From Riemann's metric to the graph metric, or applying Occam's razor to entanglements

Abstract

In 1854 Riemann prophesied: ‘Either the reality underlying space must form a discrete manifold, or the reason for measurability must be sought externally, through forces acting thereon to constrain it.’ Matter is in a graph-like state if its particle coordinates have been or are being randomized in all dimensions but one, by Brownian motions induced by random forces which maintain equipartition of kinetic energy. The graph-like particle generates a graph metric, which is one-dimensional and discrete. Physical theories of amorphous polymers generally evolve intuitively by embedding the particles in a three-dimensional continuum regarded as physically real, or in a configuration space regarded as an auxiliary. After integration over random coordinates, it appears that the end results no longer depend on the embedding space at all. The historical theories of amorphous polymer systems by Hermans and Kramers, Debye, Zimm and Stockmayer, Rouse, Fixman and some others may have the added advantage of demonstrating how the Riemann metric degenerates to the non-trivial graph-metric. The analysis of random networks, embedded in a three dimensional space with all the complications arising from chain entanglements, generally seems to result in small semi-empirical corrections which graph-like-state theories could readily develop without such complications. A programme for establishing a new axiomatic basis is suggested along these lines. The benefits would include not only a clearer relation between strikingly different theories and between their geometric and physical ingredients, but substantially simpler calculations and algorithms. Simplifications, especially some inspired by the achievements of Chompff and of Forsman, are briefly sketched. An analogy is drawn with the early programme culminating in the proof that, by virtue of their metric structures, the continum and discrete (matrix) formulations of quantum theory are isomorphic. It is suggested that the operator calculus of Rota, especially as recently perfected in his work with Roman, should be helpful in making explicit the isomorphism sought for the graph-like state.