An Algebraic Approach to Physical Scales
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An Algebraic Approach to Physical Scales Josef Janyška · Marco Modugno · Raffaele Vitolo
Received: 26 June 2008 / Accepted: 19 March 2009 / Published online: 1 April 2009 © Springer Science+Business Media B.V. 2009
Abstract This paper is aimed at introducing an algebraic model for physical scales and units of measurement. This goal is achieved by means of the concept of “positive space” and its rational powers. Positive spaces are “semi-vector spaces” on which the group of positive real numbers acts freely and transitively through the scalar multiplication. Their tensor multiplication with vector spaces yields “scaled spaces” that are suitable to describe spaces with physical dimensions mathematically. We also deal with scales regarded as fields over a given background (e.g., spacetime). Keywords Semi-vector spaces · Scales · Units of measurement Mathematics Subject Classification (2000) Primary 15A69 · Secondary 12K10 · 16Y60 · 70Sxx
1 A Mathematical Approach to Physical Scales 1.1 Informal Approach to Scales in Physics Units of measurement, coupling constants, scales and scale dimensions are very standard and basic objects in all fields and formulations of physics. Usually, these objects appear in a very informal way from a mathematical viewpoint: so, we find a gap.
J. Janyška Department of Mathematics and Statistics, Masaryk University, Janáˇckovo nám 2a, 602 00 Brno, Czech Republic e-mail: [email protected] M. Modugno Department of Applied Mathematics, Florence University, Via S. Marta 3, 50139 Florence, Italy e-mail: [email protected] R. Vitolo () Department of Mathematics “E. De Giorgi”, Via per Arnesano, 73100 Lecce, Italy e-mail: [email protected]
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On one hand, nowadays many areas of physics have been formulated in a rigorous and modern mathematical language, in particular in a geometric and algebraic language. Thus, most objects of physics are described by specific and well defined mathematical objects, such as manifolds, bundles, connections, functions, tensor fields and so on. On the other hand, the concept of physical scales are usually treated intuitively in standard literature. Actually, a rigorous mathematical analysis of relations between such objects can be found in the literature concerning the dimensional analysis. But, still, it is usually omitted to specify the notion itself of physical scale in terms of algebraic objects in a way mathematically homogeneous to other geometric objects such as bundles representing physical fields. The reason of this gap is that every physicist knows how to deal with scales in a practical way, hence he feels that an intuitive approach is sufficient for his purposes. 1.2 Examples of Scales in Physical Literature Just as an example of the standard way of dealing with units of measurement and related questions in physics, we quote a few well known textbooks among possible thousands, which are well established references in the area of physics they deal with. The book “Classical electricity and magnetism” by W.K.H. Panowsky
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