Six New Mechanics corresponding to further Shape Theories
SSix New Mechanics corresponding to further Shape Theories
Edward Anderson
DAMTP, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 OWA.
Abstract
A suite of relational notions of shape are presented at the level of configuration space geometry, with correspondingnew 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 relationalspace alias metric scale-and-shape space to which corresponds Barbour–Bertotti mechanics. This paper’s new theoriesinclude, using the invariant and group namings, 1)
Angle alias conformal shape mechanics . 2)
Area ratio alias affineshape mechanics . 3)
Area alias affine scale-and-shape mechanics . 1) to 3) rest respectively on angle space, area-ratiospace, 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 considerablyfrom GR-based conceptions of Background Independence, some of the new supersymmetric shape mechanics arecompared with both. These reveal compatibility between supersymmetry and GR-based conceptions of BackgroundIndependence, at least within these simpler model arenas. a r X i v : . [ g r- q c ] A p r Introduction
Newtonian Mechanics, and the Newtonian paradigm of Physics more generally, are based on absolute space and time.The immovable external character of absolute space and time led to these being opposed by relationalists , most notablyby Leibniz [1] and Mach [2]. On the other hand, the Newtonian paradigm of Physics sufficed to explain humankind’sobservations of nature until the end of the 19th century.Indeed, a satisfactory relational alternative to the foundations of Mechanics was not found until 1982 by Barbourand Bertotti [3]. This is a theory in which Euclidean transformations are held to be physically irrelevant. The next RPMwas not formulated until 2003; it is Barbour’s ‘mechanics of pure shape’ [4]; in this case, it is similarity transformationswhich are held to be physically irrelevant. It is useful to term this a Shape Mechanics, and the preceding a Scale-and-Shape Mechanics. I then considered what the configuration spaces are for these two theories. It turns out that thisShape Mechanics’ configuration spaces (‘shape spaces’) are simpler, with the corresponding Shape-and-Scale Mechanics’configuration spaces then being the cones over these [5]. E.g. the shape spaces for N particles in 1- d are { N − } -spheres S N − , and those in 2- d are complex projective spaces CP N − ; furthermore for 3 particles in 2- d , CP = S : the shapesphere of all possible triangular shapes. Furthermore, these shape spaces turned out to have already arisen in Kendall’sstudies of the Geometry, Probability and Statistics of shapes ([6, 7, 8]). This is a very useful interdisciplinary connectionwhich I pointed out in the Theoretical Physics literature in [9]. I further studied these two RPMs in [10], eventuallysummarizing the key properties of their configuration spaces within my review [11] on configuration spaces in TheoreticalPhysics more generally.RPMs have a number of foundationally valuable applications in Theoretical Physics [10], including the following.1) RPMs have a number of features in common with GR as viewed as a dynamical system [12, 13, 14, 10]. In particular,they have an energy constraint analogous to the GR Hamiltonian constraint; both are quadratic in the momenta. Theyalso have constraints linear in the momenta (and first-class) which are analogues of the GR momentum constraint.2) RPMs are useful in analyzing which aspects of Background Independence GR possesses. Difficulties with thesethen become facets of the notorious Problem of Time in Quantum Gravity [18, 19, 20, 21, 10, 22, 23, 24, 25]. Eachconstraint that is quadratic in the momenta can be taken to arise from the corresponding theory’s reparametrizationinvariance, which implements primary-level timelessness for whole-universe models. Each constraint that is linear inthe momenta (and first-class) can be taken to arise from a group g of physically irrelevant transformations acting onthe configuration space q of the theory. In the case of GR, g = Dif f ( Σ ) : the spatial diffeomorphisms on the 3-spaceof fixed topology Σ , whereas q = Riem ( Σ ) : the spatial 3-metrics on Σ . The corresponding quotient space of these isWheeler’s Superspace ( Σ ) = Riem ( Σ ) /Dif f ( Σ ) [15, 16]; RPMs then offer model arenas of other such quotient spacesas well: shape spaces or relational spaces alias scale-and-shape spaces. Some of these are then more closely analogousto further GR configuration spaces such as conformal superspace [17]; see Appendix C for more detailed comparisonbetween RPM and GR configuration spaces.3) I also applied [26] Kendall’s own case of Shape Statistics to Timeless Records Theory [19]. This is one of the variousapproaches to the Problem of Time; in this case, one sees how far one can get by addressing timeless propositions. Inparticular, my application makes it concrete that Shape Statistics provides the machinery requisite for rendering theclassical version of Timeless Records Theory mathematically sharp.4) RPMs are useful for modelling closed universe and quantum cosmological effects [15, 10].5) RPMs are also useful models [38, 10] in the study of geometrical quantization [28], and affine quantum geometrody-namics [29, 30].The current paper represents a major extension of the scope of RPMs: from one theory three decades ago [3] anda second theory one decade ago [4] to now presenting a large number of RPM theories, and moreover derived in asystematic manner. This is based on knowing the standard set of geometries on flat space, standard group theory of thecorresponding transformation groups and the interplay between the two [31]. The expansion in scope then rests uponthe similarity group Sim ( d ) being a subgroup of both the affine group Af f ( d ) and the conformal group Conf ( d ) . Theseare in fact two mutually incompatible extensions, so in any given model one has to make a choice between these ‘apexgroups’.Each of these ‘apex groups’ has numerous subgroups, including some shared between them. While this is elsewherewell-known mathematics, it was not brought into consideration by Barbour, nor by any of the other authors working inthis field. However, upon making various reconceptualizations as laid out in Sec 2, one can readily tap into this materialto produce more general notions of shape, corresponding notions of shape space and corresponding RPM theories.Sec 2 covers this expansion at the level of groups, invariants and the correseponding configurations. In Sec 3, Ifollow this up by consider the corresponding configuration spaces’ geometry. In particular, in Sec 2 I provide the minimal relationally nontrivial unit . This is concurrently the smallest relationally nontrivial i) whole-universe model, ii)dynamical subsystem and iii) Shape Statistics sampling unit. The archetype of minimal relationally nontrivial unit isthe relational triangle; e.g. Barbour’s seminars have often involved demonstrations involving shuffling wooden triangles.Furthermore, the corresponding crucial technical tools are based on knowledge of the topology and geometry of the1orresponding configuration space of relational triangles. This shape space is the shape sphere , or a portion thereof,depending on the exact modelling assumptions [11]. E.g. Kendall’s spherical blackboard (Fig 6.a), which is well knownin the Shape Geometry and Shape Statistics literature [6, 7]. This consists of 1/3 of a hemisphere corresponding toindistinguishable particles with mirror image triangles identified. On the other hand, a whole hemisphere is required ifthe former assumption is dropped, or a whole sphere if both are dropped; this furthermore becomes Montgomery’s ‘ pairof pants ’ [32] (in the Celestial Mechanics literature) if both assumptions are dropped and double collisions are excised.In Sec 3, I then introduce, name and consider the configuration spaces of minimal relationally nontrivial units for a largerange of further Shape and Scale-and-Shape Theories. These are also very much expected to be a technically importantnucleus for the corresponding theories of RPM and of (Scale-and-)Shape Statistics, for which I also provide matchingnames. Due to this, the summary tables in Figs 2 and 3 are of substantial importance in all of Shape Theory, ShapeStatistics and RPMs alongside its applications to understanding the foundations of GR-like theories, of BackgroundIndependence and of modelling whole(universe quantum cosmological features. This justifies presenting a number offrontiers for subsequent research directions.Sec 4 outlines known examples of topological and geometrical structure for configuration spaces. This illustratessome of the detail that one can eventually expect in the study of the new shape spaces and scale-and-shape spacesintroduced in the current paper. Sec 5 considers the issue of configuration comparers for (Scale-and-)Shape Theories.This includes three way comparison between Barbour’s approach, Kendall’s and DeWitt’s – the last of these havingbeen foundational in the study of GR itself as a dynamical system.This paper’s main worked-out application involves new theories of RPM corresponding to further notions of shape,or of scale-and-shape. The new such theories I present in Sec 6 are as follows.1) Conformal Shape Mechanics : a theory of angles alone which most readily generalizes [4] from a structural perspectivedue to the continued availability of the Euclidean norm.2)
Area Mechanics in 2- d – tied to equiareal geometry [33]. I show that Area Mechanics requires its kinetic term be builtwithout evoking the Euclidean norm, which ceases to be a licit structure in this context.3) Area Ratio alias
Affine Shape Mechanics (then Area Mechanics’ alias as
Affine Scale-and-Shape Mechanics becomesclear).For each of 2) and 3), I also provide a d -dimensional generalization of the underlying shape theory. I also then show inSec 7 that Barbour’s Best Matching comparer used in building RPMs continues to thrive in the complex plane C . I firstuse this to reformulate [4] (now renamed as metric Shape Mechanics, given all the other theories of Shape Mechanicsprovided in the current paper!) I then indicate how the indirect formulation of Mechanics runs into difficulty for theShape Theory in which the Möbius group is taken to be physically irrelevant.Sec 8 then outlines new frontiers of research in Shape Mechanics, with Sec 9 following this up by outlining corre-sponding new frontiers of research in Shape Statistics. This is due to the current paper’s range of notions of shapeexceeding that considered by Kendall, while remaining amenable to parallel shape space geometry based investigations.I.e. I generalize Kendall’s own illustrative ‘standing stones’ problem for metric Shape Statistics to affine, conformal andMöbius Shape Statistics and other projective Shape Statistics besides.Kendall and collaborators’ topological, geometrical, probabilistic and statistical work [6, 47, 8, 7], indicates theconsiderable value of the interdisciplinarity connection [9, 10, 26] between this and Metric Shape Mechanics. It isfurthermore to be expected that such interdisciplinary connections will extend to the range of other shape and scale-and-shape theories considered in this paper. Moreover, this Shape Geometry, Probability and Statistics took Kendall’sgroup 20 years to develop for one notion of shape; thus it should in no way be expected for the current publication towork all of this out. The current paper already multiplies by a sizeable factor the number of known Relational Mechanicstheories. The list of Shape Statistics frontiers and other interdisciplinary applications provided is then to be regardedas a source of future research papers, Such interdisciplinarities involving metric (scale-and-)shapes have indeed alreadybeen established. The current paper then points out that these have conceptual-level analogues for other theories of(scale-and-)shape as outlined in the Frontier sections. As well as the above-mentioned interdisciplinarities, further suchwhich are established for shape theories include withi) Robotics [34] ((for the reasons given in Secs 3 and 7.1).ii) Image Analysis [35] (e.g. shapes photographed from whichever direction).iii) Biology [36] (e.g. animal morphology).iv) Astronomy (e.g. galaxies, CMB patterning, microlensing).Additionally, I do not stop at affine and conformal type theories of Classical Mechanics. In Sec 10, I give furthermorethe first ever treatment of supersymmetric RPM, alongside frontier questions concerning supershapes. This is in thecontext of enlarging
Eucl ( d ) and Sim ( d ) firstly to superEucl ( d ) and superSim ( d ) and then onward to two competingsuper-apex groups: the superconformal and superaffine groups, alongside many other subgroups of these. Some contextand motivation for this development is as follows. 