arXiv subjects

Edward Anderson

Publications and source records attributed to Edward Anderson.

At least 55 records · Page 3Linked to original sources

arXiv2018

The Smallest Shape Spaces. I. Shape Theory Posed, with Example of 3 Points on the Line

This treatise concerns shapes in the sense of constellations of points with various automorphisms quotiented out: continuous translations, rotations and dilations, and also discrete mirror image identification and labelling indistinguishability of the points. We consider in particular the corresponding configuration spaces, which include shape spaces and shape-and-scale spaces. This is a substantial model arena for developing concepts of Background Independence, with many analogies to General Relativity and Quantum Gravity, also with many applications to Dynamics, Quantization, Probability and Statistics. We also explain the necessity of working within the shape-theoretic Aufbau Principle: only considering larger particle number $N$, spatial dimension $d$ and continuous group of automorphisms $G$ when all the relatively smaller cases have been considered. We show that topological shape spaces are graphs, opening up hitherto untapped combinatorial foundations both for these and for topological features of the more usually-considered spaces of metric shapes. We give a conceptual analysis of inhomogeneous Background Independence's clustering and uniformness aspects. We also consider the fate of shape spaces' (similarity) Killing vectors upon performing the mirror image and particle indistinguishability quotientings; this is crucial for dynamical and quantization considerations. For now in Part I we illustrate all these topological, combinatorial, differential-geometric and inhomogeneity innovations with the example of 3 points in 1-$d$. Papers II to IV then extend the repertoire of examples to 4 points in 1-$d$, triangles (3 points in 2- and 3-$d$) and quadrilaterals (4 points in 2-$d$) respectively. The quadrilateral is a minimal requirement prior to most implementations of the third part of the shape-theoretic Aufbau Principle: adding further generators to the automorphism group $G$.

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arXiv2018

The Smallest Shape Spaces. II. 4 Points in 1-d Suffices to have a Complex Background-Independent Theory of Inhomogeneity

The program of understanding Shape Theory layer by layer topologically and geometrically -- proposed in Part I -- is now addressed for 4 points in 1-$d$. Topological shape space graphs are far more complex here, whereas metric shape spaces are (pieces of) spheres which admit an intricate shape-theoretically significant tessellation. Metric shapes covers a far wider range of notions of inhomogeneity: collisions, symmetric states, mergers and uniform states are all distinctly realized in this model. We furthermore provide quantifiers for the extent to which various ways which configurations maximally and minimally realize these. Some of the uniform states additionally form cusps and higher catastrophes in the indistinguishable-particle and Leibniz shape spaces. We also provide shape-theoretically significant notions of centre for the indistinguishable-particle and Leibniz shape spaces. 4 points in 1-$d$ constitutes a useful and already highly nontrivial model of inhomogeneity and of uniformity -- both topics of cosmological interest -- and also of background independence: of interest in the foundations of physics and in quantum gravity. We finally give the automorphism groups of the topological shape space graphs and the metric shape space (pieces of) manifolds, which is a crucial preliminary toward quantizing the indistinguishable-particle and Leibniz versions of the model.

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arXiv2018

The Smallest Shape Spaces. III. Triangles in 2- and 3-d

This is an innovative treatise on triangles, resting upon 1) 3-body problem techniques including mass-weighted relative Jacobi coordinates. 2) Part I's detailed layer-by-layer topological and geometrical study of Kendall-type shape spaces - configuration spaces of all possible shapes - which, for triangles, are (pieces of) spheres. 3) Hopf mathematics. Triangles are moreover prototypical through being the smallest models which carry relative-angle as well as length-ratio information. Both 1) and 3) produce insightful new versions of Heron's formula, 3)'s simultaneously providing new foundations for 2). Medians, and regular triangles bounding between tall and flat triangles, also play prominent roles. Right triangles form three kissing cap-circles on the shape sphere, from which a shape-theoretic answer to the well-known conundrum of what is the probability that a triangle is obtuse very readily follows: 3/4. The differential-geometric aspects of this answer moreover generalize to numerous variant problems. Hopf mathematics additionally gives a general bundle section interpretation to Kendall's iconic spherical blackboard of vertex-unlablelled mirror-image-identified triangles, and of its two variants where one of these two conditions are dropped. We attribute a monopole to each of these spaces and to the full shape sphere, one due to Dirac, one to Iwai and the other two are new to this paper. We finally make insightful comparison of triangles in 2-$d$ with a) Part II's 4 points on the line. b) Triangles in 3-$d$, which are particularly significant as the smallest model exhibiting stratification. Stratified manifold-sheaf pairs - sheaves adding useful local and global structure to general bundles - lie at the heart of Shape Theory's future development.

