Analogy between scale symmetry and relativistic mechanics. I. Lagrangian formalism

Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics

Volumn 56, Issue 6, 1997, Pages 6427-6434

  Graner, F ^a   Dubrulle, B ^b,c  

^a LIPHY   (France)

^b CNRS   (France)

^c OBSERVATOIRE MIDI PYRÉNÉES   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0041318196     PISSN: 1063651X     EISSN: None     Source Type: Journal    
DOI: 10.1103/PhysRevE.56.6427     Document Type: Article

References (25)

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7. A first immediate consequence of this postulate (not a supplementary hypothesis, as is sometimes believed), is that physical quantities must be expressed as products of monoms (power laws) of physical dimensions: e.g., (mass)×(length)(Formula presented)×(time)(Formula presented) or (area)(Formula presented)×(time)(Formula presented) is acceptable, while exponentials, sums, or other combinations of monoms are not.
8. The original meaning of “gauge invariance” is invariance of the gauge used to define the unit of a measurement. Weyl coined this term, referring to the possible variations of a measuring rod (or clock period) with respect to the position, and hence its nonintegrability along a path in a curved space; see H. Weyl, Raum, Zeit, Materie, 3rd ed. (Teubner, Leipzig, 1920), Chaps. II and IV, p. 242ff, and references therein;
9. Phys. Z. 21, 649 (1920). PHZTAOThis theory was invalidated by experiments, as Weyl himself recognized later [H. Weyl, Gruppentheorie und Quantenmechanik (Teubner, Leipzig, 1928)];W. Pauli, Theory of Relativity (Pergamon, London, 1958). However, in quantum mechanics, the same term was used again for a formally analogous problem, namely, the wave equation for an electrically charged particle. Here we come back to the original meaning of “gauge” as a measuring rod, but consider its variation with scale and not with position.
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15. B. Dubrulle (unpublished).
16. the physical quantity is the intermittency function (Formula presented); the ratio of resolutions is a constant number, hidden and incorporated in the value of (Formula presented).
17. In fact, all equations in the text can equivalently be written using only (Formula presented) e.g., (Formula presented)
18. B. Dubrulle, F.-M. Bréon, F. Graner, and A. Pocheau, Phys. Rev. Lett. (to be published).
19. showed that, for a certain class of problems, all scales must be renormalized together, and that the interaction between scales is nonlocal. This does not apply to isotropic turbulence, but could apply, e.g., to turbulent front analysis. This situation is similar to quantum relativistic theory, and would necessarily involve backwards running time, i.e., values of (Formula presented) or (Formula presented) equal to or less than (Formula presented), as in mechanics; see, e.g., the necessity for antiparticles in R. Feynman, Elementary Particles and the Laws of Physics (Cambridge University Press, Cambridge, 1987).
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