A Topos Foundation for Theories of Physics: III. The Representation of Physical Quantities With Arrows

This paper is the third in a series whose goal is to develop a fundamentally new way of viewing theories of physics. 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. In paper II, we studied the topos representations of the propositional language PL(S) for the case of quantum theory, and in the present paper we do the same thing for the, more extensive, local language L(S). One of the main achievements is to find a topos representation for self-adjoint operators. This involves showing that, for any physical quantity A, there is an arrow \breve{\delta}^o(A):\Sig\map\SR, where \SR is the quantity-value object for this theory. The construction of $\breve{\delta}^o(A)$ is an extension of the daseinisation of projection operators that was discussed in paper II. The object \SR is a monoid-object only in the topos, $\tau_\phi$, of the theory, and to enhance the applicability of the formalism, we apply to \SR a topos analogue of the Grothendieck extension of a monoid to a group. The resulting object, \kSR, is an abelian group-object in $\tau_\phi$. We also discuss another candidate, \PR{\mathR}, for the quantity-value object. In this presheaf, both inner and outer daseinisation are used in a symmetric way. Finally, there is a brief discussion of the role of unitary operators in the quantum topos scheme.Comment: 38 pages, no figure

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

Correlations of πN\boldsymbol{\pi N} Partial Waves for Multi-Reaction Analyses

In the search for missing baryonic resonances, many analyses include data from a variety of pion- and photon-induced reactions. For elastic $\pi N$ scattering, however, usually the partial waves of the SAID or other groups are fitted, instead of data. We provide the partial-wave covariance matrices needed to perform correlated $\chi^2$ fits, in which the obtained $\chi^2$ equals the actual $\chi^2$ up non-linear and normalization corrections. For any analysis relying on partial waves extracted from elastic pion scattering, this is a prerequisite to assess the significance of resonance signals and to assign any uncertainty on results. The influence of systematic errors is also considered.Comment: 7 pages, 3 figures; Acknowledgements update

Charge fluctuations and electric mass in a hot meson gas

Net-Charge fluctuations in a hadron gas are studied using an effective hadronic interaction. The emphasis of this work is to investigate the corrections of hadronic interactions to the charge fluctuations of a non-interacting resonance gas. Several methods, such as loop, density and virial expansions are employed. The calculations are also extended to SU(3) and some resummation schemes are considered. Although the various corrections are sizable individually, they cancel to a large extent. As a consequence we find that charge fluctuations are rather well described by the free resonance gas.Comment: 32 pages, 18 figure

Ramsey interferometry with an atom laser

We present results on a free-space atom interferometer operating on the first order magnetically insensitive |F=1,mF=0> -> |F=2,mF=0> transition of Bose-condensed 87Rb atoms. A pulsed atom laser is output-coupled from a Bose-Einstein condensate and propagates through a sequence of two internal state beam splitters, realized via coherent Raman transitions between the two interfering states. We observe Ramsey fringes with a visibility close to 100% and determine the current and the potentially achievable interferometric phase sensitivity. This system is well suited to testing recent proposals for generating and detecting squeezed atomic states.Comment: published version, 8 pages, 3 figure

Pulsed pumping of a Bose-Einstein condensate

In this work, we examine a system for coherent transfer of atoms into a Bose-Einstein condensate. We utilize two spatially separate Bose-Einstein condensates in different hyperfine ground states held in the same dc magnetic trap. By means of a pulsed transfer of atoms, we are able to show a clear resonance in the timing of the transfer, both in temperature and number, from which we draw conclusions about the underlying physical process. The results are discussed in the context of the recently demonstrated pumped atom laser.Comment: 5 pages, 5 figures, published in Physical Review

Equivalent Radar Cross Section: What Is It and Why Is It Important?

The goal of radiometric calibration in synthetic aperture radar (SAR) is to achieve comparability between measurement results acquired with different systems (e.g. RADARSAT-2 and Sentinel-1), at different times (e.g. image stacks over many years), or with different system settings (e.g. center frequency or polarization). At the beginning of the calibration process stands the definition of the measurement quantity. We argue that the currently accepted measurement quantity for point targets, radar cross section (RCS), is not actually the quantity that a SAR system measures, and propose to replace the quantity with equivalent radar cross section (ERCS): The equivalent radar cross section (ERCS) shall be equal to the radar cross section of a perfectly conducting sphere which would result in an equivalent pixel intensity if the sphere were to replace the measured target. The concept “ERCS” has been introduced before [1]. In this presentation, we attempt to communicate the problem from different angles to make it more easily comprehensible and to contribute to a discussion in the CEOS community on a new definition of the radiometric measurement quantity in SAR. These topics are: • Mathematical view: By reviewing the basic SAR convolution integral and the definition of RCS it becomes obvious why RCS cannot be the radiometric measurement quantity in SAR. • Historical view: Considering early, comparably low resolution SAR systems with narrow bandwidths and small angular ranges, it is apparent why RCS has been an acceptable quantity in the past. The advancement towards higher accuracy and higher resolution systems makes a distinction between RCS and ERCS paramount though for today’s and tomorrow’s systems. • Comparison with black bodies: The radiation characteristics of certain surfaces are completely specified if their temperature is known. These black bodies do not exist in nature; they are an idealization. The blackbody radiation at a given wavelength depends only on the temperature. A single number (the brightness temperature) is therefore sufficient to summarize the complex Planck spectrum of a blackbody. This is similar to a large, perfectly conducting sphere in SAR which is used as an idealized object in the ERCS definition. A single number (the sphere’s cross sectional area) summarizes its properties. • Comparison with stellar photometry: In the 18 th century, different astronomers used different optical equipment to measure and compare the brightness of stars. Due to varying passbands (transfer functions) of lenses and photographic film, the results were not comparable. The problem was later solved by introducing standard photometric systems, where the passbands of the used equipment is standardized and calibrated. A comparable interaction exists between a SAR instrument and the measured terrain reflectivity due to the convolution operation in the processor. This we call the SAR passband problem [2]. We propose a similar approach to resolve the SAR passband problem by introducing standardized passbands (weighting/apodization functions at defined bandwidths). By adopting ERCS as the measurement quantity in the future, calibration and measurement results become truly compatible across current and future narrow and particularly wideband, high-resolution, and high- accuracy SAR systems

Modelling the ability of legumes to suppress weeds

The ability of different legume cover crops to suppress annual weeds during the early establishment phase was compared using a simulation model of inter-plant competition and field observations. Height, partitioning parameters, extinction coefficients, crop density and time of emergence were recorded for 11 species sown in monocultures. A naturally occurring population of fat hen (Chenopodium album) was present on the experiment. The competition model was run to compare the expected suppressive ability of the different species on this weed. Samples of C. album were also taken from each plot immediately prior to cutting to provide some empirical observations. Predicted suppressive ability was correlated with seed size and height with large seeded, tall species such as white sweet clover being the most competitive. However, these species may recover poorly from mowing compromising their potential to suppress perennial weeds and a mixture of contrasting species may provide the optimum weed control