2) Whether Relationalism and Background Independence more generally have a similar characterization in Supergravity.The answer is no ([22, 82] and Appendix D). Due to this, it was a substantial oversight to consider only GR in [18, 37]on the assumption that other theories of gravity would be similar in this regard.2) Whether Relationalism is compatible with Supersymmetry [22, 82]. The current paper shows that the answer to thisis yes.I end in Sec 11 by providing the Dirac quantizations for a representative four among the current paper’s new RPMs. Thisparallels how e.g. Rovelli [38] and Smolin [39] quantized [3] around a decade after its inception (whereas I subsequentlyconsidered both Dirac and reduced quantizations of both [3] and [4]).Appendix A supports the text by outlining the group-based approach to the foundations of flat theories of geometry.Appendix B outlines those parts of the theory of Lie algebras and Lie groups that are used in this paper, includingsupersymmetric counterparts. Appendix C compares the foundational variety of flat space notions of geometry withthat of notions of differential geometry, Appendix C also gives an outline of GR’s Dirac algebroid of constraints. Thisis provided firstly because RPMs’ constraint algebras share a number of features with it. Secondly, it is providedfor for comparison with Appendix D’s significantly distinct supergravity algebroid of constraints. This contrast isof foundational interest as regards canonical quantization, Quantum Gravity and differences in the form in whichBackground Independence can be manifested. Finally, the latter is also contrasted with supersymmetric RPMs’ ownconstraint algebras. First consider configuration space q [40]: the space of generalized configurations Q A for a physical system. The rest ofthis Sec considers finite flat-space Mechanics examples of configurations and configuration spaces. N.B. that these infact promptly come to involve mass-weighted configurations.This paper considers the case of point configurations. Here q = q ( N, d ) = × NI =1 a ( d ) , for a ( d ) the absolute space.For these, the relevant principles of Configurational Relationalism is that Physics involves not only a q but also a g oftransformations acting upon q that are taken to be physically redundant. Then two a priori distinct conceptualizationsof Configurational Relationalism are then possible for point configurations.a) g acts on absolute space a ( d ) (usually R d ).b) g acts on configuration space q ( N, d ) , i.e. acting rather on a material entity of at least some physical content.However, in the case of q = q ( N, d ) = × NI =1 a ( d ) , the group action takes, particle by particle, the form of a groupaction on inividual particles in a ( d ) . Due to this, each kind of geometry that can be considered for a ( d ) corresponds toa realtionalism imposed on the whole of q ( N, d ) .Thus b) can be resolved by resolving a), which amounts to addressing the groups acting on R d has been well-addressed,e.g. in Klein’s Erlangen approach to geometry. In this way, Appendix A’s well-known mathematics can be straightfor-wardly commandeered to settle b) also.Some limitations on the choice of g , q pairs are as follows.A) Nontriviality . g cannot be too large, by a degrees of freedom counting criterion. Then using q := dim ( q ) and l := dim( g ), a theory on q / g is inconsistent if l > q , trivial if l = q and relationally trivial if l = q − . The last ofthese is because relational nontriviality requires for one degree of freedom to be expressed in terms of another. This isas opposed to it being meaningfully expressible in terms of some external or elsewise unphysical ‘time parameter’.B) Further structural compatibility is required. A simple example of this is that in considering d -dimensional particleconfigurations, g is to involve the same d (or less, but certainly not more).C) A more general structural compatibility criterion is that g is to have a group action. A group action’s credibilitymay further be enhanced though its being ‘natural’, and some further mathematical advantages are conferred from it This is trivial in the sense that there are no independent degrees of freedom in the principal stratum of the orbit space. Moreover, thegroup action can be such that non-principal strata retain nontrivial dynamical content. Such non-principal strata occur e.g. in the standardaction of SO (3) on R and on the configuration space of × Ni =1 R of N -particle configurations. This is one way in which the counting argumentis ‘local’ rather than‘global’. Another is how 2 + 1 GR manages to be trivial as regards local degrees of freedom but none the less is capableof possessing global degrees of freedom. A group action α on a set X is a map α : g × X → X such that i) { g ◦ g } x = g ◦{ g x } (compatibility) and ii) ex = x ∀ x ∈ X (identity).This terminology continues to apply if X carries further structures. Examples include left action gx , right action xg , and conjugate action gxg − . By a natural group action, I just mean one which does not require a choice as to how to relate X and g due to there being one‘obvious way’ in which it acts. E.g. Perm( X ) acts naturally on X , or an n × n matrix group acts naturally on the corresponding n -vectors.An action is faithful if g (cid:54) = g ⇒ g x (cid:54) = g x for some x ∈ X , whereas it is free if this is so for all x ∈ X . A map is proper if the inverse mapof each compact set is itself compact; a particular subcase of this used below is proper group action . g for a given q so as to eliminate all trace of any extraneous background entities. The automorphism group Aut ( a ) of absolute space a is then an obvious possibility for g . However some subgroup[41] of Aut ( a ) might also be desirable, not least because the inclusion of some such automorphisms depends on whichlevel of mathematical structure σ is to be taken to be physical. I.e. g ≤ Aut ( (cid:104) a , σ (cid:105) ) is a more general possibility. Suchsubgroups also comply with A) and stand a good chance of suitably satisfying criteria B) and C). See Sec 5 for examples. Suppose we are in possession of a q , g candidate pair. Then seek to represent the generators of g in terms of Q A , ∂∂Q A which manifestly acts on q , We can then check whether some candidate objects O ( Q A ) are g -invariants by explicitlychecking that these are indeed preserved by g ’s generators. For particles configurations, invariants are plentiful andintuitively clear for a wide range of g , as a direct follow-on of the forms taken by the corresponding g -invariantsinvariants in R d . E.g. for g = Eucl ( d ) , separations between particles and angles between particles are of this nature;Fig 4 tabulates such for further g . g -act g -all’ method A given set of objects can also be interesting in a wider range of ways than being g -invariants. Moreover, in some cases,invariants are unknown or nonexistent. There are more general concepts of ‘good g objects, such as g -tensors (of which g -invariants are but one example: g -scalars) or objects which are one of the preceding modulo a linear function of thegenerators. Here one is representing auxiliaries in terms of q E and tangent bundle auxiliary quantities d g G .Configurational Relationalism’s broad strategy is the ‘ g - act g - all method [42]. Consider here consider some object O belonging to some space of objects o . The O may be composites of some kind of more basic variables b ( Q A alone in thisSec’s q -only setting, though further Secs such as Sec 5 extend this). Such composites indeed cover far more than just s : also e.g. notions of distance, information, correlation, and also quantum operators and quantum versions of all ofthe preceding. In whichever case, start by applying g -act; this can initially be conceived of as o g × −→ g × o , O (cid:55)→ → g g O .End by applying g -all: some operation S g ∈ g is applied, which makes use of all of the g G ∈ g . This has the effect ofcancelling out g -act’s use of g G , so overall a g -invariant version of each O is produced, which I denote by O g − inv := S g ∈ g ◦ → g g O . (1)Examples of S g ∈ g include summing, integrating, averaging (group averaging is an important basic technique in Groupand Representation Theory), taking infs or sups, and extremizing, in each case indeed meaning over g . For the first twoexamples, S g ∈ g include (cid:88) g ∈ g , (cid:90) g ∈ g D g . (2)Barbour’s Best Matching’s own S g ∈ g is extremization over g (see Secs 4–5).Finally, ‘Maps’ can additionally be inserted between g -act and g -all to produce an even more general O g − inv := S g ∈ g ◦ Maps ◦ → g O . (3)‘Maps’ covers a very general assortment of maps, though these are to all be g -invariant; if not, g would act on a newtype of object O (cid:48) = Maps ◦ O . For the most commonly considered case of a = R d , Appendices A and B provide many suitable g ’s; see also Fig 2.What physical considerations enter these choices? A case of note is whether scale is to be physically meaningless.Moreover, in directly modelling nature, disregarding scale jeopardizes standard cosmological theory without providinga viable replacement [44]. Additionally, retaining scale may enable time provision [45]. On the other hand, the metricShape Theory is both mathematically simpler and recurs as a subproblem within the metric Scale-and-Shape theory.Moreover, it is quite commonplace to consider physical theories with scale that possess a distinct scale-invariant phase inan ‘unbroken’ higher-energy regime, by which not matching everyday experience is not necessarily the end to a theory’srelevance. 4igure 1: Coordinate systems for 3 particles (underline denotes spatial vectors and bold font denotes configuration space quantities). a)and b) Absolute particle position coordinates ( q , q , q ) in 1- and 2- d . These are reckoned with respect to fixed axes A and a fixed originO. c) and d) Relative inter-particle (Lagrange) coordinates { r IJ , I > J } . Their relation to the q I are obvious: r IJ := q J − q I . In the caseof 3 particles, any 2 of these form a basis; I use upper-case Latin indices A, B, C for a basis of relative separation labels 1 to n . No fixedorigin enters their definition, but they are in no way freed from fixed coordinate axes A. e) and f) It is more convenient to work with relativeJacobi coordinates because these diagonalize the ‘mass matrix’ or kinetic metric that the moment of inertia, kinetic term and kinetic arcelement are built from. Relative Jacobi coordinates attain this through in general being relative separations of particle clusters . × denotesthe centre of mass of particles 2 and 3. Convenience furthermore dictates that I take the mass-weighted relative Jacobi coordinates ρ A . Thenthe kinetic metric is just an identity array with components δ ij δ AB . Jacobi coordinates are indeed widely used in Celestial Mechanics [51]and Molecular Physics [52]. One possibility which has hitherto not been mentioned in setting up RPMs is that preservation of inner products · isnot the only possibility: an alternative to this is preservation of × products (or in dimension-independent language offorms, of exterior products ∧ ). This corresponds to whether infinitesimal Procrustean stretches ( d -volume top formpreserving in dimension d ) and shears are to be physically meaningless. These distort relative angles and of ratiosof relative separations respectively. They combine with translations and rotations to form the ‘equi-top-form group’ Equi ( d ) , or furthermore with the dilations to form the affine group Af f ( d ) . This represents one way of extending Sim ( d ) through its being a subgroup within a larger group Each of these groups corresponds to a further known typeof geometry as per Appendix B.1.Further issues involve whether the configurations are to be mirror image identified, and whether the particles are to bedistinguishable. Both of these issues translate to the form taken by the configuration space topology [10]. This involvesusing not necessarily q but q = (cid:80) NI =1 a / g (cid:48) more generally, in particular for discrete group g (cid:48) = Z , Z N , Z × Z N though partial indistinguishability is also possible.A distinct further possibility which has not yet been mentioned in setting up RPMs stems from whether to allow the‘inversion in the sphere’ transformation (73), which also preserves angles. If so, special conformal transformations exist,providing a distinct extension of Sim ( d ) . Note furthermore that the affine and conformal extensions are incompatiblewith each other as per (91), so these two extensions cannot furthermore be composed. They correspond to two different‘apex groups’ within each of which the Sim ( d ) hitherto used in RPMs and Shape Statistics sits as a subgroup.This is a good point at which to note that 1- d is too simple to support distinctions between a number of types ofgeometry. As well as having no continuous rotations (and its only discrete rotation coinciding with inversion), 1- d hasno nontrivial volume forms and so is bereft of an affine extension. Thus rotations and the further possibility of extensionto affine transformations require dimension ≥ . On the other hand, 2- d has an infinite- d conformal group. In fact 1- d does too, though that one is less interesting through coinciding with the reparametrizations. See Appendix B for anoutline of both of these workings). These invalidate 1 and 2- d configurations of a finite number of particles from havinga nontrivial relational theory by the counting argument A) of Sec 2. Thus dimension ≥ is required to investigate thispossibility, though e.g. dimension 2 also provides finite subgroups of the conformal group within which Sim (2) sits as asubgroup. E.g. the Möbius group considered in Sec 5.6; this group in turn corresponds to a type of projective geometry. This is also a good point at which to discuss the Relationalism of the above two extensions. The conformal extension’sinvolvement of inversion amounts to replacing a ( d ) = R d by R d ∪ ∞ ; this is a new consideration in the context of RPMsand Background Independence more generally. On the one hand, this is adding an extra structure which might beinterpreted as absolute: Riemann’s notion of the ‘point at infinity’. On the other hand, appending this one point allowsfor a rather more general class of angle-preserving transformations to be well-defined. The affine extension, on theother hand, remains within the usual a ( d ) = R d . It amounts to modelling situations in which configurations have no This does not mean that relationalists necessarily discard structures, but rather that they are prepared to consider the outcome ofentertaining more minimalist ontologies. A homomorphism is a map µ : s → s that is structure-preserving. In particular, if a such is invertible (equivalently bijective) it is an isomorphism , if s = s it is an endomorphism , and, if both apply, it is an automorphism . I use a Gothic font for spaces so as to not confuse configurations with the configuration spaces they belong to. Also Z a is here the cyclicgroup of order a . Further real projective variants additionally exhibit point-to-line duality, which further unusual property heralds departure from Relational
Particle
Mechanics. Moreover, the C case of projective geometry additionally does not discern between lines and circles. Invariant corresponding to each group, dimension of the group and the minimal relationally nontrivial unit in spatial dimensions1, 2 and 3. O is an absolute origin; the axis and ruler logos denote absolute orientation and absolute scale respectively. In particular, notethat the bottom six rows are interesting and new. In this way, the current paper covers enough new material, with a broad enough range ofnew applications for further researchers to work on, to justify introducing this useful shorthand notation for the invariants/observables, andthe namings of all the corresponding configurations and of theories of Mechanics, and of Statistics in this and the next Figure. overall meaning of either relative angle (by equivalence under global shears), or of relative ratio (by equivalence underglobal Procrustean stretches). Considering these in the context of RPMs is also new to the current paper. See Sec 5 fordiscussion of issues of direct realization, and of applications for which that is not a consideration.In the by now well studied relational space and (metric) shape space cases, in both 2- and 3- d the minimal relationallynontrivial unit is the triangle. Barbour’s well-known demonstrations of Best Matching with wooden triangles, andKendall’s method of sampling in threes leading to his spherical blackboard methodology – Fig 6.a – follow. Theminimal relationally nontrivial unit is all of that type of Relationalism’s smallest whole universe relational model, smallestrelationally nontrivial subsystem, and smallest relationally nontrivial sampling unit for Shape Statistics.
