Abstract
We study a long-recognised but under-appreciated symmetry called dynamical similarity and illustrate its relevance to many important conceptual problems in fundamental physics. Dynamical similarities are general transformations of a system where the unit of Hamilton’s principal function is rescaled, and therefore represent a kind of dynamical scaling symmetry with formal properties that differ from many standard symmetries. To study this symmetry, we develop a general framework for symmetries that distinguishes the observable and surplus structures of a theory by using the minimal freely specifiable initial data for the theory that is necessary to achieve empirical adequacy. This framework is then applied to well-studied examples including Galilean invariance and the symmetries of the Kepler problem. We find that our framework gives a precise dynamical criterion for identifying the observables of those systems, and that those observables agree with epistemic expectations. We then apply our framework to dynamical similarity. First we give a general definition of dynamical similarity. Then we show, with the help of some previous results, how the dynamics of our observables leads to singularity resolution and the emergence of an arrow of time in cosmology.
Highlights
1.1 Similarity and conceptual problems in modern physicsIn Henri Poincaré’s magistral Science and Method (Poincaré 2003) we’re invited to image a universe like our own that is identical in every respect except that it is a thousand times larger
We study a long-recognised but under-appreciated symmetry called dynamical similarity and illustrate its relevance to many important conceptual problems in fundamental physics
We develop a general framework for symmetries that distinguishes the observable and surplus structures of a theory by using the minimal freely specifiable initial data for the theory that is necessary to achieve empirical adequacy
Summary
In Henri Poincaré’s magistral Science and Method (Poincaré 2003) we’re invited to image a universe like our own that is identical in every respect except that it is a thousand times larger. In applying the PESA in light of the empirical adequacy of different physical theories, we will find that when dynamical similarities act on approximately (but not completely) isolated subsystems of the universe, the removal of dynamically similar structure is not justified This applies to many well-studied cases in physics including the so-called Runge–Lenz symmetries of the Kepler problem [see Belot (2013)]. To be more concrete about the implications of removing dynamically similar structure, we will develop a conceptual framework, leveraged on previous formal results, in which pathologies in the evolution equations of certain cosmological systems will be seen to be due to the surplus structure introduced by dynamical similarity These pathologies can be removed if the symmetry is understood as relating indiscernible states of the universe. Removing the “extra mathematical hooks” of dynamical similarity has significant implications for many empirical and conceptual problems in modern physics
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