Exploring Probability Spheres in Fractional Dimension

[bchui]

I have heard of such idea:

A sphere of fractional dimension 0<s<1 is understood as a probability sphere with probability s to have an electron at a certain position

for example the volume of the sphere S^{n-1} in \Re^n has volume

Vol(S^{n-1})= (2\Pi^{n/2})/(\Gamma(n/2)

and we can apply the result to non-integer values of n

Anyone have heard of this idea and show me the link for further information?

 

[leonhardeuler1]

I have been searching for the same thing, and a few places say that it is given by:

\Gamma^2(1/2)/\Gamma(n/2)

for any dimension, even fractional ones, but I am trying to find a way to prove it myself, perhaps using integration with respect to the Hausdorff measure (since it recognizes non-integer dimensions). And just a note- the formula you gave is actually the area measure of the unit S^{n-1} sphere, the volume of the unit S^{n-1} sphere is actually:

Vol(S^{n-1})=[2\pi^{(n-1)/2}]/[\Gamma((n-1)/2+1)],

**Note that the denominator can be rewritten {(n-1)/2}\Gamma((n-1)/2)}.

 

[bchui]

The proof for integer [tex]n[/tex] is simple and done by induction. It could be found for example in

Chapter 5.9 of W Fleming: "Functions of Several Variables", Springer-
Verlag 1977

We generalize [tex]n![/tex] to [tex]\Gamma(n+1)[/tex] and the formula is obtained.
My problem is the physical aspect. What is the physical mean of a "fractional sphere" and could it possibly be related to "probability sphere" in Quantum Mechanics?