Acta Applicandae Mathematicae

, Volume 110, Issue 3, pp 1249–1276 | Cite as

An Algebraic Approach to Physical Scales

  • Josef Janyška
  • Marco Modugno
  • Raffaele VitoloEmail author


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).


Semi-vector spaces Scales Units of measurement 

Mathematics Subject Classification (2000)

15A69 12K10 16Y60 70Sxx 


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  1. 1.
    Abraham, R., Marsden, J.: Foundations of Mechanics, 2nd edn. Benjamin, Elmsford (1978) zbMATHGoogle Scholar
  2. 2.
    Barenblatt, G.I.: Scaling. Cambridge Texts in Applied Mathematics. Cambridge University, Cambridge (2003) zbMATHGoogle Scholar
  3. 3.
    Bogoliubov, N.N., Shirkof, D.V.: Quantum Fields. The Benjamin/Cummings, Redwood City (1983) zbMATHGoogle Scholar
  4. 4.
    Butler, Kim Ki-Hang: The number of idempotents in (0,1)-matrix semigroups. Linear Algebra Appl. 5, 233–246 (1972) zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Canarutto, D.: Possibly degenerate tetrad gravity and Maxwell-Dirac fields. J. Math. Phys. 39(9), 4814–4823 (1998) zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Canarutto, D.: Two-spinors, field theories and geometric optics in curved spacetime. Acta Appl. Math. 62(1), 187–224 (2000) zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Canarutto, D.: “Minimal geometric data” approach to Dirac algebra, spinor groups and field theories. Int. J. Geom. Met. Mod. Phys. 4(6), 1005–1040 (2007) zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Canarutto, D., Jadczyk, A., Modugno, M.: Quantum mechanics of a spin particle in a curved spacetime with absolute time. Rep. Math. Phys. 36(1), 95–140 (1995) MathSciNetGoogle Scholar
  9. 9.
    Cohen-Tannoudji, C., Diu, B., Laloë, F.: Méchanique Quantique, vols. I–II. Collection Enseignement des Sciences, vol. 16. Hermann, Paris (1977/1986) Google Scholar
  10. 10.
    Darton, M., Clark, J.O.E.: The Dent Dictionary of Measurement. Dent, London (1994) Google Scholar
  11. 11.
    Duff, M.J., Okun, L.B., Veneziano, G.: Trialogue on the number of fundamental constants. ArXiv:physics/0110060v3 [physics:clas-ph], 13 Sep. 2002
  12. 12.
    Gähler, W., Gähler, S.: Contributions to fuzzy analysis. Fuzzy Sets Syst. 105, 201–224 (1999) zbMATHCrossRefGoogle Scholar
  13. 13.
    Golan, J.S.: Semirings and Their Applications. Kluwer Academic, Dordrecht (1999) zbMATHGoogle Scholar
  14. 14.
    Greub, W.: Multilinear Algebra, 2nd edn. Springer, New York (1978) zbMATHGoogle Scholar
  15. 15.
    Hebisch, U., Weinert, H.J.: Semirings—Algebraic Theory and Applications in Computer Science. World Scientific, Singapore (1993) zbMATHGoogle Scholar
  16. 16.
    Husemoller, D.: Fibre Bundles. GTM, vol. 20. Springer, New York (1975) zbMATHGoogle Scholar
  17. 17.
    Janyška, J., Modugno, M.: Covariant Schrödinger operator. J. Phys.: A, Math. Gen. 35, 8407–8434 (2002) zbMATHCrossRefGoogle Scholar
  18. 18.
    Janyška, J., Modugno, M.: Hermitian vector fields and special phase functions. Int. J. Geom. Meth. Mod. Phys. 3(4), 1–36 (2006) Google Scholar
  19. 19.
    Janyška, J., Modugno, M.: Geometric structure of the classical relativistic phase space phase functions. Int. J. Geom. Meth. Mod. Phys. 5(5), 1–56 (2008) Google Scholar
  20. 20.
    Janyška, J., Modugno, M., Vitolo, R.: Semi-vector spaces and units of measurement. Commutative Algebra (math.AC), 5 Oct. 2007. arXiv:0710.1313
  21. 21.
    Jerrard, H.G., McNeill, D.B.: A Dictionary of Scientific Units, 6th edn. Chapman and Hall, London (1992) Google Scholar
  22. 22.
    Ketov, S.V.: Conformal Field Theory. World Scientific, Singapore (1995) zbMATHGoogle Scholar
  23. 23.
    Matzas, G.E.A., Pleitez, V., Saa, A., Vanzella, D.A.T.: The number of dimensional fundamental constants. ArXiv:0711.4276v2 [physics. class-ph], 4 Dec. 2007
  24. 24.
    Misner, C.W., Thorne, K.S., Wheeeler, J.A.: Gravitation. Freeman, New York (1973) Google Scholar
  25. 25.
    Modugno, M., Saller, D., Tolksdorf, J.: Classification of infinitesimal symmetries in covariant classical mechanics. J. Math. Phys. 47, 1–27 (2006) CrossRefMathSciNetGoogle Scholar
  26. 26.
    Olver, P.J.: Applications of Lie Groups to Differential Equations, 2nd edn. Springer, New York (1991) Google Scholar
  27. 27.
    Panofsky, W.K.H., Phillips, M.: Classical Electricity and Magnetism. Addison-Wesley, Reading (1956) Google Scholar
  28. 28.
    Pap, E.: Integration of functions with values in complete semi-vector space. In: Measure Theory, Oberwolfach 1979. Lecture Notes in Mathematics, vol. 794, pp. 340–347. Springer, Berlin (1980) CrossRefGoogle Scholar
  29. 29.
    Prakash, P., Sertel, M.R.: Topological semivector spaces, convexity and fixed point theory. Semigroup Forum 9, 117–138 (1974) zbMATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Prakash, P., Sertel, M.R.: Hyperspaces of topological vector spaces: their embedding in topological vector spaces. Proc. AMS 61(1), 163–168 (1976) CrossRefGoogle Scholar
  31. 31.
    Saller, D., Vitolo, R.: Symmetries in covariant classical mechanics. J. Math. Phys. 41(10), 6824–6842 (2000) zbMATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Takahashi, M.: On the bordism categories III. Math. Sem. Notes Kobe Univ. 10, 211–236 (1982) zbMATHMathSciNetGoogle Scholar
  33. 33.
    Uzan, J.P.: The fundamental constants and their variation: observational status and their theoretical motivations. ArXiv:hep-ph0205340v1, 30 May 2002
  34. 34.
    Vitolo, R.: Quantum structures in Galilei general relativity. Ann. Inst. Henri Poincaré 70 (1999) Google Scholar
  35. 35.
    Vitolo, R.: Quantum structures in Einstein general relativity. Lett. Math. Phys. 51, 119–133 (2000) zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  • Josef Janyška
    • 1
  • Marco Modugno
    • 2
  • Raffaele Vitolo
    • 3
    Email author
  1. 1.Department of Mathematics and StatisticsMasaryk UniversityBrnoCzech Republic
  2. 2.Department of Applied MathematicsFlorence UniversityFlorenceItaly
  3. 3.Department of Mathematics “E. De Giorgi”LecceItaly

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