynamical, fluctuating quantity in quantum gravity. We show how this problem can be solved in the histories formulation of general relativity. We implement the 3+1 decomposition using metric-dependent foliations which remain spacelike with respect to all possible Lorentzian metrics. This allows us to find an explicit relation of covariant and canonical quantities which preserves the spacetime character of the canonical description. In this new construction, we also have a coexistence of the spacetime diffeomorphisms group, and the Dirac algebra of constraints.Comment: 23 pages, submitted to Class. Quant. Gra

Above and beyond? How businesses can drive sustainable development by promoting lasting pro-environmental behaviour change: an examination of the IKEA Live Lagom project

Current global changes require new business approaches driving sustainable development on all fronts. To date, most business approaches have focused on sustainable marketing and corporate social responsibility initiatives. In this field study, we examine IKEA’s Live Lagom project, a three year behaviour change initiative that aimed to explore how to go above and beyond conventional approaches demonstrating how businesses could support sustainable development by supporting their customers’ attempts to live more sustainable lifestyles. We examined the effectiveness of the project involving multi-faceted behaviour change interventions, testing for behavioural changes both during and after the project period. In addition, we explored changes in participants’ attitudes towards the company. Findings show that the extensive set of interventions led to changes in pro-environmental behaviours across all three participant groups with potentially positive impacts on the customer-company relationship. The article thus provides a call for further businesses to engage in similar behaviour change projects that would allow citizens to engage in more sustainable lifestyles and behaviours across contexts

Persistence of Tripartite Nonlocality for Non-inertial Observers

We consider the behaviour of bipartite and tripartite non-locality between fermionic entangled states shared by observers, one of whom uniformly accelerates. We find that while fermionic entanglement persists for arbitrarily large acceleration, the Bell/CHSH inequalities cannot be violated for sufficiently large but finite acceleration. However the Svetlichny inequality, which is a measure of genuine tripartite non-locality, can be violated for any finite value of the acceleration.Comment: 4 pages, pdflatex, 2 figure

Topos-Theoretic Extension of a Modal Interpretation of Quantum Mechanics

This paper deals with topos-theoretic truth-value valuations of quantum propositions. Concretely, a mathematical framework of a specific type of modal approach is extended to the topos theory, and further, structures of the obtained truth-value valuations are investigated. What is taken up is the modal approach based on a determinate lattice \Dcal(e,R), which is a sublattice of the lattice \Lcal of all quantum propositions and is determined by a quantum state $e$ and a preferred determinate observable $R$. Topos-theoretic extension is made in the functor category \Sets^{\CcalR} of which base category \CcalR is determined by $R$. Each true atom, which determines truth values, true or false, of all propositions in \Dcal(e,R), generates also a multi-valued valuation function of which domain and range are \Lcal and a Heyting algebra given by the subobject classifier in \Sets^{\CcalR}, respectively. All true propositions in \Dcal(e,R) are assigned the top element of the Heyting algebra by the valuation function. False propositions including the null proposition are, however, assigned values larger than the bottom element. This defect can be removed by use of a subobject semi-classifier. Furthermore, in order to treat all possible determinate observables in a unified framework, another valuations are constructed in the functor category \Sets^{\Ccal}. Here, the base category \Ccal includes all \CcalR's as subcategories. Although \Sets^{\Ccal} has a structure apparently different from \Sets^{\CcalR}, a subobject semi-classifier of \Sets^{\Ccal} gives valuations completely equivalent to those in \Sets^{\CcalR}'s.Comment: LaTeX2