Specific PDEs for Preserved Quantities in Geometry. I. Similarities and Subgroups

Abstract

We provide specific PDEs for preserved quantities $Q$ in Geometry, as well as a bridge between this and specific PDEs for observables $O$ in Physics. We furthermore prove versions of four other theorems either side of this bridge: the below enumerated sentences. For the generic geometry - in the sense of it possessing no generalized Killing vectors, i.e.\ continuous geometrical automorphisms - the $P$ form a smooth space of free functions over said geometry. If a geometry possesses the corresponding type of Killing vectors, the $P$ must Lie-brackets commute with `sums-over-points of the automorphism generators', $S$. The observables counterpart of this is that in the presence of first-class constraints $F$, the $O$ must Poisson-brackets commute with these. Then 1) defining $Q$, $O$ requires closed subalgebras of $S$, $F$. 2) The $Q$, and the $O$, themselves form closed algebras. 3) The subalgebras of $Q$, $O$ form bounded lattices dual to those of $S$, $F$ respectively. Both $S$, $Q$ and $F$, $O$ commutations can moreover be reformulated as first-order linear PDEs, treated free-characteristically. The secondmost generic case has just one $S$ or $F$, and so just one PDE, which standardly reduces to an ODE system. The more highly nongeneric case of multiple $S$ or $F$, however, returns an over-determined PDE system. 4) We prove that nonetheless these are always integrable. This is significant by being mostly-opposite to how the more familiar generalized Killing equations themselves behave. We finally solve for the preserved quantities of similarity geometry and its subgroups; companion papers extend this program to affine, projective and conformal geometries.

arXiv comment30 pages, including 6 figures. References updated, minor typos removed, and notational changes

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• A Local Resolution of the Problem of Time. IX. Spacetime Constructability
• Geometry from Brackets Consistency
• Specific PDEs for Preserved Quantities in Geometry. III. 1-d Projective Transformations and Subgroups
• Specific PDEs for Preserved Quantities in Geometry. II. Affine Transformations and Subgroups
• A Local Resolution of the Problem of Time. XIV. Grounding on Lie's Mathematics
• Nambu variant of Local Resolution of Problem of Time and Background Independence
• Quadrilaterals in Shape Theory. II. Alternative Derivations of Shape Space: Successes and Limitations
• Isotropy Groups and Kinematic Orbits for 1 and 2-$d$ $N$-Body Problems
• A Local Resolution of the Problem of Time. VIII. Assignment of Observables