These are waysin which minimal relationally nontrivial units are important. Thus in the last three columns of Fig 2, I depict andexplain the form these take, for each of this paper’s suite of subgroups of the affine group, the full conformal group in3- d and the Möbius group in 2- d , In perusing this Figure, it may be useful to bear in mind how the simplest affine andprojective geometry theorems also require use of more points than the simplest Euclidean geometry ones [33].More generally, yet further models of absolute space a might be considered, such as S d or T d . E.g. for GR, both S and T have been substantially studied as compact models for space; T d is of course also a compactification of R d ; Cosmology also makes use of hyperbolic space H , which admits compactifications of its own. RPMs based onsuch are then closer to GR than RPMs based on flat space. RPMs based on S d in place of R d in the role of absolutespace have started to be considered elsewhere [10, 46]. Not that S admits the additional interpretation and realizationas an observed space model: the sky, due to which Shape Theory and Shape Statistics for it had already previouslyappeared in the literature [47, 7]. The current paper does not further explore this paragraph’s additional possibilities,nor have RPM on T d , H d or compactifications thereof started to be investigated to date. None the less, it is clear6hat the current paper’s systematics concerning invariants, groups, families of subgroups, minimal relationally nontrivialunits, configuration spaces, configuration comparers, and constructions of Mechanics and Statistics upon (generalized)shape spaces, furthermore carries over to all these other cases as well. In particular, the current paper considers thedifferent levels of geometry on open infinite flat space, but if one passes to whichever curved space instead, one canagain contemplate a comparable range of geometrical structures thereupon.] In setting up a large number of new theories of Shape Mechanics, with underlying Shape Theories and configurationspaces, many of which are new too, if is rather necessary to create names and notations for the configuration spacesin question (Figure 3). These are conceptually important entities, whose precise mathematical nature shall one day beknown in detail, much as [7, 10] lay this out in the case of redundant similarity group.Figure 3:
For each group–invariant pair, we give name and notation for the corresponding configuration space, and the names for thecorresponding Mechanics and Statistics. Note the two alternative namings: by group and by the theory’s invariant objects. Note manyconfiguration space geometry notions reduce to others, and that ones of known geometry are indicated; see [7, 10, 11] for more about these.N.B. that the last 6 rows are both new and of particular significance.
Relative space r ( N, d ) = q ( N, d ) /T r ( d ) = R nd for n := N − . Bases of relative inter-particle separation vectors– Lagrange coordinates – and of cluster separation vectors – relative Jacobi coordinates – are then natural thereupon(Fig 1.c)–f). In an absolute worldview, these correspond to passing to centre of mass frame, whereas in a relationalworldview they correspond to absolute absolute origin being meaningless. Useful Lemma (Jacobi pairs) . Within the subgroups of the affine group, the number of relational configurationspaces requiring independent study is halved, since each version with translations is the same as the version withoutwith one particle more.Proof. For these groups taking out the centre of mass is always equally trivial. Moreover, the diagonal form in Jacobi’srelative ρ A is identical in every respect with that of the mass-weighted point particles bar there being one ρ A less. (cid:50) q and ρ are rendered appropriateby the assumption of material point particles [displayed in column 1 of Figs 2–3 and further laid out in Fig 5]. These arewhat then firstly serve as potential functional dependence, and subsequently end up being wavefunction dependencies.Figs 3 and 4 then name the corresponding configuration spaces.If absolute axes are also to have no meaning, the remaining configuration space is relational space R ( N, d ) := q ( N, d ) /Eucl ( d ) , (4)of dimension nd – d { d − } / = d{2 n + 1 – d }/2: in particular, N − in 2- d , N − in 2- d and N − in 3- d . If, instead,absolute scale is also to have no meaning, the configuration space is Kendall’s preshape space [7] p ( N, d ) := r ( N, d ) /Dil ,of dimension nd − . If both absolute axes and absolute scale are to have no meaning, then the configuration space isKendall’s [7] shape space s ( N, d ) := q ( N, d ) / Sim ( d ) . (5)This is of dimension N d − { d { d + 1 } / } = d { n + 1 − d } / − ; in particular N − in 1- d , N − in 2- d and N − in3- d ; as well as featuring in accounts of RPMs [9, 10], it is well-known from the Shape Geometry and the Shape Statisticsliteratures [6, 47, 7]. Also note that p ( N,
1) = s ( N, , since there are no rotations in 1- d . The above quotient spacesare taken to be not just sets but also normed spaces, metric spaces, topological spaces, and, where possible, Riemanniangeometries. Their analogy with GR’s configuration spaces is laid out in Fig 16.Note that the Jacobi pairs simplification does not apply within those further groups that include the special conformaltransformations K i . This is because of the commutation relation (88), by which translations cease to be so triviallyremovable. Also contrast the conformal case’s pure angle information with the similarity case’s mixture of angle andratio information.See Fig 4 for an outline of further subgroups of Conf( d ), alongside indication of other combinations of generatorswhich also fail to close as groups for the reasons stated.The configuration space level can have a metric geometry (or in reduced cases generally a stratified such), even if theoriginal configurations sit in a geometry with less structure than that. This is entirely possible because the map fromspace to the space of spaces need not be category-preserving.Finally, while two area ratios have the range of a quadrant or a whole plane if signed, it is not a priori likely forthese to carry a flat metric.Figure 4: Layout of g -invariant contents, in the pattern following on from Fig 14’s layout of which combinations of generators aregroup-theoretically allowed. Note that four of the subgroups of Conf ( d ) have the same invariants; this is due to incorporating the specialconformal transformation being rather restrictive. Considering larger units than the minimal one is valuable not only since furtherly relational theories need some such, butalso because some applications need more system complexity. It is insightful here to point out that Montgomery’s fallingcat and the relational triangle are fully minimal robotic models. E.g. envisage the space of triangles not as a bunchof rigid wooden shapes but as the shapes that can be formed by a flexible, extendible entity. This perspective involves paths in configuration space . Models along the above lines rather quickly acquire complexity. Indeed [49, 50] consideredthe K-shaped clustering of three relative Jacobi coordinate vectors presentation of quadrilaterals as axe configurations. Bold font here denotes configuration space objects, with indices iI and iA respectively. This involves 2-rod configurations, corresponding to configuration space RP . Whereas Fig 3 mentions the most commonly considered quotients, here are a number of further possibilities. These are once againdisplayed in parallel with Fig 14’s layout of which combinations of generators are group-theoretically allowed. Useful simplifying relations bywhich some of these configuration spaces are mathematically very similar to others are also indicated, such as the Jacobi parings and whicharise from quotienting out subgroups of whichever of the affine and conformal groups.
The underlying shape space in this case is CP : Fig 6.b), and the relational space is C ( CP ) . I now point out thataxes are already complex enough tools to have very different applications according to angles and proportions. Thus a‘robotic axe’ reinterpretation of the model shape space of axes is already a model of a significantly adaptable robotictool. Finally robotic models may also eventually expected to enter foundational Theoretical Physics, along the lines ofHartle’s IGUS [53]. I.e the Information Gathering and Utilizing System model concept could well receive a classical andthen quantum robotic implementation.
Various such can be built from q ’s kinetic metric’s M inner product and norm [6, 7, 3, 54, 10](Kendall Dist) = ( Q , Q ) M , (6)(Barbour Dist) = || d Q || M , (7)(DeWitt Dist) = ( d Q , d Q (cid:48) ) M . (8)Next, if there is additionally a physically irrelevant g acting upon q ,(Kendall g -Dist) = ( Q · −→ g g Q (cid:48) ) M , (9)(Barbour g -Dist) = || d g Q || M and (10)(DeWitt g -Dist) = ( −→ g d g Q , −→ g d g Q (cid:48) ) M . (11)Then g -all moves – such as integral, sum, average, inf, sup or extremum – can be applied, after insertion of Maps ifnecessary. This is the first publication to consider this three-way comparison, DeWitt’s own approach being well-knownfrom the foundations of GR as a dynamical system [54].Then e.g. (10) subjected to the ×√ W and integration maps before a g -all extremum move gives Best Matching;this can furthermore now be recognized as a subcase of a weighted path metric. (9) itself differs from the other twocases in using a finite group action to the other two cases’ infinitesimal ones. In another sense, it is (9) and (11) whichare akin: compare two distinct inputs versus (10) working around a single input. ‘Comparers’ then have a further issue:if M = M ( Q ) , does one use Q or Q in evaluating M itself? This situation does not arise in the R n shapes contextof Kendall, but it does in DeWitt’s GR context; he resolved it in the symmetric manner, i.e. using Q and Q to equalextents. This exists independently of whether it is contracted into velocities or changes; e.g. moment of inertia is this metric contracted intomechanical configurations themselves. It only provides a norm if it is positive-definite. a) Unlabelled triangles’ spherical blackboard (or half of it; triangleland is either S or some regular portion thereof, dependingon exactly how the configurations are being modelled [11]). Here ‘regular’ means that the base and median partial moments of inertia areequal, D is a double collision and M is a merger (third particle at the centre of mass of the other two). If the triangles are labelled and withmirror images distinct, the whole sphere is realized. b) Complex projective chopping board of axe configuration. This figure concentrates onthe two ratio coordinates β and χ , suppressing the further variety in relative angle coordinates φ and ψ ( β, χ, φ ψ are Gibbons–Pope typecoordinates, see e.g. [49]). Figure 7:
Furthermore, how good the ‘best fit’ is can be assessed in substantially geometrically general cases by making set ofrelational objects out of primed and unprimed vertices (Fig 8.a), to which the corresponding notion of Shape Statisticsis to be applied. In the present case, these are triangles that one can test against the (cid:15) -bluntness criterion.As a first new example, consider affine space. Its d -volume top form construct is not amenable to a notion of distanceout of being antisymmetric and thus not obeying symmetry, separation, or, if signed, positivity. Affine space can howeverbe equipped as a metric space, giving a Dist ( q , G q ) style finite comparer [35] for G ∈ GL ( d, R ) . For instance this is auseful tool in image recognition, which is affine to leading order [35]. This corresponds to how images can be stretchedand sheared as well as enlarged, translated and rotated, depending on the exact modelling assumptions made. Oneapplication of this is to nearby objects, for which many viewpoints can be attained ‘by walking to the other side’ of theobject. One interesting proposition then is whether two pictures feature the same object viewed from different angles.A second application is that the former proposition also arises within hypothetical smaller closed-topology universes[55], and within the setting of multiple images from gravitational lensing [56]. However, the first of these has additionalevolution effects (the multiple images would generally correspond to the object’s configuration in different aeons). Thesecond has additional distortion effects (the geometry around a gravitational lens is not flat, taking one outside thescope of the current paper). A third application concerns studying populations of anisotropic astrophysical objects(most obviously galaxies). Then does one’s collection of images feature a single such viewed from multiple angles, or is ita mixture of objects from distinct populations? Furthermore, are we seeing these objects at entirely random orientations,or do they exhibit a statistically significant pattern of orientations?As further new examples, see the next two Secs for many new instances of Best Matching. This application is to whole-universe models, for which Temporal Relationalism applies. I.e. there is no time at theprimary level for the universe as a whole. This is best implemented by a geometrical action, which is mathematically10igure 8: a) Assessing matched triangles. b) Assessing matched fish (2- d image version), corresponding to Thompson’s observation that thereare two species of fish whose shapes are approximately related by a scale and shear transformation [36]. Since matching has already occurred,the images being compared can be taken to involve equal-area polygons, thus involving equiareal, rather than full affine, mathematics. Ineach case, consider the minimum relationally nontrivial figures formed between the two matched figures. E.g. are these a) significantly blunt,b) significantly lacking in area? dual to an action making no use of parametrization, which upon introducing a parametrization is then invariant underreparametrization. At least one primary constraint must then follow from any reparametrization-invariant action due toa well-known argument of Dirac’s [59]. In the case of Mechanics, this gives an equation which is usually interpreted asan energy conservation equation. However in the present context this is to be reinterpreted as an equation of time [57].Indeed rearranging it gives an expression for emergent Machian time: a concrete realization resolution of primary-leveltimelessness by Mach’s ‘time is to be abstracted from change’. In the case of metric Scale-and-Shape Mechanics, thisemergent time amounts to a relational recovery of a quantity which is more usually regarded as Newtonian time. Notethat more generally handling Mechanics models in a temporally relational manner requires a modified version of thePrinciples of Dynamics as laid out in [58]. The particular cases of geometrical actions in the next two sections are all Jacobi-type action [40], corresponding to a Riemannian notion of geometry.The case of Relationalism most usually considered as an RPM is metric Scale-and-Shape Mechanics [3], as motivatedby the direct approximate physical regime in which Mechanics is used. Note however that Sec 4’s example of the affinetransformations arising in the study of images shows how the ‘direct physical regime’s geometry’ may not be the onlyone of relevance. Finally, if fundamental theory is under consideration, the defects of Newtonian theory itself may wellnot be the only useful guideline. E.g. scale-invariant and conformal models are often considered in High-Energy Physics(c.f. Sec 2.3). Also RPM has been argued to be of substantial value [18, 19, 21, 10, 11] as a model arena for GR’s owndynamical structure, configuration spaces, and of some of GR’s Background Independence aspects and correspondingProblem of Time facets. Viewed in this way, the mathematically simpler 1- and 2- d RPMs – which already exhibitmany of the features of GR that are emulated by RPMs – are often preferable to the 3- d RPMs, many of whose extracomplexities are not aligned with GR’s extra complexities. Moreover, this GR model arena perspective also leaves anumber of other modelling assumptions for RPMs open. See Appendix C in this regard, both for metric shape RPMand for the new RPMs presented below.
