Pure shape dynamics: General framework

Abstract

We put forward a general framework for describing relational physical theories. This framework-which we call Pure Shape Dynamics (PSD)-represents a conceptual evolution of the original insights brought about by the Shape Dynamics programme. PSD's novel take on relationalism is its insistence on describing any dynamical system by means of the intrinsic geometry of its associated curve on the suitable relational configuration space of the theory, namely shape space, whereby the corresponding equation of state of the curve expresses the ratio of change of its geometric degrees of freedom. The mathematical structure underlying the equation of state is the suitable action of a local section in some unit tangent bundle over a natural generalisation of yet another unit tangent bundle, which we call shape phase space.

A proposal for a metaphysics of self-subsisting structures. I. Classical physics

“…This can be done by banishing any concept whatsoever of parametrization of the dynamical curve γ, hence taking it as an unparametrized curve γ 0 . 6 This is the essential idea underlying Pure Shape Dynamics (PSD), which can be considered a natural evolution of SD (see Koslowski et al, 2021, for a thorough technical introduction to this new relational framework).…”

“…The major novelty of the relationalist picture advocated by PSD is exactly its insistence on using only intrinsic geometric properties of γ 0 in S in the description of the evolution of a given physical system, which is expressed in terms of the equation of state of γ 0 : A point q a ∈ γ 0 corresponds to the full configuration of the system, which by construction is its shape qua objective data, to which it is added the set {α a I } of any intrinsic geometric properties of γ 0 needed to fully specify the evolution. The mathematical structure underlying this manifestly intrinsic nature of the description is the directional action of a local section A(q a , α a I ) in a suitable unit tangent bundle over S: A(q a , α a I ) = dq/ds dα a I /ds = dq dα a I , i.e., the equation of state of γ 0 expresses the ratio of change of its intrinsic geometric degrees of freedom (see, again, Koslowski et al, 2021 for the technical details).…”

The Physics and Metaphysics of Pure Shape Dynamics

“…After this brief tour on the development of relational approaches to dynamics, we finally get to the latest refinement of the theory, dubbed Pure Shape Dynamics (PSD; see Koslowski et al, 2021, for a general technical introduction to the framework). In a nutshell, the qualifier "Pure" means that PSD describes any dynamical theory exclusively in terms of the intrinsic geometric properties of the unparametrized curve γ 0 traced out by the physical system in shape space Q ss , hence ensuring that there are no external reference strucconjugate variable, the so-called York time.…”

“…In short, the dynamical evolution as described by PSD does not fundamentally rely on any notion of parametrization whatsoever and, hence, it is a genuinely intrinsic description of a physical system. Remarkably enough, PSD is capable of reproducing known physics despite its decidedly intrinsic nature (see Koslowski et al, 2021, for the E = 0 N-body problem; the case of dynamical geometry will be the subject of another paper, currently in preparation). On the other hand, the original formulation of SD needs some monotonically increasing parameter-be it the ratio of dilatational momenta or York time/spatial volume, as discussed earlier-to be defined on a dynamical curve in order to make sense of the physical evolution.…”

“…A Fascinating Metaphysical DiversityIn the previous section, we have seen the fundamental theoretical elements that enter the dynamical description of a system according to PSD. Simply 15 Ephemeris equations are heavily model-dependent.Explicit expressions exist for the N -body system(Koslowski et al, 2021) and Bianchi IX cosmological model(Koslowski et al, 2018).…”