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edited Jan 4, 2013 at 16:08

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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.

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.

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.

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created Jan 3, 2013 at 20:26

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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.