Let us begin with the most familiar case of RPM: the Barbour–Bertotti 1982 theory, albeit reformulated in terms ofrelative Jacobi coordinates. We will then show how the other RPMs arise from various sources of geometrical variety inthis setting. The action for this RPM is [10] S [ ρ A , b ] = √ (cid:90) d s √ W , d s = || d b ρ || , d b ρ A := d ρ A − d b × ρ A , (12)for W := E − V for E the total energy and V the potential, which in this case is of the form V ( − · − ) . Also the index A runs from to n := N − : the number of independent inter-particle cluster vectors, where N is the number of particlesitself.The quadratic constraint E := || π || / V ( − · − ) = E (13)then follows as a primary constraint. This is often interpreted as an energy constraint in the nonrelational and subsystemcontexts, but is to be interpreted as an equation of time in the relational whole-universe context. Also, the(zero total angular momentum of the universe) , L := (cid:88) nA =1 ρ A × π A = 0 (14) A previous such theory existed [48] but was built in another manner. This previous theory fails to fit mass anisotropy bounds, and ceasesto look natural once cast in reduced variables. [3] also remains simple all the way down its reduction process [10], whereas [48] does not.Instead there is a distinct [48] type theory [10] which is simple in each stage of the reduction process’ variables. These not matching up givesa further theoretical reason to favour [3], and theories built similarly to it for other g , rather than [48] type theories. In this paper, I use the calligraphic font to denote constraints. b ; thus it results from implementing ConfigurationalRelationalism.[The original version [3] has the ρ A to q I of the above, with an extra translational best matching correction − d a resulting in an extra ( zero total momentum constraint ) P := (cid:88) NI =1 p I = 0 , (15)for p I the momentum conjugate to q I .] In this case – the theory originally due to Barbour 2003 [4] and somewhat reformulated as per [9, 10] – S [ ρ A , b, c ] = √ (cid:90) d s √ W , d s = || d b,c ρ || /ρ , d b,c ρ A := d ρ A − d b × ρ A − d c ρ A (16)for V = V ( − · − / − ·− ) and ρ := √ I for I the total moment of inertia.This gives E := I || π || / V ( − · − / − ·− ) = E (17)as a primary constraint, (14) again from variation with respect to b , and, from variation with respect to c [4],(zero total dilational momentum of the universe) , D := (cid:88) NI =1 ρ A · π A = 0 . (18) d conformal Shape Mechanics, alias local angle Mechanics In fact, the further special conformal Best Matching can be appended into the q I version of (16) to give ( I runs from to N ) S [ q I , a, b, c, k ] = √ (cid:90) d s √ W , d s = || d a,b,c,k q || / √ I , d a,b,c,k q Ia := d q Ia − d a a − ( d b × q I ) a − d c q Ia −{ q I δ ab − q Ia q Ib } d k b (19)for V now of the form V ( ∠ ) . This is a first RPM theory that is new to this paper. Then variations with respect to a , b and c yield (15), and the q I counterparts of (14) and (18) respectively. But now also variation with respect to k produces the( zero total special conformal momentum constraint ) K a := (cid:88) NI =1 { q I δ ab − q Ia q Ib } p Ib = 0 . (20)The quadratic constraint arising as a primary constraint is now E := I || π || / V ( ∠ ) = E . (21)Then the linear constraints close as per the conformal algebra, and K a manages to commute with E also. This set-upworks similarly for d > ; this just requires a different presentation for the larger amounts of Rot ( d ) .One further motivation for this model is that, while Similarity Mechanics is already to some extent a useful model ofGR’s conformal superspace (Appendix C), Conformal Shape Mechanics is surely a better model of this. d , alias area mechanics As a second theory of RPM new to this paper, suppose that one proceeds by extending Sec 5.2’s construct to includeaffine Best Matching, so S [ ρ A , b, e, f ] = √ (cid:90) d s √ W , d s = || d SL ρ || (22)for d SL ρ A := d ρ A − d s S ρ A the SL (2 , R ) best-matched derivative, for d s the 3-vector [ d f, d e, d b ] of auxiliaries and S := [( ) , ( ) , ( )] T , (23)alongside V = V ( − × − ) in this case. Then E := || π || / V ( − × − ) = E (24)12rises as a primary constraint, and the linear zero total SL (2 , R ) momentum constraint S := (cid:88) nA =1 ρ A S π A (25)arises from variation with respect to s . This includes L again from b -variation alongside new zero total Procrusteanmomentum P r and shear momentum S h constraints. Explicitly, P r := ρ x π x − ρ y π y , S h := ρ x π y + ρ y π x . (26)Then these last two linear constraints fail to Poisson-brackets close with the candidate theory’s E . Thus this constitutesan example of Best Matching failing as an implementation due to the outcome of the Dirac Algorithm [59]. In suchcases, one is to decide which part(s) of the triple q , g , S are to be modified. The present case has a clear solution: the || || structure ceases to have any business in an affine theory! In this way, the id to Conf family of further Best Matchingappendings within the normed form of kinetic line element is not a general procedure. What is more general is my ‘good g objects’ approach (Sec 2.2), by which || || is recognized to be illicit at the outset for a g as redundant as Equi (2) .According to that, take instead d s = (cid:88) nA, B =1 ( d SL ρ A × d SL ρ B ) . (27)Then e.g. focusing on the smallest relationally nontrivial case, (cid:80) cycles A, B =1 ( d SL ρ A × d SL ρ B ) inverts nicely in thechange to momentum sense, giving the primary constraint E := (cid:88) cycles A, B =1 π A × π B / V ( − × − ) = E , (28)alongside the same linear S as before. This mechanical model is indeed consistent. d , alias Area Ratio Mechanics A third theory of RPM new to this paper has S [ ρ A , b, c, e, f ] = √ (cid:90) d s √ W , d s = (cid:88) nA, B =1 d GL ρ A × d GL ρ B / (cid:88) nC, D =1 ρ C × ρ D . (29)Here, d GL ρ A := d ρ A − d g G ρ A the GL (2 , R ) best-matched derivative, for d g := [ d f, d e, d b, d c ] auxiliaries and G := [( ) , ( ) , ( ) , ( )] T . (30)Also V = V ( − × − / − × − ) .This results in a the same linear constraints as in the previous Subsec plus D , the four of which can be packaged asthe zero total GL (2 , R ) momentum constraint G := (cid:88) nA =1 ρ A G π A . (31)In the smallest relationally nontrivial case n = 3 , using once again the sum of cycles combination, the primary constraintis E := (cid:88) cycles A, B =1 ( ρ A × ρ B ) (cid:88) cycles C, D =1 ( π C × π D ) / V ( − × − / − × − ) = E . (32) d relational configuration also admit a C formulation As well as the two components of
T r (2) being an obligatory pairing in this setting (keep both or none),
Rot (2) and
Dil are also an obligatory pairing: as the modulus and phase parts of a single complex number.The simplest notions of relational configuration that admit a C formulation have configuration spaces forming thediamond array C N , C n , CP n , CP n − corresponding to quotienting out by none, one or both of T r (2) and
Rot (2) × Dil
In one sense, complex representability extends to the 2- d affine case: its Procrustean stretch and shear can be co-represented in complex form On the other hand, this is a γ ¯ z representation, i.e. antiholomorphic, and furthermore areais not particularly natural in the complex plane, since it is an Im part rather than a whole complex entity: Im ( z A ¯ z B ) .For sure, Möbius Configurational Relationalism continues to lie firmly within the complex domain.Complex examples of Best Matching include the following. The translation correction is d A = d a x + i d a y and therotation-and-dilation correction is d B z I for d B = d c + i d b . (33)Affine best matching – new to this paper – completes this with the Procrustean-and-shear correctiond C ¯ z I for d C = d f + i d e . (34)13igure 9: Complex suite of a) invariants and b) the corresponding groups g . The Möbius group has further subgroups amenable tocomplex formulation that are not considered here. On the other hand, Möbius best matching – also new to this paper – completes the above with a holomorphic quadraticcorrection: d M ¯ z I for M ∈ C . (35)The new geometrical entity of particular interest are the cross-ratio spaces c ( N, , In particular the minimal relationallynontrivial unit is c (4 , . Whereas each cross ratio is itself a complex number-valued quantity, it is not yet clear whichgeometrical structure is natural to cross-ratio space. The zero total momentum constraint, if present, is just P z = (cid:80) NI =1 p I , for each p I of form p xI + ip yI . On the otherhand, the complex zero total dilational-and-angular momentum constraint is (new to this paper) Q z := (cid:88) NI =1 ¯ z I p I = 0 . (36)A complex action for Sim (2) is then (using complex vector norm, and new to this paper, S = √ (cid:90) || d A,B zz || √ E − V , (37)for V = V ( − / − ) . The corresponding form of the quadratic primary constraint is E := || z || || p || / V ( − / − ) = E . (38)Finally, with
Conf ( d ) of course becoming infinite-dimensional in 2- d , I also briefly consider a finite Lie subgroup inthat case, i.e. the Möbius group as a finite substitute. [An infinite group of physically irrelevant transformations wouldtrivialize any finite particle configuration.] The zero total Möbius momentum constraint – new to this paper – is M z := (cid:88) NI =1 ¯ z I p I = 0 . (39)Moreover, provision of an indirectly formulated action for this is blocked. I.e. in this case, for now we reach a newimpasse rather than a new theory of RPM. This occurs due to the following reason. Whereas one can readily enterchanges into a ratio so as to produce a function that is homogeneous linear in change, in a cross ratio everything whichenters in the numerator also features in the denominator. This forces the change to feature homogeneously with degreezero. Frontier 1. An important next step is to work out reduced versions of the RPM actions. For metric shape and scale-and-shape theories, this was covered in [9, 5, 10]. The GR counterpart of this procedure is also well known, in generalleading to the impasse known as the Thin Sandwich Problem [14, 60]. Moreover, reduction is a means of getting at leastsome candidates for the (generalized) shape space geometries upon which to base the both the corresponding ShapeStatistics and the geometrical reduced quantization scheme.Frontier 2. Distinct direct considerations can on some occasions permit finding the relational configuration space andthen building a Mechanics thereupon. E.g. this occurs for 1- and 2- d metric shape RPM [9, 10]; moreover, in this case14irect consideration and indirect consideration followed by reduction coincide. To what extent does this coincidenceextend among the new theories laid out in this paper?For instance, the direct implementation of Möbius RPM remains open in this case, due to the inherent problem withmaking a homogeneous linear function out of cross-ratios. I.e. that converting any number of z i in [ z , z ; z , z ] intod w i quantities does not succeed in giving a quantity homogeneous linear in the d’s.Frontier 3. Some projective geometries have point-to-line duality. An interesting question for the foundations of Mechan-ics then is if one sets up a ‘point particle Mechanics’ here, can it be reinterpreted as a dual ‘line Mechanics’ thereupon?How does this impact upon our preconceptions of classical dynamics? Clumping Statistics investigates hypotheses concerning ratios of relative separations (detailed information which can beattributed locally and to subsystems). These already exist in 1- d and in settings simpler than metric shape spaces, sothis topic is well-known. Astrophysical situations modelled by this include tight binary stars, globular clusters, galaxiesand voids: absense of clumping. E.g. Roach [61] provided a discrete statistical study of clumping; this can in turn beinterpreted in terms of coarse-grainings of RPM configurations. Also note that Geometrical Probability on the shapespace s ( N,
1) = S n − provides an alternative method to this.Next, consider completing the above at the metric shape space level to probing relative angle information, the existenceof which requires d ≥ . E.g. Kendall [6, 47] investigated the relative angle question of whether the locations of thestanding stones of Land’s End in Cornwall contained more alignments than could be put down to to random chance. This involves the following procedures.1) Sample in threes.2) Consider whether there were a statistically significant number of almost collinear triangles quantified by a bluntnessangle < (cid:15) some small value (Fig 10.a).3) Use probability distributions based on the corresponding shape space geometry (i.e. on Kendall’s spherical black-board).Another application of Relative Angle Statistics is disproving claims of quasar alignment. Shape Statistics has alsobeen applied to biological modelling of spoecific 3- d objects (e.g. skulls) viewed as ratios and relative angles based uponsome approximating collection of ‘marker points’ [8].Significant results for different values of (cid:15) carry different implications [62]. Were the standing stones laid out skillfullyby the epoch’s standards for e.g. astronomical or religious reasons ( (cid:15) ≤ minutes of arc), or were they just the markersof routes or plots of land ( (cid:15) ≤ degree)?I also point out here that metric Shape Statistics (whether or not scaled) is likely to be a useful tool for Robotics. Forinstance, this can be used to analyze the extent to which robots adopt approximately the same configuration in responseto similar external conditions. Or as regards the extent to which there is success in robots mimicking (sequences of)animal configurations (fall like a cat, run like a horse...). I.e. (toy model) robotic configuration spaces not only involvedetermining the path properties outlined in Sec 3, but are also the basis for theories of firstly Geometrical Probabilityand secondly of Shape Statistics.Frontier 4. As regards specific Mechanics questions to be settled by Shape Statistics, consider whether a given CelestialMechanics configurations exhibit a shape statistically significant number of eclipses: clumping-based questions includewhether it exhibits a shape statistically significant number of tight binaries. Whether it contains a shape statisticallysignificant globular cluster. Finally, whether two images of globular clusters have sufficiently similar clustering detail tobe of the same system or to be members of similarly formed populations.N.B. that the metric case of Shape Statistics is a well-established approach [6, 47, 8, 7]. Whereas the current paperconcentrates on laying out many new theories of background-independent Mechanics, these are equally tied to being Kendall’s work is a solution to Broadbent’s [62] previous posing of the standing stones problem as one to be addressed by some kind of
Geometrical Probability. Moreover for ‘observed sky’ applications, the space of spherical triangles [47] is even better. See [7] for a review of the correspondingshape geometry and Shape Statistics, and [46] for the related topic of Shape Mechanics built from stripping down S d rather than R d absolutespaces. a) (cid:15) -bluntness in probing metric shapes. b) and c) are assessments of almost-colinearity in fours in the conformal and affinesettings. In the Möbius setting, cross ratios are to be used in this assessment and the method has to consider almost-cocircularity in foursalongside almost collinearity in fours. geometrical theories of shape with statistical applications. If the reader wishes to see what these look like for metricshapes – i.e. similarity-redundant geometry – they should cast a look through [7]. In the current article, I point outthere is an analogue of this where it is each of conformal, affine, equivoluminal groups that are redundant, or wherethese, the Euclidean and the similarity group are supersymmetrized. It is on this basis that I point to a number offrontiers. These are likely to be of interest to applied topologists, to geometers, to people working in Probability andStatistics, as well as to people working in Theoretical Mechanics and on the Foundations of Physics, in particular onmodels of Background Independence for use in Quantum Gravity. I next entertain the idea of the above Shape Statistics being but the first of a larger family, each corresponding to adistinct notion of shape. A more generalized methodology is as follows.1) Find the probe unit, as tabulated in the last three columns of Fig 2.2) Find the geometry of the space of the probe unit (the next paper in this series); by the discussion in Sec 4 this extendsto comparison of two ’best fit’ configurations.3) Build geometrical probability theory thereupon. Note that this being geometrical is not always necessary since someshape spaces turn out to be flat (see [11] for examples).4) Build Shape Statistics – using a restricted region of 2) – corresponding to the probe unit taking some particulardistinctive form.For each Shape Statistics, one application is to the classical Records Theory of the corresponding Shape Mechanics.
