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The mathematical foundations of dimensional analysis and the question of fundamental units

Abstract

The algebra of physical (dimension-carrying) quantities is axiomatized in a scheme called a Φ algebra. Systems of units (gauges) are readily described in this algebra, and the set of transformation of units (the gauge group) is discussed. The notion of the (gauge) invariance group of a function on the Φ algebra is described, and all functions having a given invariance group are canonically deduced in what is here called the Φ theorem. The scheme is applied to classical mechanics in order to understand better in what sense mass, length, and time are “irreducible” physical quantities. The latter two are found to be so, but the former is not. Other theories, relativity and quantum mechanics, are also briefly discussed and are shown to reduce this fundamental set of units even further.

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Szekeres, P. The mathematical foundations of dimensional analysis and the question of fundamental units. Int J Theor Phys 17, 957–974 (1978). https://doi.org/10.1007/BF00678423

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Keywords

  • Field Theory
  • Elementary Particle
  • Quantum Field Theory
  • Quantum Mechanic
  • Gauge Group