Physics – Mathematical Physics
Scientific paper
2005-07-07
Int.J.Geom.Meth.Mod.Phys.3:285-314,2006
Physics
Mathematical Physics
33 pages, Latex, no figures, identical to published version
Scientific paper
10.1142/S0219887806001156
In various areas of modern physics and in particular in quantum gravity or foundational space-time physics it is of great importance to be in the possession of a systematic procedure by which a macroscopic or continuum limit can be constructed from a more primordial and basically discrete underlying substratum, which may behave in a quite erratic and irregular way. We develop such a framework within the category of general metric spaces by combining recent work of our own and ingeneous ideas of Gromov et al, developed in pure mathematics. A central role is played by two core concepts. For one, the notion of intrinsic scaling dimension of a (discrete) space or, in mathematical terms, the growth degree of a metric space at infinity, on the other hand, the concept of a metrical distance between general metric spaces and an appropriate scaling limit (called by us a geometric renormalisation group) performed in this metric space of spaces. In doing this we prove a variety of physically interesting results about the nature of this limit process, properties of the limit space as e.g. what preconditions qualify it as a smooth classical space-time and, in particular, its dimension.
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