Frontier 5. One likely application of the 2- d case – Area and Area Ratio Statistics – is image recognition. In looking atimages of point configurations, or approximating images using sampling points, the flat space affine Shape Statistics ofimages only makes sense if probing at least in fours. The minimal equivalent of the spherical blackboard is the affineshape space A (4 , , of dimension 2.Our illustrative question then is whether Thompson’s fish are affine shape statistically significantly coincident underaffine transformations, or whether they ‘just look that way’ much like the standing stones of Land’s End ‘look to theeye’ to have a lot of collinearities in threes. For now I present a 2- d analogue of the fish problem (as per Fig 8.b), whichpoints to use of equiareal Shape Statistics in this case. This concerns quantitative analysis of propositions concerning local angles exhibited by a point distribution. In particularin 3- d , the minimum sampling unit here is probing in sixes, corresponding to a 3- d angle space a (4 , . This is but one Cross-Ratio Statistics, since cross ratios are invariants in a wider range of projective geometries. Inthe present case in the complex plane, the minimum probing size is 4 points, corresponding to the 2- d cross-ratio space c (4 , .As per Fig 10.b-c), this case has a very natural analogue of the Euclidean case’s ‘collinearity in threes’ hypothesis.Firstly note that in this geometrical setting lines and circles can be mapped to each other, so collinear is to be upgradedto that, and four points are needed. Secondly, with these changes made, the question becomes how many almost-realcross ratios there are. This makes use of the well-known fact that real cross ratios correspond to a line-or-circle passingthrough the four points, with ‘almost real’ in the role of bluntness parameter.16rontier 6. The above may be used to test whether a given picture was drawn according to the rules of perspective.Many pictures might naïvely look like perspective drawings, but the standing stones problem has illustrated the gapbetween that and statistical significance. The above method can firstly be used to settling this issue, and secondly – byusing diverse tolerance parameters – can be used to assess how accurately a confirmed perspective drawing was crafted.This could in turn be used as a means of dating the drawing, identifying the equipment used, or helping to identifythe artist by comparison with the value of ‘almost real’ in the paintings reliably attributed to them. Conversely, thismethod may have the capacity to spot wrongly attributed pictures and forgeries. Note that this can be considered tobe another type of image analysis. Frontier 7. Investigate whether some instances of evolving objects – e.g the shape of a particular animal’s skull as thatanimal ages, of skulls over the course of the evolution of species, or of galaxies – preserve relative angle information toa greater extent than ratio information. Investigating whether this is significant to the extent of each such sequenceinvolving conformal maps would require setting up a conformal Shape Statistics. Other hypotheses would include thatassessing angle information within metric Shape Statistics suffices, or that ageing and evolution are better modelledaffinely. For instance, what type of geometrical transformation is general enough to model each of the three examplesgiven above?In conclusion, having a Shape Statistics for each type of geometry multiplies opportunities of spotting significant patternsin nature.
GR has a clear-cut Temporal and Configurational Relationalism split as laid out in Appendix C. On the other hand,Appendix D explains how Supergravity does not. This difference stems from Supergravity’s constraint algebraic structurebeing more complicated than GR’s. The above distinction between GR and Supergravity then has further knock-oneffects as regards other aspects of Background Independence, such as 1) as regards which notions of observables remainmeaningful. 2) Supergravity does not have a direct analogue of Superspace. Moreover, Supersymmetry is itself asource of constraints, and the example of Supergravity shows that capable of overriding the importance of other sourcesof constraints such as Temporal Relationalism and Configurational Relationalism. Due to these observations, whether itis possible for Supersymmetry to be compatible with Relationalism and with Background Independence more generallyis an interesting question. Below, we settle this matter in the affirmative by constructing supersymmetric RPMs for thefirst time.Absolute space is here a Z -graded version of R d : { a = R ( d | p ) } N for N = p Supersymmetry, meaning with p superchargeseach accompanied by conjugates. The a → × Ni =1 a construct then continues to apply: q ( d | p ) = R ( Nd | p ) . Then seeFig 15 for eleven supersymmetric g which are subgroups of one or both of the superconformal and superaffine ‘super-apex groups’. Each provides a corresponding notion of Relationalism and a reduced configuration space geometry. Itis possible also at least in principle to consider a supersymmetric q subject to a ‘merely bosonic’ g (such as thosetabulated in Fig 5).Finite models including fermions attain Temporal Relationalism [10] through being homogeneous linear geometriesd s = (cid:112) m AB d Q A d Q B √ W + l C d Q C . (40)Note that the action (40) is no longer of Jacobi type but rather of Randers type [65] (a subcase of Finsler geometry ifit is additionally nondegenerate, and of Jacobi–Synge type action). Here m is a quadratic ‘bosonic’ contribution to theoverall notion of metric involved, and l is a linear ‘fermionic’ contribution. This is in the further context of the speciesindexed by A , B on the one hand and by C on the other are not to be overlapping in this setting (i.e. a partition intodistinct bosonic and fermionic species respectively). (40) is also a model arena for the relational form of Einstein–DiracTheory [66]. The above presentation is prior to applying Best Matching with respect to the g in question.Let us next apply that specifically in the case of N = 1 , d = 1 and g = superT r (1) . Here the particles indexed by I each have coordinates q I , θ I , ¯ θ I , due to the usual 1- d x being accompanied by two Grassmann coordinates θ and ¯ θ . Then S susy [ q I , θ I , ¯ θ I , a, α ] = √ (cid:90) (cid:110) || d a,α q ||√ W + i (cid:88) NI =1 { ¯ θ I d a,α θ I − d a,α θ I θ I } (cid:111) (41) For sure, this is meant here in Wheeler’s sense, rather than in the entirely technically different sense used in the Supersymmetry literature. In 1- d , the version with only one Grassmann coordinate is also possible. The non-relational version of Supersymmetric Mechanics withtwo Grassmann coordinates was first studied by Nicolai [63] and Witten [64]. a,α θ I := d θ I − d a + i d α , d a,α θ I := d ¯ θ I − d a − i d ¯ α (42)and bosonic best-matched derivatives d a,α q I := d q I − d a − ¯ θ I d α − d ¯ α θ I . (43)The d α and d ¯ α corrections to the fermionic species are ‘Grassmann translations’. Furthermore, upon imposing Super-symmetry these also feature as corrections to the bosonic changes in the Grassmann-linear manner indicated.Then variation with respect to a gives a new form of 1- d zero total momentum of the universe constraint S susy := (cid:88) NI =1 { p I + p θI − p ¯ θI } = 0 . (44)The new form just reflects that fermions also carry momentum. On the other hand, variation with respect to α and ¯ α give the zero total supersymmetric exchange momentum of the universe constraints S := − (cid:88) NI =1 (cid:8) p θI + i ¯ θ I p I (cid:9) = 0 , S † := (cid:88) NI =1 (cid:8) p ¯ θI + iθ I p I (cid:9) = 0 . (45)Note that these gain one piece from the fermionic sector and one piece from the bosonic sector. These constraints areaccompanied by the standard quasi-bosonic E , except that now V contains fermionic species also: E := || p || / V ( q I , θ I , ¯ θ I ) = E . (46)Taking for now the stance of not knowing the supersymmetric analogues of shape, the incipient form of V is V ( q I , θ I , ¯ θ I ) = V B ( q K ) + (cid:88) NI =1 (cid:110) θ I u I ( q K ) − ¯ θ I v I ( q K ) + (cid:88) NJ =1 θ I ¯ θ J w IJ ( q K ) (cid:111) , (47)by virtue of the automatic truncation in Grassmann polynomials afforded by the underlying anticommutativity. Thendemanding algebraic closure gives the conditions on V for V to be a function of the superT r (1) notion of shape as (cid:88) NI =1 ¯ θ I (cid:26) ∂V B ( q K ) ∂q I + (cid:88) NJ =1 θ J ∂u J ( q K ) ∂q I (cid:27) = 0 , (cid:88) NI =1 θ I (cid:26) − ∂V B ( q K ) ∂q I + (cid:88) NJ =1 ¯ θ J ∂ ¯ v J ( q K ) ∂q I (cid:27) = 0 . (48)Frontier 8. Gain an understanding of the notion of ‘supershapes’.Furthermore, the above implementation of best matched Configurational Relationalism readily extends to all dimensionsand, concurrently, to apply to all the other supersymmetric g listed whose Supersymmetry is tied to the momentumgenerators P i .Frontier 9. Work in ecess to that presented here is required for those theories in which Supersymmetry is tied to specialconformal transformations K i instead, with the principal remaining question of interest being the production of anexplicit superconformal RPM.The above implementation proceeds firstly by correcting terms with spatial vector indices d q Ii by subtracting productsof ¯ θ I ˙ α and d α α , and of θ Iα and d ¯ α ˙ α , necessitating ‘interconversion arrays’ A iα ˙ α . Then familiarity with standardspinorial formulations points to these arrays being e.g. the vector of Pauli matrices in dimension 3, and correspondinggeneralizations in further dimensions based on that dimension’s corresponding Dirac and Clifford mathematics. This casealso requires use of the well-known distinction between † and ¯ . Additionally, it is clear from super-brackets relationsat the level of the algebra that fermionic species are rotational spinors and also carriers of nontrivial homotheticweight. These give corresponding matching corrections to the fermionic changes. Upon variation with respect tothe translational and rotational auxiliaries, the preceding two best matching terms contribute, respectively, fermionicangular momentum to the zero total angular momentum constraint and fermionic dilational momentum to the zerototal dilational momentum constraint. In the SL ( d, R ) case, fermionic species are indeed SL ( d, R ) spinors, givingcorresponding matching conditions, and contributing fermionic angular momentum, shear and Procrustean stretch tothe corresponding zero total angular momentum, shear and Procrustean stretch constraints.The given superT r (1) theory suffices to establish that Supersymmetry is another setting in which a simple removal ofthe centre of mass by passing to relative coordinates is not possible. Nor does preemptively taking out the centre of massallow for one’s philosophical worldview and subsequent physical paradigm to avoid the possibility of Supersymmetry.This is since relative translations still exist within that setting, and these are sufficiently similar mathematically toabsolute translations to enable Supersymmetry in the usual manner. Dotted and undotted Greek indices here are a standard spinorial index notation. g ’s super-shapespace?[The theory of supershapes is not well-known enough for questions about ‘supersymmetric Shape Statistics’ to be posedfor now.]Frontier 12. The range of supersymmetric RPMs sketched out in the current paper is of anticipated future value inassessing the extent to which Relationalism and other aspects of Background Independence can be reconciled withSupersymmetry. This is in anticipation of more detailed investigation of how the former are substantially altered inpassing form GR to Supergravity [22].Finally N.B. that in the above theories { S , ¯ S } ∼ P and not E , (49)signifying that these supersymmetric RPMs are not models of Supergravity’s principal alteration (Appendix D) ascompared to GR (Appendix C) as regards aspects of Background Independence and subsequent Problem of Time facets.This is relevant to the discussion of differences between GR and Supergravity in Appendices C and D. In particular,at least the super-RPM arena in which the Supersymmetry is tied to P i is one in which Configurational Relationalismcan indeed include Supersymmetry, and do so without interfering with the ‘usual’ separate provision of TemporalRelationalism. All of this paper’s relational theories of Mechanics make for interesting quantum schemes. In each case, if a modelis relational to this extent, how is the corresponding QM affected? It is not that different [10] for metric RPM withand without scale! This is due to relative angular momentum (and relative dilational momentum [70], and mixtures[71, 49]) having the same mathematics as angular momentum. Moreover, the quantum metric shape quadrilateral [50]did produce a more unusual and distinctive combination of features of the Periodic Table and of Gell-Mann’s eightfoldway. Whereas the conformal case is well known for one absolutist particle and in QFT setting, it is not known as an N -body relational problem as posed here.I make use of Dirac quantization, much as Smolin [39] did for the original RPM [3]. Within Dirac quantization, oneworks with the standard position coordinates or Jacobi coordinates based kinematical quantization [28] (first done byRovelli [38] for the original RPM). [In contrast, kinematical quantization becomes a nontrivial geometrical issue inreduced quantization, though understanding the geometry and topology of the reduced configuration space in questionleads to this being resolved also.] I also make use of the Laplacian operator ordering, which in the current flat redundantconfiguration space case is equivalent also to the conformal operator ordering and the ξ operator orderings more generally.