1202.4186 (Edward Anderson)
Edward Anderson
Relational particle models (RPM's) are toy models of many aspects of GR in
geometrodynamical form, suitable as toy models for studying 1) strategies for
the problem of time in quantum gravity, in particular timeless, semiclassical,
histories and observables approaches and combinations of these. 2) Various
other quantum-cosmological issues: structure formation/inhomogeneity,
significance of uniform states... They are relational in that only relative
ratios of separations, relative angles and relative times are significant; more
widely, this is a `Leibniz--Mach--Barbour' brand of relationalism. The
relational quadrilateral's usefulness is via it simultaneously possessing
linear constraints and nontrivial subsystems; also its configuration space is
now a nontrivial complex-projective space. This paper studies
quadrilateralland's configuration space. In particular, what the relational
quadrilateral counterparts of triangleland's A) Dragt-type coordinates
(ellipticity, anisoscelesness, and triangle area, which is also a democracy
invariant), B) subsystem-split parabolic coordinates and C) the most
blockwise-simple coordinates (spherical polars). These were key to unlocking
the dynamics, QM and problem of time calculations for the triangle, and their
counterparts turn out to be likewise for the quadrilateral in Papers II, III
and IV respectively. I show these are now A) a hexuplet of shape coordinates
(which now exclude the democracy invariant square root of the sum of squares of
areas), B) a linear combination of these termed Kuiper coordinates, and C) the
Gibbons--Pope-type coordinates. Each of these is given a lucid new
interpretation in terms of quadrilaterals. I furthermore investigate
qualitatively-significant regions of the configuration space of quadrilaterals,
in anticipation of timeless and Halliwell-type combined Problem of Time
strategies and of uniformity in Quantum Cosmology.
View original:
http://arxiv.org/abs/1202.4186
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