The principal objective of this paper is to provide a torsor theory of physical quantities and basic operations thereon. Torsors are introduced in a bottom-up fashion as actions of scale transformation groups on spaces of unitized quantities. In contrast, the shortcomings of other accounts of quantities that proceed in a top-down axiomatic manner are also discussed. In this paper, quantities are presented as dual counterparts of physical states. States serve as truth-makers of metrological statements about quantity values and are crucial in specifying alternative measurement units for base quantities. For illustration and ease of presentation, the classical notions of length, time, and instantaneous velocity are used as primordial examples. It is shown how torsors provide an effective description of the structure of quantities, systems of quantities, and transformations between them. Using the torsor framework, time-dependent quantities and their unitized derivatives are also investigated. Lastly, the torsor apparatus is applied to deterministic measurement of quantities.
[1] Arapura, D. (2012). An introduction to etale cohomology. Lecture notes. https://www.math.purdue.edu/~dvb/preprints/etale.pdf
[2] Baez, J. (2009). Torsors made easy. http://math.ucr.edu/home/baez/torsors.html
[3] Clifford, A. H. (1958). Totally ordered commutative semigroups. Bulletin of the American Mathematical Society, 64, 305–316.
[4] Domotor, Z., Batitsky, V. (2016). An algebraic approach to unital quantities and their measurement. Measurement Science Review, 16 (3), 103–126.
[5] Drobot, S. (1958). On the foundations of dimensional analysis. Studia Mathematica, 14, 84–99.
[6] Emerson, W. H. (2008). On quantity calculus and units of measurement. Metrologia, 45, 134–138.
[7] Hölder, O. (1901). Die Axiome der Quantität und die Lehre vom Mass. Berichte über die Verhandlungen der Königlich Sächsischen Gesellschaft der Wissenschaften zu Leipzig, Mathematisch-Physikaliche Classe, 53, 1–64.
[8] International Organization for Standardization. (1992). Quantities and units – Part 0: General principles. ISO 31-0:1992.
[9] National Institute of Standards and Technology (NIST). (2008). The International System of Units (SI): NIST Special Publication 330. http://physics.nist.gov/Pubs/SP330/sp330.pdf
[10] Kasprzak, W., Lysik, B., Rybaczuk, M. (2004). Measurements, Dimensions, Invariant Models and Fractals. Poland, Wroclaw: Spolom.
[11] Kitano, M. (2013). Mathematical structure of unit systems. Journal of Mathematical Physics, 54, 052901-1 – 052901-17.
[12] Krantz, D. H, Luce, R. D., Suppes, P., Tversky, A. (1971). Foundations of Measurement, Volume I. Academic Press.
[13] Luce, R. D., Suppes, P. (2002). Representational measurement theory. In Stevens’ Handbook of Experimental Psychology: Third Edition. Wiley, 1–41.
[14] Maxwell, J. C. (1873). Treatise on Electricity and Magnetism. Oxford University Press.
[15] Michell, J. (1999). Measurement in Psychology: A Critical History of a Methodological Concept. Cambridge University Press.
[16] Suppes, P., Zinnes, J. L. (1963). Basic measurement theory. In Handbook of Mathematical Psychology, Volume 1. John Wiley & Sons, 1–76.
[17] Whitney, H. (1968). The mathematics of physical quantities. Part I: Mathematical models for measurement. Part II: Quantity structures and dimensional analysis. The American Mathematical Monthly, 75, 115–138 & 227–256.