[These differ from the Laplacian by − ξR for R the Ricci scalar of the configuration space. A particular configuration spacedimension dependent value of ξ renders the overall operator conformally invariant, hence constituting the conformaloperator ordering.] These operator orderings were originally proposed by DeWitt [67] for use in what became GRQuantum Cosmology through the pioneering works of Misner [68] (see also [69] for uses of such operator orderings).Example 1) The metric scale-and-shape RPM has (cid:98) L Ψ = (cid:126) i (cid:88) nA =1 ρ A × ∂ Ψ ∂ρ A = 0 , (50)meaning that Ψ = Ψ( − · − ) . The ‘main wave equation’ is (cid:98) E Ψ = − (cid:126) (cid:52) R nd Ψ + V ( − · − )Ψ = E Ψ , (51)(in this case, conformal operator ordering collapses to Laplacian ordering since R nd is flat). This example is but a slightupgrade of Smolin’s [39] (which involved particle coordinates rather than Jacobi coordinates).Example 2) Metric shape RPM has (50) and [10] (cid:98) D Ψ = (cid:126) i (cid:88) nA =1 ρ A · ∂ Ψ ∂ρ A = 0 ; (52)together, these mean that Ψ = Ψ( − · − / − ·− ) . The ‘main’ wave equation can then be expressed as (cid:98) E Ψ = − (cid:126) (cid:52) R nd Ψ + V ( − · − / − ·− )Ψ = E Ψ (53)19y use of conformal flatness of the configuration space metric.Example 3) New to the current paper, 3- d Conformal shape RPM has (cid:98) K i Ψ = (cid:126) i (cid:88) NI =1 { q I δ ij − q i q j } ∂ Ψ ∂q iI = 0 , (54)which in conjunction with (cid:98) P Ψ = (cid:126) i (cid:88) NI =1 ∂ Ψ ∂q I (55)and the q -versions of (50) and (52) signify that Ψ = Ψ( ∠ ) . The main wave equation is then (cid:98) E Ψ = − (cid:126) I (cid:52) R Nd Ψ + V ( ∠ )Ψ = E Ψ . (56)Example 4) Also new to the current paper, 2- d Area RPM has (cid:98) S Ψ = (cid:126) i (cid:88) nA =1 ρ A S ∂ Ψ ∂ρ A = 0 . (57)This means that Ψ = Ψ( − × − ) . Noting that this and the next two examples lie outside the levels of structure uponwhich Laplacians and conformal Laplacians are defined, so it is even less clear for now in these cases how to operatororder, the ‘main wave equation’ for 4-particle area RPM is (cid:98) E Ψ = − (cid:126) (cid:88) cycles A, B =1 (cid:0) ∂∂ρ A × ∂ Ψ ∂ρ B ) + V ( − × − )Ψ = E Ψ . (58)Example 5) New again to the current paper, 2- d affine shape RPM has linear constraints (57) and (52), or, formulatingthem together, (cid:98) G Ψ = (cid:126) i (cid:88) nA =1 ρ A G ∂ Ψ ∂ρ A = 0 . (59)This means that Ψ = Ψ( − × − / − × − ) . The main wave equation for 4-particle affine shape RPM is (cid:98) E Ψ = − (cid:126) (cid:88) cycles C, D =1 ( ρ C × ρ D ) (cid:88) cycles A, B =1 ( ∂∂ρ A × ∂ Ψ ∂ρ B ) Ψ + V ( − × − / − × − )Ψ = E Ψ . (60)Example 6) As a final new Dirac quantization in this paper, in the case of superT r (1) RPM, (cid:98) S = − (cid:126) i (cid:88) NI =1 (cid:8) ∂∂θ I + i ¯ θ I ∂∂q I (cid:9) Ψ = 0 , (61) (cid:98) S † = (cid:126) i (cid:88) NI =1 (cid:8) ∂∂ ¯ θ I + iθ I ∂∂q I (cid:9) Ψ = 0 , (62) (cid:98) P susy Ψ = (cid:126) i (cid:88) NI =1 (cid:8) ∂∂q I + ∂∂θ I − ∂∂ ¯ θ I (cid:9) Ψ = 0 , (63) (cid:98) E Ψ − (cid:126) (cid:52) R N Ψ + V ( q I , θ I , ¯ θ I )Ψ = E Ψ . (64)with V within the form allowed by (48).Frontier 13. Study this paper’s new theories’ Dirac quantization schemes.Frontier 14. Obtain the reduced quantization schemes, from first obtaining the underlying corresponding geometry andthen proceeding along the lines of i) Isham’s geometrical kinematical quantization and ii) formulation and solution ofthe wave equations.See Appendices C and D for this paper’s final Frontiers. Acknowledgements
To those close to me gave me the spirit to do this. And with thanks to those who hosted me andpaid for the visits: Jeremy Butterfield, John Barrow and the Foundational Questions Institute. Thanks also to ChrisIsham, Julian Barbour and Niall ó Murchadha for a number of useful discussions over the years, and to the AnonymousReferee for useful comments.
A Flat R d geometries A.1 Real geometries
I develop this here from a simple Kleinian position – based on invariants corresponding to transformation groups – byconsidering g ≤ Aut ( (cid:104) R d , σ (cid:105) ) for various layers of mathematical structures σ . σ could be · (scalar products, i.e. the See [33] for an in-depth account of the foundations of geometry, albeit not based on group theory. See also [31] for comparison betweenfour approaches to the foundations of geometry, including the group-theoretic approach. Finally, I use (cid:104) , (cid:105) to demarcate a space (prior tothe comma) that is equipped with further structures (after the comma). δ ij ), but also / denoting ratios, − denoting differences, as feature e.g. in the Euclidean notion ofdistance, or ∠ denoting angles. σ could also be ∧ : the top form wedge product supported in dimension d , e.g. areabuilt out of cross products × in 2- d or volumes built out of scalar triple products [ × · ] in 3- d . Some geometriesadditionally allow for a number of combinations of these structures; see in particular column 1 of Fig 2.To be clear about the above shorthands’ definitions, let u , v , w , y ∈ R d . Then the scalar product is a 2-slot operation u · v . The Euclidean norm alias magnitude is then a special case of the square root of this: || v || := √ v · v . Also( Euclidean distance between u and w ) := || u − w || , (65)i.e. the Euclidean norm of the difference between the two vectors u − w . Ratio is then a 2-slot operation acting onscalars, e.g. a ratio of two components of vectors( ratio of magnitudes of u and w ) := || u |||| w || , (66)( ratio of distances ) := || u − v |||| w − y || , (67)( ratio of scalar products ) := ( u · v )( w · y ) . (68)The angle between u and w is then the arccos of the particular combination( scalar product of unit vectors (cid:98) u and (cid:98) v ) = ( (cid:98) u · (cid:98) v ) = ( u · v ) || u || || v || , (69)which is a product of square roots of 2 subcases of (68). Finally, the d -volume top form is( areas of parallelograms formed by vectors u , v ) = ( u × v ) in 2- d , (70)and ( volumes of parallelepipeds formed by vectors u , v , w ) = [ u × v · w ] in 3- d . (71)Then possible g include the following; cases whose corresponding geometry is well-known are indicated. See Figs A and12 for the meanings of the types of transformations involved, and columns 1 and 2 of Fig 2 for a summary. g = id :a trivial limiting case corresponding to no transformations being available. g = Aut ( (cid:104) R d , −(cid:105) ) = T r ( d ) : translations x → x + a , which form a d -dimensional Abelian group (cid:104) R d , + (cid:105) . g = Aut ( (cid:104) R d , / (cid:105) ) = Dil : dilations alias homotheties x → kx , which form a 1- d Abelian group (cid:104) R + , ·(cid:105) . g = Aut ( (cid:104) R d , − / −(cid:105) ) = T r ( d ) (cid:111) Dil . g = Aut ( (cid:104) R d , ·(cid:105) ) = Rot ( d ) : rotations x → Bx forming the special orthogonal group SO ( d ) := { B ∈ GL ( d, R ) | B T B = I , det B = 1 } , which is ofdimension d { d − } / g = Aut ( (cid:104) R d , − · −(cid:105) ) := Isom ( R d ) = T r ( d ) (cid:111) Rot ( d ) =: Eucl ( d ) : the d { d + 1 } / -dimensional Euclidean group of isometries, corresponding to
Euclidean geometry itself. g = Aut ( (cid:104) R d , · / ·(cid:105) ) = Rot ( d ) × Dil . g = Aut ( (cid:104) R d , −·− / −·−(cid:105) ) = T r ( d ) (cid:111) { Rot ( d ) × Dil } =: Sim ( d ) : the d { d +1 } / dimensional similarity group correspondingto similarity geometry . g = Aut ( (cid:104) R d , ∧ (cid:105) ) = SL ( d, R ) : the d − dimensional special linear group, consisting of the d { d − } / rotations, d { d − } / shears and d − ‘Procrustean stretches’. g = Aut ( (cid:104) R d , ∧ −(cid:105) ) = T r ( d ) (cid:111) SL ( d, R ) : the d { d +1 }− dimensional‘equi-top-form group’ corresponding to ‘equi-top-form geometry’ (for d = 2 , ∧ = × and this is the quite well known equiareal geometry ). g = Aut ( (cid:104) R d , ∧ / ∧ (cid:105) ) = GL ( d, R ) : the d -dimensional general linear group, consisting of rotations,shears and Procrustean stretches now alongside dilations. g = Aut ( (cid:104) R d , ( ∧ − ) / ( ∧ − ) (cid:105) ) = T r ( d ) (cid:111) SL ( d, R ) =: Af f ( d ) the d { d + 1 } -dimensional affine group of linear transformations, corresponding to affine geometry .So far, the above transformations can all be summarized within the form of the eq at the top of Fig 11 The mostgeneral case included here is affine geometry, within which all the other g above are realized as subgroups. Reflections could also be involved in each case. These are a third elementary type of isometry about an invariantmirror hyperplane (line in 2- d , plane in 3- d ). Unlike translations and rotations, they are a discrete operation. For amirror through the origin, characterized by a normal n , the explicit form for the corresponding reflection is the lineartransformation Ref : v → v − v · n ) n . (72)Moreover, a further direction in d -dimensional geometry can be taken by introducing inversions in S d − , Inv : v → v || v || . (73)These also preserve angles – but not other ratios of scalar products (Fig 12.b) – paving the way to the yet larger groupof transformations that specifically preserves just angles. The semidirect product group g = N (cid:111) H is given by ( n , h ) ◦ ( n , h ) = ( n ϕ h ( n ) , h ◦ h ) for N (cid:1) g : ‘ N is a normal subgroup of g ’, H a subgroup of g and ϕ : H → Aut ( N ) a group homomorphism. Compare the direct product’s ( g , k ) ◦ ( g , k ) = ( g ◦ g , k ◦ k ) : thishas no normal group specification, and trivial automorphism. Elementary transformations. 2- d illustration of translation, rotation, dilation, shear, and Procrustean stretch (i.e. d -volume topform preserving stretches, in particular area-preserving in 2- d and volume-preserving in 3- d ). I also indicate the relation of the last four ofthese to the irreducible pieces of the general linear matrix G , and which geometrically illustrious groups these transformations form part of.The T superscript denotes ‘tracefree part’. Note that Procrustean stretches do not respect ratios and shears do not respect angles. Figure 12: d renditions of a) reflection, which in this case is about a mirror line. b) Inversion in the circle. This transformation requiresa grid of squares to envisage – rather than a single square – since it has a local character which differs from square to square. N.B. also thatthis can map between circles and lines, with the sides of the squares depicted often mapping to circular arcs. Another perspective on geometry involves weakening the five axioms of Euclidean geometry [33, 31]. The best-known such weakening is absolute geometry , which involves dropping just
Euclid’s parallel postulate. This leads firstlyto hyperbolic geometry arising as an alternative to Euclid’s, and then more generally to such as Riemannian differentialgeometry. In contrast with this, affine geometry is that this retains
Euclid’s parallel postulate, and indeed places centralimportance upon developing its consequences (‘parallelism’). This approach drops instead Euclid’s right-angle andcircle postulates. These two initially contrasting themes continues to run strong in the eventual generalization to affinedifferential geometry.Two furtherly primary types of geometry are, firstly, ordering geometry , which involves just a ‘intermediary point’variant of Euclid’s line postulates. By involving neither the parallel postulate nor the circle and right-angle pair ofpostulates, this can be seen as serving as a common foundation for both absolute and affine geometry [33]. On the otherhand, projective geometry involves ceasing to be able to distinguish between lines and circles in addition to angles beingmeaningless and no parallel postulate. From a group-theoretic perspective, this is evoking the projective linear group
P GL ( d, R ) = GL ( d, R ) /Z ( GL ( d, R )) . Probably the best-known example of projective group is the
Möbius group