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arXiv2018

Monopoles of Twelve Types in 3-Body Problems

We consider twelve different ways of modelling the 3-body problem in dimension $\geq 2$. These can be viewed as models of classical and quantum background independence. We show that a different type of monopole is realized in each's relational space: a type of reduced configuration space. 8 cases occur in 2-$d$, and 4 distinct ones in 3-$d$; these reflect counts of non-equivalent subgroup actions of $S_3 \times C_2$ and $S_3$ respectively. The $S_3$ acts on particle labels; the extra $C_2$ corresponds to the purely 2-$d$ option of whether or not to identify mirror images. The non-equivalent realization is due to a suite of subgroup, orbit space and stratification features. Our 2-$d$ monopoles include 4 known ones: a realization of Dirac's monopole in relational space rather than its more habitual setting of space, the 2-$d$ version of Iwai's monopole, and indistinguishable particle monopoles with and without mirror image identification. The 4 new ones are indistinguishable under a 2-particle label switch or under even permutations, in each case with optional mirror image identification. Our 4 3-$d$ monopoles are 2 known ones: the actual Iwai monopole and its already-announced indistinguishable-particles counterpart, and 2 new ones: the two-particle label switch and even permutation cases. All 4 3-$d$ cases are stratified. The three even-permutation cases are orbifolds, two with boundary, the 3-$d$ case's boundary constituting a separate stratum, giving a stratified orbifold. We document each of the 12 cases' underlying shape space and relational space, and each monopole's Hopf mathematics, global-section versus topological quantization dichotomy, Dirac string positioning, and Chern integral concordance with topological contributions form of Gauss--Bonnet Theorem.

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arXiv2017

Maximal Angle Flow on the Shape Sphere of Triangles

There has been recent work using Shape Theory to answer the longstanding and conceptually interesting problem of what is the probability that a triangle is obtuse. This is resolved by three kissing cap-circles of rightness being realized on the shape sphere; integrating up the interiors of these caps readily yields the answer to be 3/4. We now generalize this approach by viewing rightness as a particular value of maximal angle, and then covering the shape sphere with the maximal angle flow. Therein, we discover that the kissing cap-circles of rightness constitute a separatrix. The two qualitatively different regimes of behaviour thus separated both moreover carry distinct analytic pathologies: cusps versus excluded limit points. The equilateral triangles are centres in this flow, whereas the kissing points themselves -- binary collision shapes -- are more interesting and elaborate critical points. The other curves' formulae and associated area integrals are more complicated, and yet remain evaluable. As a particular example, we evaluate the probability that a triangle is Fermat-acute, meaning that its Fermat point is nontrivially located; the critical maximal angle in this case is 120 degrees.

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arXiv2017

Shape (In)dependent Inequalities for Triangleland's Jacobi and Democratic-Linear Ellipticity Quantitities

Sides and medians are both Jacobi coordinate magnitudes, moreover then equably entering the spherical coordinates on Kendall's shape sphere and the Hopf coordinates. This motivates treating medians on the same footing as sides in triangle geometry and the resulting Shape Theory. In this paper, we consequently reformulate inequalities for the medians in terms of shape quantities, and proceed to find inequalities on the mass-weighted Jacobi coordinates. This work moreover identifies the $4/3$ -- powers of which occur frequently in the theory of medians -- as the ratio of Jacobi masses. One of the Hopf coordinates is tetra-area. Another is anisoscelesness, which parametrizes whether triangles are left-or-right leaning as bounded by isoscelesness itself. The third is ellipticity, which parametrizes tallness-or-flatness of triangles as bounded by regular triangles. Whereas tetra-area is clearly cluster choice invariant, Jacobi coordinates, anisoscelesness and ellipticity are cluster choice dependent but can be `democratized' by averaging over all clusters. Democratized ellipticity moreover trivializes, due to ellipticity being the difference of base-side and median second moments, whose averages are equal to each other. Thus we introduce a distinct `linear ellipticity' quantifier of tallness-or-flatness of triangles whose democratization is nontrivial, and find inequalities bounding this. Some of this paper's inequalities are shape-independent bounds, whereas others' bounds depend on the isoperimetric ratio and arithmetic-to-geometric side mean ratio shape variables.

