The Future Shape of Theoretical Physics and Beyond

摘要

A suite of relational notions of shape are presented at the level of configuration space geometry, with corresponding new theories of shape mechanics and shape statistics. These further generalize two quite well known examples: -1) Kendall's (metric) shape space with his shape statistics and Barbour's mechanics thereupon. 0) Leibnizian relational space alias metric scale-and-shape space to which corresponds Barbour--Bertotti mechanics. This paper's new theories include, using the invariant and group namings, 1) $Angle$ alias $conformal$ $shape$ $mechanics$. 2) $Area$ $ratio$ alias $affine$ $shape$ $mechanics$. 3) $Area$ alias $affine$ $scale$-$and$-$shape$ $mechanics$. 1) to 3) rest respectively on angle space, area-ratio space, and area space configuration spaces. Affine shape matching and affine shape statistics are argued to be of value to the theory of image analysis, as are in another sense their projective counterparts which rest on the geometry of cross-ratio space (another configuration space). The shape statistics of -1) and 0) are argued to be of value in robotics. 4) Various supersymmetric counterparts of -1) to 3) are considered. Since supergravity differs considerably from GR-based conceptions of background independence, some of the new supersymmetric shape mechanics are compared with both. These reveal compatibility between supersymmetry and GR-based conceptions of background independence, at least within these simpler model arenas.