P GL (2 , C ) acting upon C ∪∞ as the fractionallinear transformations z −→ az + bcz + d , for a, b, c, d ∈ C such that ad − bc (cid:54) = 0 . This is 6- d , because there is one complex For a matrix group g = GL ( v ) (for v a vector space) or subgroups thereof, the centre Z ( g ) of g consists of whichever k I are allowedby the definition of that subgroup and the field F that v is based upon. λ ∈ C , λ az + λ bλ cz + λ d = λλ az + bcz + d = az + bcz + d . (74)For practical use within Euclidean theories of space, note in particular that ‘spatial’ measurements in our experience liewithin the forms (67) and (69), i.e. measuring tangible objects against a ruler and measuring angles between tangibleentities. On the other hand, more advanced, if indirect, physical applications make use of (extensions of) the othernotions of geometry above. B Lie groups and Lie algebras A Lie group [75] is simultaneously a group and a differentiable manifold; its composition and inverse operations aredifferentiable. Working with the corresponding infinitesimal (‘tangent space’) around g ’s identity element – the Liealgebra g – is more straightforward due to vector spaces’ tractability, whilst very little information is lost in doing so.[E.g. the representations of g determine those of g .] More formally, a Lie algebra is a vector space equipped with aproduct (bilinear map) |[ , ]| : g × g −→ g that is antisymmetric and obeys the Leibniz (product) rule and the Jacobiidentity |[ g , |[ g , g ]| ]| + cycles = 0 (75) ∀ g , g , g ∈ g . This is an example of algebraic structure : equipping a set with one or more product operations. In thepresent case, Lie brackets are exemplified by Poisson brackets and commutators. Particular subcases of Lie bracketsthen include the familiar Poisson brackets and quantum commutators.Moreover, a Lie algebra’s generators (Lie group generating infinitesimal elements) τ p obey |[ τ p , τ q ]| = C rpq τ r , (76)where C rpq are the structure constants of that Lie algebra. It readily follows that the structure constants with all indiceslowered are totally antisymmetric, and also obey C o [ pq C rs ] o = 0 . (77)Next suppose that it is hypothesized that some subset of the generators, K k , is significant. Denote the rest of thegenerators by H h . On now needs to check to what extent the algebraic structure in question actually complies with thisassignation of significance. Such checks place limitations on how generalizable some intuitions and concepts which holdfor simple examples of algebraic structures are. In general, the split algebraic structure is of the form |[ K k , K k (cid:48) ]| = C k (cid:48)(cid:48) kk (cid:48) K k (cid:48)(cid:48) + C hkk (cid:48) H h , (78) |[ K k , H h ]| = C k (cid:48) kh K k (cid:48) + C h (cid:48) kh H h (cid:48) , (79) |[ H h , H h (cid:48) ]| = C khh (cid:48) K k + C h (cid:48)(cid:48) hh (cid:48) H h (cid:48)(cid:48) . (80)Denote the second to fifth right hand side terms by 1) to 4). 1) and 4) being zero are clearly subgroup closure conditions.2) and 3) are ‘interactions between’ h and K . The following cases of this are then realized in this paper.I) Direct product . If 1) to 4) are zero, then g = K × h .II) Semi-direct product . If 2) alone is nonzero, then g = K (cid:111) h .III) Thomas integrability . If 1) is nonzero, then K is not a subalgebra: attempting to close it leads to some K k arediscovered to be integrabilities. I denote this by K →(cid:13) h ; the arrow points to the part of the split which arises as anintegrability of the other part. A simple example of this occurs in splitting the Lorentz group’s generators up intorotations and boosts; this is indeed the group-theoretic underpinning [76] of Thomas precession (see Appendix B.1).IV) Two-way integrability
If 1) and 4) are nonzero, neither K nor h are subalgebras, due to their imposing integrabilitieson each other. I denote this by K ↔(cid:13) h , with the double arrow indicating that the two parts of the split are integrabilitiesof each other. In this case, any wishes for K to play a significant role by itself are almost certainly dashed by themathematical reality of the algebraic structure in question.Note that this classification is important as regards understanding how GR’s constraints are more subtle than GaugeTheory’s, and Supergravity’s than GR’s; this is further developed in Appendices C and D. B.1 Examples of Lie groups and Lie algebras
For Abelian Lie groups, the structure constants are all zero. Examples of this include
T r ( d ) and Dil ( d ) – which areboth noncompact – and the compact Rot (2) = SO (2) . The corresponding Lie algebras’ generators are P i = − ∂∂x i , D = − x i ∂∂x i and L = y ∂∂x − x ∂∂y .On the other hand, for d > the Rot ( d ) = SO ( d ) are non-Abelian Lie groups. Compare these with the O ( d ) Liegroups: SO ( d ) is a Lie subgroup of O ( d ) . The former has 2 connected components related by a discrete reflection. The23orresponding Lie algebra sees only the connected component that contains the identity, so is the same for each of O ( d ) and SO ( d ) . GL ( v ) and SL ( v ) are also Lie groups [non-Abelian for dim ( v ) > ].Some Lie algebras used in this paper are as follows. The general linear algebras gl ( v ) of d × d matrices over F , thereal cases of which have dimension d . The special linear algebras sl ( v ) are the zero-trace such, the real cases of whichhave dimension d − . The generators for gl ( d, R ) are, very straightforwardly, G ij = x i ∂∂x j . The special orthogonalalgebras so ( d ) := { A ∈ gl ( d, R ) | A + A T = 0 } of dimension d { d − } / . These are generated by M ij = x i ∂∂x j − x j ∂∂x i subject to the schematic noncommutation relation |[ M , M ]| ∼ M . (81)Among these, so (2) is the above-mentioned Abelian algebra and so (3) has the alternating symbol (cid:15) ijk for its structureconstants. The 3- d case also simplifies by the duality between M ij and L i . SO (4) [and, more famously, SO (3 , : theLorentz group] satisfy accidental relations linking them to a direct product of two copies of SO (3) . Moreover, in the SO (3 , case – whereas linear combinations can be taken so as to obtain this split, the original presentation’s generatorsdiffer in physical significance (3 rotations and 3 boosts) – which physical meanings are not preserved by taking said linearcombinations. Adhering then to the physically meaningful split into J i and K i is the setting of the Thomas precessionmentioned in the previous sub-appendix. Schematically, this decomposes (81) into |[ J , J ]| ∼ J , |[ J , K ]| ∼ K , |[ K , K ]| ∼ K + J , (82)the key bracket being the last one by which the boosts are not a subalgebra, the precession in question referring to therotation arising thus from a combination of boosts.Some composite Lie groups of particular relevance to this paper are then
Eucl ( d ) and Sim ( d ) . Here e.g. Eucl ( d ) ’ssemidirect product structure is due to the bracket |[ J , P |] ∼ P , signifying that P is a Rot ( d ) -vector. Also, T r ( d ) (cid:111) Dil ’ssemidirect product structure rests upon |[ P , D ]| ∼ P . On the other hand, Rot ( d ) – Dil independence as found e.g.within the family of subgroups of
Af f ( d ) [and thus in particular Eucl ( d ) and Sim ( d ) ] is based upon rotation anddilation generators commuting: a direct product split |[ M , D ]| = 0 ( |[ L , D ]| = 0 in 3- d and |[ L , D ]| = 0 in 2- d ) . (83) B.2 Killing vectors and isometries: plain, homothetic and conformal
Figure 13:
Decomposition of special conformal transformation into an inversion, translation and another inversion.
One way of getting at
Eucl ( d ) which usefully extends to further groups is by solving Killing’s equation in flat space. Inthis case, given Killing’s Lemma [77], the form ξ i = a i + B ij x j (84)for the Killing vectors readily follows. Repeating for the homothetic Killing equation in flat space, ξ i = a i + B ij x j + cx i (85)ensues. Finally, repeating for the conformal Killing equation in flat space ξ i = a i + B ij x j + cx i + { k j x i − k i x j } δ ik x k . (86)ensues for d ≥ [75]. The k i correspond to special conformal transformations x i −→ x i − k i x − k · x + k x (87)formed from an inversion, a translation and then a second inversion. The infinitesimal generator is K i := x ∂∂x i − x i x j ∂∂x j . Thus conformal group Conf ( d ) of dimension d { d + 2 } / (in particular 10 for d = 3 ) arises.24or d = 2 , the conformal Killing equation famously collapses to the Cauchy–Riemann equations, causing an infinityof solutions: any holomorphic function f ( z ) will do. In 1- d , the conformal Killing equation collapses to d ξ/ d x = φ ( x ) ,amounting to reparametrization by a 1- d coordinate transformation v = Φ( x ) + a for Φ := (cid:82) φ ( x ) d x . This case is not subsumed within (86): Conf (1) is also infinite-dimensional, albeit rather less interesting than the 2- d version. (1- d hasno angles to preserve, though conformal factors can be defined for it none the less; note also that the metric drops out of the 1- d conformal Killing equation.)Due to an integrability of the schematic form |[ K , P ]| ∼ M + D , (88)the conformal algebra is ( P, K ) →(cid:13) ( M, D ) Thomas. I.e. a translation and an inverted translation compose to give botha rotation (‘conformal precession’) and an overall expansion. Elsewise K i behaves much like P i does. B.3 Some further subgroups acting upon R d The above three Sections can be viewed as introducing P i , M ij , D and K i generators.One can furthermore consider e.g. shears and d -dimensional volume preserving stretches ( G T ( ij ) generators); each ofthese are only nontrivial for d ≥ . Alongside the rotations, these form SL ( d, R ) ; then the equi-top-form group E ( d ) := T r ( d ) (cid:111) SL ( d, R ) , corresponding to the eponymous geometry (equiareal in 2- d [33]). Also, GL ( d, R ) = Dil × SL ( d, R ) ;then the affine group E ( d ) := T r ( d ) (cid:111) GL ( d, R ) , corresponding to affine geometry [33]. dim( E ( d ) ) = d { d + 1 } − anddim( A ( d ) ) = d { d + 1 } . The unsplit nonzero affine brackets are, schematically, [ G , G ] ∼ G , [ G , P ] ∼ P , (89)signifying closure of the GL ( d, R ) subgroup and that P i is a GL ( d, R ) vector. As regards general sl ( d, R ) generators,perform the antisymmetric–symmetric and trace–tracefree splits on gl ( d, R ) ’s generators (Fig A). Then the antisym-metric part is just the rotations, the tracefree symmetric part is E ij = x i ∂∂x j + x j ∂∂x i − n δ ij x k ∂∂x k and the trace part(the usual dilation) is discarded. E.g. Corresponding infinitesimal matrices for sl (2 , R ) these are (cid:16) (cid:17) and (cid:16) (cid:17) which form the triple (23) with the infinitesimal rotation matrix. The corresponding generators are = x ∂∂x − y ∂∂y forProcrustean stretches and = x ∂∂y + y ∂∂x for shears. It is also important to note that [ Shear , Shear (cid:48) ] ∼ Rotation (90)by which the non-rotational parts of SL ( d, R ) cannot be included in the absense of the rotations.Figure 14: Summary sketch, of groups including further groups acting upon R d . These are arrived at by adding generators as per thelabelled arrows. Moreover, the group relations involved do not permit all combinations of generators to be included. In particular, absencesmarked X are due to integrability (88). Absences marked ∗ are due to integrability (90). Finally, absences marked † are due to obstruction(91). Figures 4 and 5 then use a matching layout. Moreover, the K i and G T ( ij ) generators are not compatible with each other, as is clear fromconformal transformations only preserving angles whereas shears do not preserve angles . (91)25hus there are two distinct ‘apex groups’: Conf ( d ) from including K i and Af f ( d ) from including S ij . ‘Apex’ is usedhere in the sense that the other possibilities are contained within as Lie subgroups. These include a number of subgroupsnot yet considered (Fig 13).Finally, some of the above groups also support nontrivially distinct projective versions, obtained by quotienting outby the centre of the group in question. E.g. if this is performed upon the affine group, projective geometry [33, 31]ensues. B.4 Superconformal and superaffine algebras and some of their subalgebras
Some algebraic structures involve anticommutators |[ A , B ]| + := AB + BA These enter models of fermionic species. Seee.g. [78, 79] for an extensive ‘mathematical methods for physicists’ treatments of these and of the ensuing notion ofspinors.As compared to the conformal group
Conf ( d ) , the superconformal group superConf ( d ) [80] has additional fermionicgenerators S and Q (and conjugates), in which sense it is doubly supersymmetric (denoted N = 2 ). Here, |[ S , ¯ S ]| + ∼ P (92)and |[ Q , Q ]| + ∼ K , (93)so both the momentum and the special conformal transformation arise as integrabilities.On the other hand, the superaffine algebra superAf f ( d ) has just the one additional fermionic generator S (andconjugate). Then (92) holds, by which superAf f ( d ) is S →(cid:13) ( P, G ) Thomas (composition of supersymmetries results ina translation). Examining the table of subgroups in Fig 15, the exhibited supersymmetric subgroups of superAf f ( d ) share this property. In contrast, the superconformal group has a more complicated sequence of integrabilities, with ( S, Q ) →(cid:13) ( P, K ) →(cid:13) ( M, D ) ; this is a tripartition to (78-80)’s bipartition. Finally note that all the groups in the table bar superConf ( d ) have just the one supersymmetric generator (and conjugate): N = 1 . 8 are supersymmetric subgroups of superConf ( d ) and 5 of superAf f ( d ) [4 of which are also among the previous 8].Figure 15: Some subgroups, and overruled combinations, of supersymmetric groups, within the superconformal and superaffine apex groups.These are again laid out according to 5, except now with at least one supersymmetric generator as well. The cases marked with an F fail tosupport Supersymmetry due to Supersymmetry requiring at least one of P i and K i as an integrability: (92-93). C Flat to differential geometry modelling of space, and GR-level counter-part of this paper