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arXiv2017

Two new versions of Heron's Formula

Recollect that Heron's formula for the area of a triangle given its sides has a counterpart given the medians instead, which carries an extra factor of $\frac{4}{3}$. On the one hand, we formulate the pair of these in Linear Algebra terms, showing that they are related by a sides-to-medians involution $J$, which we find to furthermore commute with the `Heron map' $H$ as visible in the expanded version of Heron. Upon further casting the pair of these in terms of mass-weighted Jacobi coordinates, we find moreover that they are placed on an exactly equal footing, the factor of $\frac{4}{3}$ having now cancelled out. This motivates the `Heron--Jacobi' version of Heron's formulae, for mass-weighted area in terms of mass-weighted sides and mass-weighted medians respectively. On the other hand, we show that diagonalizing the Heron map $H$ provides new derivations of, firstly, the famous Hopf map, and, secondly, of Kendall's Theorem that the space of triangles is a sphere. This occurs by the `Heron--Hopf' version of Heron's formula simplifying down to none other than the on-sphere condition. Thus we establish that -- both an important fibre bundle model, and a foundational theorem of Shape Theory: a widely applicable Differential Geometry and Topology topic -- arise together as consequences of just Heron's formula, Stewart's Theorem, and some elementary Linear Algebra manipulations. This working also accounts for the extra factor of 4 in the Hopf coordinate that is elsewise equal to the mass-weighted area in the 3-body problem context. It finally offers a new interpretation of the shape-theoretic ellipticity and anisoscelesness which realize the other two Hopf quantities: as eigenvectors shared by the Heron map $H$ and the sides-to-medians involution $J$.

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arXiv2017

Alice in Triangleland: Lewis Carroll's Pillow Problem and Variants Solved on Shape Space of Triangles

We provide a natural answer to Lewis Carroll's pillow problem of what is the probability that a triangle is obtuse, Prob(Obtuse). This arises by straightforward combination of a) Kendall's Theorem - that the space of all triangles is a sphere - and b) the natural map sending triangles in space to points in this shape sphere. The answer is 3/4. Our method moreover readily generalizes to a wider class of problems, since a) and b) both have many applications and admit large generalizations: Shape Theory. An elementary and thus widely accessible prototype for Shape Theory is thereby desirable, and extending Kendall's already-notable prototype a) by demonstrating that b) readily solves Lewis Carroll's well-known pillow problem indeed provides a memorable and considerably stronger prototype. This is a prototype of, namely, mapping flat geometry problems directly realized in a space to shape space, where differential-geometric tools are readily available to solve the problem and then finally re-interpret it in the original `shape-in-space' terms. We illustrate this program's versatility by posing and solving a number of variants of the pillow problem. We first find Prob(Isosceles is Obtuse). We subsequently define tall and flat triangles, as bounded by regular triangles whose base-to-median ratio is that of the equilateral triangle. These definitions have Jacobian and Hopfian motivation as well as entering Kendall's own considerations of `splinters': almost-collinear triangles. We find that Prob(Tall) = 1/2 = Prob(Flat) is immediately apparent from regularity's symmetric realization in shape space. However, Prob(Obtuse and Flat), Prob(Obtuse is Flat) and all other nontrivial expressions concerning having any two of the properties mentioned above, or having one of these conditioned on another, constitute nontrivial variants of the pillow problem, and we solve them all.

