Who can explain the proof of the formula (2.12) given here: J. Phys. A: Math. Gen. 20 (1987) 3861-3875. Printed in the UK http://ru.scribd.com/doc/118425928/Svozil-Quantum-Field-Theory-on-Fractal-Space-time It is easy to prove that in general case this formula is wrong!!!
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4$\begingroup$ Welcome to MO! Both your questions are very brief. For the former one this still somehow worked fine. For the present one I am a bit worried that this is too vague and/or broad a question to admit a reasonable answer (in the restricted format of this website). Thus, I would like to ask you to include some more details; for example, the context in which this question arose. For advice on how to write questions for this website efficiently see 'how to ask' for general information on the site 'faq' (links at the top). Please, also note that you can expand the existing question (use 'edit'). $\endgroup$– user9072Commented Jan 3, 2013 at 19:39
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2$\begingroup$ Who can in details explain the proof of the formula (2.12) given here: J. Phys. A: Math. Gen. 20 (1987) 3861-3875. ru.scribd.com/doc/118425928/… $\endgroup$– Jaykov FoukzonCommented Jan 3, 2013 at 19:54
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1$\begingroup$ This is just the integral of a spherically symmetric function $s$ in polar coordinates. (2.10) and the discussion preceding it spell this out. $\endgroup$– Steve HuntsmanCommented Jan 3, 2013 at 20:14
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As Steve Huntsman points out in his comment this is just writing out the integral against the Hausdorff measure on the fractal domain in polar coordinates over the same fractal. Since the author does not specify what the fractal domain actually and merely asserts that it is distributed throughout $\mathbb{R}^{4}$ such that it is "regular and homogenous" with respect to the Hausdorff measure he is able to claim that for the right sort of fractal domain the formulas that we are used to when integrating over $\mathbb{R}^{d}$ hold even when $d$ is not an integer. That is what looks like the tricky part to me, just assuming the existence of such a fractal. The discussion of why the coefficient is what it is appears just after (2.6) and arises simply by analogy to the volume of the unit sphere that usually appears in polar coordinate transforms.
ADDENDUM (4 Jan 2013): The formula for the volume of the unit sphere that the author uses is wrong if you take it out of the context that the author takes for granted. And the author never makes the claim that it should. The logic of the paper is: given a fractal subset of $\mathbb{R}^{4}$ such that this formula for integration in polar coordinates and the volume of the unit sphere holds then the following physics can be done. It is not fair to the author to complain that when you take a fractal such that his assumptions don't hold that the formula is wrong. That being said, I still wish he had actually exhibited one of these fractals.
answered Jan 3, 2013 at 20:26
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$\begingroup$ BSteinhurst.I agree. It is easy to prove that in general case this formula is wrong $\endgroup$ Commented Jan 4, 2013 at 4:12
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$\begingroup$ Sometimes wrong formulas are acceptible in physics papers. Strange as it seems to mathematicians. $\endgroup$ Commented Jan 4, 2013 at 14:30