A Topos Foundation for Theories of Physics: IV. Categories of Systems

This paper is the fourth in a series whose goal is to develop a fundamentally new way of building theories of physics. The motivation comes from a desire to address certain deep issues that arise in the quantum theory of gravity. Our basic contention is that constructing a theory of physics is equivalent to finding a representation in a topos of a certain formal language that is attached to the system. Classical physics arises when the topos is the category of sets. Other types of theory employ a different topos. The previous papers in this series are concerned with implementing this programme for a single system. In the present paper, we turn to considering a collection of systems: in particular, we are interested in the relation between the topos representation for a composite system, and the representations for its constituents. We also study this problem for the disjoint sum of two systems. Our approach to these matters is to construct a category of systems and to find a topos representation of the entire category.Comment: 38 pages, no figure

A Topos Foundation for Theories of Physics: II. Daseinisation and the Liberation of Quantum Theory

This paper is the second in a series whose goal is to develop a fundamentally new way of constructing theories of physics. The motivation comes from a desire to address certain deep issues that arise when contemplating quantum theories of space and time. Our basic contention is that constructing a theory of physics is equivalent to finding a representation in a topos of a certain formal language that is attached to the system. Classical physics arises when the topos is the category of sets. Other types of theory employ a different topos. In this paper, we study in depth the topos representation of the propositional language, PL(S), for the case of quantum theory. In doing so, we make a direct link with, and clarify, the earlier work on applying topos theory to quantum physics. The key step is a process we term `daseinisation' by which a projection operator is mapped to a sub-object of the spectral presheaf--the topos quantum analogue of a classical state space. In the second part of the paper we change gear with the introduction of the more sophisticated local language L(S). From this point forward, throughout the rest of the series of papers, our attention will be devoted almost entirely to this language. In the present paper, we use L(S) to study `truth objects' in the topos. These are objects in the topos that play the role of states: a necessary development as the spectral presheaf has no global elements, and hence there are no microstates in the sense of classical physics. Truth objects therefore play a crucial role in our formalism.Comment: 34 pages, no figure

A Topological Study of Contextuality and Modality in Quantum Mechanics

Kochen-Specker theorem rules out the non-contextual assignment of values to physical magnitudes. Here we enrich the usual orthomodular structure of quantum mechanical propositions with modal operators. This enlargement allows to refer consistently to actual and possible properties of the system. By means of a topological argument, more precisely in terms of the existence of sections of sheaves, we give an extended version of Kochen-Specker theorem over this new structure. This allows us to prove that contextuality remains a central feature even in the enriched propositional system.Comment: 10 pages, no figures, submitted to I. J. Th. Phy

Antibody responses to nasopharyngeal carriage of Streptococcus pneumoniae in adults: A longitudinal household study

Background. Natural immunity to Streptococcus pneumoniae is thought to be induced by exposure to S. pneumoniae or cross-reactive antigens. No longitudinal studies of carriage of and immune responses to S. pneumoniae have been conducted using sophisticated immunological laboratory techniques.Methods. We enrolled 121 families with young children into this study. Nasopharyngeal (NP) swabs were collected monthly for 10 months from all family members and were cultured in a standard fashion. Cultured S. pneumoniae isolates were serotyped. At the beginning (month 0) and end (month 10) of the study, venous blood was collected from family members 118 years old. Serotype-specific antipolysaccharide immunoglobulin G (IgG) and functional antibody and antibodies to pneumolysin, pneumococcal surface protein A (PspA), and pneumococcal surface antigen A (PsaA) were measured in paired serum samples.Results. Levels of anticapsular IgG increased significantly after carriage of serotypes 9V, 14, 18C, 19F, and 23F by an individual or family member. For serotype 14, a higher level of anticapsular IgG at the beginning of the study was associated with reduced odds of carriage (P = .0006). There was a small (similar to 20%) but significant increase in titers of antibodies to PsaA and pneumolysin but no change in titers of antibody to PspA.Conclusions. Adults respond to NP carriage by mounting anticapsular and weak antiprotein antibody responses, and naturally induced anticapsular IgG can prevent carriage

Contextual logic for quantum systems

In this work we build a quantum logic that allows us to refer to physical magnitudes pertaining to different contexts from a fixed one without the contradictions with quantum mechanics expressed in no-go theorems. This logic arises from considering a sheaf over a topological space associated to the Boolean sublattices of the ortholattice of closed subspaces of the Hilbert space of the physical system. Differently to standard quantum logics, the contextual logic maintains a distributive lattice structure and a good definition of implication as a residue of the conjunction.Comment: 16 pages, no figure

A logic road from special relativity to general relativity

We present a streamlined axiom system of special relativity in first-order logic. From this axiom system we "derive" an axiom system of general relativity in two natural steps. We will also see how the axioms of special relativity transform into those of general relativity. This way we hope to make general relativity more accessible for the non-specialist

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

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

Is Gliese 581d habitable? Some constraints from radiative-convective climate modeling

The recently discovered exoplanet Gl581d is extremely close to the outer edge of its system's habitable zone, which has led to much speculation on its possible climate. We have performed a range of simulations to assess whether, given simple combinations of chemically stable greenhouse gases, the planet could sustain liquid water on its surface. For best estimates of the surface gravity, surface albedo and cloud coverage, we find that less than 10 bars of CO2 is sufficient to maintain a global mean temperature above the melting point of water. Furthermore, even with the most conservative choices of these parameters, we calculate temperatures above the water melting point for CO2 partial pressures greater than about 40 bar. However, we note that as Gl581d is probably in a tidally resonant orbit, further simulations in 3D are required to test whether such atmospheric conditions are stable against the collapse of CO2 on the surface.Comment: 9 pages, 11 figures. Accepted for publication in Astronomy & Astrophysic

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