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arXiv2016

'Shape Dynamics': Foundations Reassessed

`Shape dynamics' is meant here in the sense of a type of conformogeometrical reformulation of GR, some of which have of late been considered as generalizations of or alternatives to GR. This note concerns in particular cases based on the notion of volume-preserving conformal transformations (VPCTs), in the sense of preserving a solitary global volume of the universe degree of freedom. The extent to which various ways of modelling VPCTs make use of group theory at all, in a congruous manner, and with minimal departure from standard Lie group theory, is considered. This points to changing conception of VPCTs from the current finite integral implementation to an infinitesimal differential implementation (or to avoiding using them at all). Some useful observations from flat-space conformal groups (well-known from CFT) concerning the existence or otherwise of VPCT groups are also provided.

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arXiv2016

Kinematical Quantization

We consider here kinematical quantization: a first and often overlooked step in quantization procedures. $\mathbb{R}$, $\mathbb{R}_+$ and the interval are considered, as well as direct (Cartesian) products thereof. Some simple minisuperspace models, and mode by mode consideration of slightly inhomogeneous cosmology, have indefinite signature versions of such kinematical quantizations. The examples in the current paper build in particular toward the case of vacuum $\mathbb{S}^3$ slightly inhomogeneous cosmology's mode configuration space, which is mathematically a finite time interval slab of Minkowski spacetime.

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arXiv2016

Configuration Spaces in Fundamental Physics

I consider configuration spaces for $N$-body problems, gauge theories and for GR in both geometrodynamical and Ashtekar variables forms, including minisuperspace and inhomogeneous perturbations thereabout in the former case. These examples include many interesting spaces of shapes (with and without whichever of local or global notions of scale). In considering reduced configuration spaces, stratified manifolds arise. Three strategies to deal with these are `excise', `unfold' and `accept'. I show that spaces of triangles arising from various interpretations of 3-body problems already serve as model arena for all three. I furthermore argue in favour of the `accept' strategy on relational grounds. This approach requires sheaf methods (which go beyond fibre bundles and general bundles, which I contrast with sheaves and presheaves in some appendices). Sheaf methods are also required for the stratifold construct that pairs some well-behaved stratified manifolds with sheaves. I apply arguing against `excise' and `unfold' to GR's superspace and thin sandwich, and to the removal of collinear configurations in mechanics. Non-redundant configurations are also useful in providing more accurate names for various spaces and theories.

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arXiv2016

On Types of Observables in Constrained Theories

The Kuchar observables notion is shown to apply only to a limited range of theories. Relational mechanics, slightly inhomogeneous cosmology and supergravity are used as examples that require further notions of observables. A suitably general notion of A-observables is then given to cover all of these cases. `A' here stands for `algebraic substructure'; A-observables can be defined by association with each closed algebraic substructure of a theory's constraints. Both constrained algebraic structures and associated notions of A-observables form bounded lattices.

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arXiv2016

Origin of Structure in the Universe: Quantum Cosmology Reconsidered

Based on a more careful canonical analysis, we motivate a reduced quantization - in the sense of superspace quantization - of slightly inhomogeneous cosmology in place of the Dirac quantization in the existing literature, and provide it in the vacuum case. This is attained through consideration of configuration space geometries at various levels of reduction. Some of these have the good fortunate of being flat. Geometrically natural coordinates for these are interpreted in terms of the original redundant formulation's well-known mode expansion coefficients.

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arXiv2016

Problem of Time in Slightly Inhomogeneous Cosmology

The Problem of Time (PoT) is a multi-faceted conceptual incompatibility between various areas of Theoretical Physics. Whilst usually stated as between GR and QM, in fact 8/9ths of it is already present at the classical level. Thus we adopt a `top-down' classical and then quantum approach. I consider a local resolution to the Problem of Time that is Machian, which was previously realized for relational triangle and minisuperspace models. This resolution has three levels: classical, semiclassical and combined semiclassical-histories-records. This article's specific model is a slightly inhomogeneous cosmology considered for now at the classical level. This is motivated by how the inhomogeneous fluctuations that underlie structure formation - galaxies and CMB hotspots - might have been seeded by quantum cosmological fluctuations, as magnified by some inflationary mechanism. In particular, I consider the perturbations about $S^3$ case of this involving up to second order, which has a number of parallels with the Halliwell-Hawking model but has a number of conceptual differences and useful upgrades. The article's main features are that the elimination part of the model's thin sandwich is straightforward, but the modewise split of the constraints fail to be first-class constraints. Thus the elimination part only arises as an intermediate geometry between superspace and Riem. The reduced geometries have surprising singularities influenced by the matter content of the universe, though the N-body problem anticipates these with its collinear singularities. I also give a `basis set' of Kuchar beables for this model arena.