A) Firstly, metric geometry is one of the outcomes of weakening the axioms of Euclidean geometry. In particular,a first route to such is through assuming only absolute geometry and then finding a large multiplicity of such toexist. Moreover, some aspects of ‘metric geometry’ remain upon ceasing to assume a metric. For instance, some suchaspects form topological manifold geometry and differentiable manifold geometry. Furthermore, many of the outcomes26igure 16: a) This Sec’s specific sequence of configuration spaces, as a useful model arena for GR’s outlined in b). CS denotes conformalsuperspace and V denotes a solitary global volume degree of freedom. of stripping down the structure of Euclidean geometry have close counterparts in differential geometry. E.g., similarity,conformal, affine and projective differential geometries exist [81, 41]. Additionally, it is possible to reconcile e.g. affineand metric (or conformal metric) notions in differential geometry. E.g. the standard modelling assumptions in GRinclude that the metric connection serve as a (and the only) affine connection.In this way, stripping away the layers of structure assumed in GR (whether to model spacetime, or to model spacewithin a geometrodynamical perspective) leads to closely analogous first ports of call to those in stripping away thelayers of structure assumed in RPM’s based upon R d . In the specific case of GR, these first ports of call [82] areconformal geometry and affine geometry. Upon Supersymmetry’s extra structure, Supergravity is another first port ofcall. In the case of studying GR configurations, physical irrelevance of Diff( Σ ) is also under consideration; the mostcommon corresponding configuration spaces are depicted in Fig 16.b). Here Σ some spatial topology taken in thecurrent paper to have some fixed compact without boundary form. The Figure juxtaposes these with analogous RPMconfiguration spaces as laid out in the main part of the current paper. One can also consider an analogous diamondof affine spaces, with variants on GR allowing for the additional possibility of both retaining a metric and introducingan affine connection other than the metric connection. Indeed, one means of such an alternative theory having torsionis through the difference of two distinct affine connections constituting a torsion tensor. Superconformal Supergravityis an example of a further combination of the elements of these first ports of call. All of these are open to relationalanalyses and yet wider consideration of Background Independence leading to whether they exhibit significant differencesfrom GR as regards the Problem of Time’s many facets.A second port of call is considering Topological Relationalism (whether in the context of variants on GR allowingfor topology change, or in the context of Topological Field Theories being a further generalization of Conformal FieldTheories). See [73] for discussion of the second and subsequent ports of call from a relational perspective. What aboutmodelling this second port of call with RPMs? In fact, this is relatively straightforward. E.g. in one sense changein particle number in RPMs is analogous to topology change in GR. (I.e. theories which retain the upper layers ofstructure whilst letting subsequent layers of structure also be dynamical rather than absolute backgrounds). In anothersense (stripping away the upper layers), topological RPMs involve distinguishing only those configurations which aretopologically distinct. Then e.g. scaled triangleland collapses to a small finite number of points: the total collision, thedouble collision (or three such if labelled) and the general configuration that is not any of the previous. This is rathersimpler than the space of all topological manifolds in some given dimension! Thus it is very plausible for RPMs to beable to model multiple layers of structure (at the cost of resembling GR rather less at the lower levels).B) Let us next return to the upper layers of structure, so as to justify some of the many GR–RPM analogies that occurthere. GR can be cast in Temporal and Configurational Relationalism form [20], the latter being tied to the group ofspatial diffeomorphisms Dif f ( Σ ) . The underlying configuration space is Riem ( Σ ) , i.e. the space of spatial (positive-definite) 3-metrics on Σ . Here the constraint provided by Temporal Relationalism [20] gives – via Dirac’s argumentthat reparametrization invariance necessarily implies a primary constraint [59] – a relational recovery of the quadratic GR Hamiltonian constraint H [12]. On the other hand, Configurational Relationalism [20, 10] provides the linear GRmomentum constraint M i [12]. In this sense, Temporal and Configurational Relationalism remain distinct BackgroundIndependence aspects in GR, much as they also are in RPMs. Each provides a constraint of its own.These constraints then close in accord with the Dirac algebroid of GR, which is schematically of the form { M , M } ∼ M , (94) { M , H } ∼ H , (95) { H , H } ∼ M . (96)These are the classical theory’s Poisson brackets; this closure is a classical realization of the further Constraint Closureaspect of Background Independence.The first bracket means that the M i – which correspond to the spatial diffeomorphisms Dif f ( Σ ) – themselvesclose to form a true infinite- d Lie algebra. The second bracket signifies that H is a Dif f ( Σ ) scalar density. The third27racket is not only the one expressing an integrability [83] but also the one containing both a structure function and thederivative of its RHS constraint. Its integrability means that in GR, Temporal Relationalism obliges the existence ofnontrivial Configurational Relationalism. This feature does not occur in RPMs, in which Temporal and ConfigurationalRelationalism can each be modelled in the absense of the other. The third bracket can furthermore be viewed as analgebroid counterpart of Thomas precession: H→(cid:13)M i . In more detail, compare how composing two boosts results ina rotation: Thomas precession, whereas composing two time evolutions – or pure hypersurface deformations in [84]’sinterpretation of H – results in a spatial diffeomorphism: Moncrief–Teitelboim on-slice Lie dragging. [Lie dragging is themotion corresponding to Diff( Σ ) in the same manner as precession is a name for a motion corresponding to Rot ( d ) .] Thisanalogy is (as far as the Author is aware) new to this paper. As well as rendering the Dirac Algebroid more pedagogicallyaccessible to students, this analogy is useful in the below revelations about Supergravity being substantially differentfrom GR as regards the form taken by its Relationalism and Background Independence more generally.Further consequences of the above algebraic form as regards background independent features are as follows.1) that Wheeler’s Superspace ( Σ ) := Riem ( Σ ) /Dif f ( Σ ) – central to geometrodynamics – is a meaningful intermediaryconfiguration space. This follows from M i forming a subalgebra of the Dirac algebroid by the first bracket, so it ismeaningful to reduce out M i by itself.2) Expression in terms of Beables or Observables is a fourth aspect of Background Independence; the general failure ofwhich in classical and quantum gravitational theories then constitutes the well-known ‘Problem of Observables’ facet ofthe Problem of Time. Observables or beables are objects which commute with constraints. In the case of commutingwith all constraints – H and M i in GR – one is dealing with Dirac observables. In the case of commuting with linearconstraints only – M i in GR – one is dealing with Kuchař observables [72]. Moreover, concepts of observables orbeables as objects which commute with subsets of constraints only make sense if the subset in question algebraicallycloses [22]. Thus M i forming a subalgebra of the Dirac algebroid by the first bracket guarantees the meaningfulness ofKuchař observables in the case of GR. See [87] for the form taken by the Kuchař observables for the current paper’snew non-supersymmetric theories.Further conformal Configurational Relationalism has also been considered in the case of GR and of alternative theoriesof conformogeometrodynamics [44, 17, 86]. The original conception of this used metric shape RPM as a model arena;however, the current paper’s conformal shape RPM is surely a comparable or enhanced source of insights.Frontier 15. Finally, returning to topic A), perform the differentiable manifold level counterpart of the current paper’scomparative relational study of the diverse levels of structure to be found in flat geometries.The current paper and Frontier 15 can furthermore be seen as part of a wider program in which an increasing number oflevels of mathematical structure assumed in Physics are taken to be dynamical rather than fixed background structures.This program was initiated by Isham [74], who considered replacing geometrical quantization based upon the usualconfiguration space with that based upon generalized configuration spaces. I then provided a classical precursor for thisprogram in [73], in particular sketching out an even wider range of classical Shape Statistics theories than the currentpaper’s (at the levels of topological manifolds and of topological spaces). That work also posed Shape Statistics on GR’sSuperspace( Σ ) and conformal superspace, CS( Σ ). Those remaining a distant dream in the general case, I also posedthe more tractable analogues in the settings of anisotropic minisuperspace and of inhomogeneous perturbations aboutminisuperspace [11, 25]. D Contrast with canonical formulation of Supergravity
Supersymmetry and split space-time GR can each separately be envisaged as Thomas-type effects at the level of algebraicstructures. Furthermore, upon considering both at once, the integrabilities involved go in opposite directions, formingthe more complicated ‘two-way’ integrability case). [Contrast also with the superconformal group’s composition ofintegrabilities being tripartite with two aligned one-way integrabilities instead.] The schematic form of the key newrelation for Supergravity is (see e.g. [88]) { S , S } C ∼ H + M . (97)This forms the second integrability of the ‘two-way’ pair, to the first integrability being of form (96). The implications of the two-way integrability case include that the linear constraints do not close by themselves; thus1) they cannot be quotiented out as a unit (in this sense no Supergravity counterpart of Wheeler’s Superspace). An algebroid allows for ‘structure functions’ – including derivative operators in the present case – of constraints to appear on theright-hand side. The C subscript stands for Casalbuoni Poisson bracket [89]. The time component P arising within (92) in the indefinite case’s super-Poincaré subgroup can be seen as a precursor of (97).
28) Temporal and Configurational Relationalsim become a fused notion as opposed to separate notions.2) There is no Supergravity counterpart of Kuchař observables.Moreover, two more possibilities for splits manifest themselves. The main idea here is that the Temporal to Configura-tional Relationalism split, the quadratic and linear constraints split and the notion of Kuchař observables correspond totreating the linear consraints
LIN differently in GR. However, in a wider range of theories including Supergravity, thisconsideration is to be supplanted more generally by splits which respect the subalgebraic structures contained withinthe constraint algebraic structure.Then as a second possibility for a split [82], one can also consider the S , H to LIN (cid:48) split (for
LIN (cid:48) the linear non-supersymmetric constraints); this is Thomas with a
LIN (cid:48) subalgebra.As a third possibility, the
LIN (cid:48) , H to S split is Thomas, exhibiting a ‘non-supersymmetric’ subalgebraic structure. Inparticular, these other splits permit a meaningful notion of quotienting out LIN (cid:48) , giving a well-defined quotient spaceand a well defined notion of observables in this modified sense. I.e.1.A) extend Riem( Σ ) to include the space of gravitino fields, and then quotient out solely by the non-supersymmetriclinear constraints.2.A) A notion of observables or beables can be defined as commuting with solely the non-supersymmetric linear con-straints.The yet further versions 1.B) and 2.B) removing the word ‘linear’ from the preceding definitions are likely to be harderto handle. I term 2) and 2.A) non-supersymmetric Kuchař and Dirac observables or beables respectively. I term 1) and1.A) non-supersymmetric Superspace and True-space respectively; True-space is a formal reference to the space of truedynamical degrees of freedom itself. Finally, whether any of the entities termed ‘non-supersymmetric’ are in violation ofthe spirit of Supersymmetry may be a matter further relevance from viewpoints which take Supersymmetry sufficientlyseriously.Furthermore, due to H ’s ties to Temporal Relationalism in GR, and of (some) linear constraints’ ties to ConfigurationalRelationalism, Supergravity’s change of status as regards which of these constraints can be entertained independently ofwhich others also concerns how Relationalism is to be viewed. Additionally, Temporal Relationalism, Configurational Re-lationalism, Constraint Closure and Expression in terms of Beables are four of the aspects of Background Independence[24] underlying four of the Problem of Time facets [18, 37, 85]. Thus Supergravity exhibits a very different realizationof these from GR [22], which could herald one or both of foundational problems for Supersymmetry or a hitherto insuf-ficiently general conceptualization of Background Independence and the Problem of Time. One possibility here is thatSupersymmetry breaks down the divide between Temporal and Configurational Relationalism as separate providers ofconstraints. Another possibility is that Supersymmetry renders constraint provision tripartite, by itself constituting athird provider of constraints. The current paper then shows that none of the above happen in superRPMs whose super-symmetry is tied to the translations. In this case, Supersymmetry is compatible with Relationalism and is implementedas a subcase of Configurational Relationalism. Of course, the current paper also points out that superRPMs lack ananalogue of (96). In this manner they realize the opposite single plank to that realized by non-supersymmetric GR. It isthen rather interesting that the existing conception of Background Independence can be combined with Supersymmetrywithout requiring modification of either , at least in these nontrivial and complementary model arenas.Frontier 16. Resolve the above matter in full Supergravity. Or at least do so in some model arena exhibiting specificallythe ‘two-way’ pair of integrabilities, by which S implies H , and H imply LIN (cid:48) , with the Temporal to ConfigurationalRelationalism divide hitherto having been between H and the full set of linear constraints. Does this have any furtherconsequences for the ‘space and configuration primality’ versus ‘spacetime primality’ debate? [13, 20, 24]. References [1] See
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