M, Duality, Brane Dimensional Analysis: A Brief Note

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Posted by OsherDoctorow on June 12, 2003 at 14:41:58:

In my Fundamental Equation 4 thread, I derived a result with dimensional analysis of rather simple type, and I'd like to urge people not to shy away from dimensional analysis in branes, M Theory, Duality. As a Real Analysis and Scalar-inclined person (although dabbling in other things like reducing other things to these), dimensional analysis especially appeals to me because of its simplicity and real-valued aspects.

I am not alone in this, although the number of adherents of any viewpoint has usually nothing to do with its accuracy (unless inversely). Paul Dirac and Pascual Jordan were quite interested in dimensional analysis, as was Sir Arthur Stanley Eddington and Garrett Birkhoff and Sedov and many, many more (although the hydrodynamics and to a lesser extent electrical and chemical engineering people have been especially involved in it). For some interesting background on this, see E. de St Q. Isaacson and M de St Q. Isaacson, Dimensional Methods in Engineering and Physics - reference sets and the possibilities of their extension, Wiley: N.Y. 1975.

The real breakthrough in Dimensional Analysis which made it rather central to the Mainstream in mathematics came, however, with G. W. Bluman (U. British Columbia) and S. Kumei (Shinshu U.), Symmetries and Differential Equations, Springer-Verlag: N.Y. 1989, and the papers on which it was based. They showed that reduction of variables through dimensional analysis for partial differential equations (PDEs) is a special case of reduction from invariance under Lie groups of one-parameter and p-parameter scaling transformations (stretching transformations) in BVPs (boundary value problems), and the one-parameter transformation invariance can be generalized to arbitrary one-parameter Lie groups of point transformations of variables which can be found algorithmically for given differential equations from properties of infinitesimal generators.

Jordan in fact applied dimensional analysis to cosmology as did several other people including one who wrote a book on the latter applications (I'll have to look up his name), and the Isaacson & Isaacson reference discusses it.