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arXiv2016

Six New Mechanics corresponding to further Shape Theories

A suite of relational notions of shape are presented at the level of configuration space geometry, with corresponding new theories of shape mechanics and shape statistics. These further generalize two quite well known examples: --1) Kendall's (metric) shape space with his shape statistics and Barbour's mechanics thereupon. 0) Leibnizian relational space alias metric scale-and-shape space to which corresponds Barbour-Bertotti mechanics. This paper's new theories include, using the invariant and group namings, 1) $Angle$ alias $conformal$ $shape$ $mechanics$. 2) $Area ratio$ alias $affine$ $shape$ $mechanics$. 3) $ Area$ alias $affine$ $scale$-$and$-$shape$ $mechanics$. 1) to 3) rest respectively on angle space, area-ratio space, and area space configuration spaces. Probability and statistics applications are also pointed to in outline. 4) Various supersymmetric counterparts of -1) to 3) are considered. Since supergravity differs considerably from GR-based conceptions of Background Independence, some of the new supersymmetric shape mechanics are compared with both. These reveal compatibility between supersymmetry and GR-based conceptions of Background Independence, at least within these simpler model arenas.

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arXiv2015

We have time because we shall never know

We argue against current proposals concerning the non-existence of time. We point out that a large number of these proposals rely, at least implicitly, on the assumption of `closure' (or `partial closure') of the laws of Physics. I.e. the assumption that laws of Physics as they are known today are either complete (and hence closed) or that they possess features that a hypothetical future `complete' theory must share (and hence are partially closed). Given that the assumption of closure of laws of Physics can never be verified operationally, it cannot justifiably be used to support the claim for non-existence of time. Some approaches against time are `timeless' at the primary level for the universe as a whole. In these approaches time arises at a secondary level, mostly in the sense of `time being abstracted from change'. On the other hand, there are other approaches that deny the existence of time altogether. We argue that metaphysical arguments of this type - similar to those based on closure - by implicitly implying the absence of history, are by their nature circular.

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arXiv2015

Problem of Time: Temporal Relationalism Compatibility of Other Local Classical Facets Completed

Temporal Relationalism (TR) is that there is no time for the universe as a whole at the primary level. Time emerges rather at a secondary level; one compelling idea for this is Mach's `time is to be abstracted from change'. TR leads to, and better explains, the well-known Frozen Formalism Problem encountered in GR at the quantum level. Indeed, abstraction from change is a type of emergent time resolution of this.Moreover, the Frozen Formalism Problem is but one of the many Problem of Time facets, which are furthermore notoriously interconnected. The other local and already classically present facets are as follows. 2) The GR Thin Sandwich involves a subcase of Best Matching, which is one particular implementation of Configurational Relationalism. 3) The Constraint Closure Problem, 4) the Problem of Observables or Beables, 5) Spacetime Relationalism, 6) the Spacetime Construction Problem, and 7) the Foliation Dependence Problem as resolved in classical GR by Refoliation Invariance. In this Article, I bring together the individual classical resolutions of these, and how these can be rendered compatible with TR. Having covered that in detail for 2) to 6) elsewhere, the rest of the current Article is dedicated to the detailed form that 7) and its TR compatible modification takes. I.e. I consider TR implementing foliations, the TR versions of Refoliation Invariance and the associated TR version of hypersurface kinematics and hypersurface deformations. These require a TR counterpart of the Arnowitt--Deser--Misner split.

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arXiv2015

Kendall's Shape Statistics as a Classical Realization of Barbour-type Timeless Records Theory approach to Quantum Gravity

I previously showed that Kendall's work on shape geometry is in fact also the geometrical description of Barbour's relational mechanics' reduced configuration spaces (alias shape spaces). I now describe the extent to which Kendall's subsequent statistical application to e.g. the `standing stones problem' realizes further ideas along the lines of Barbour-type timeless records theories, albeit just at the classical level.

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