Here's another approach I thought of a very long time ago, fwiw: Let C = R/Z be the circle group. For any d in Z+, the element [1/d] in C generates a subgroup <[1/d]> of C. Then, R^d can be thought of as the set of all functions f: <[1/d]> -> R . Since the above makes sense for any d in R, not just Z+, we can define R^d for any d in R as the set of all functions f: <[1/d]> -> R once again. Only now if d is irrational then <[1/d]> is a countable dense subgroup of C. --Dan
On Dec 24, 2014, at 2:48 PM, Mike Stay <metaweta@gmail.com> wrote:
Here's another approach to fractional dimensions.
Suppose you have a finite set X of states that a physical system can be in. Each state has a particular energy. The partition function Z(T) = sum_{x in X} exp(-H(x)/kT) gives us the normalization factor at a given temperature. The probability of being in the state x is described by the Gibbs distribution p(x) = exp(-H(x)/kT) / Z(T). When the temperature is infinite, every state is equally likely. If there are n states, the system behaves something like a particle in n dimensions: Z(∞) is n. When the temperature cools, high-energy states become exponentially unlikely, reducing the effective dimensionality of the system. Thin films allow low-energy electrons to move in two dimensions, but high-energy electrons can fly right off the film. Z(T) decreases continuously from n down to 1 (only the ground state is occupied) as T goes down from infinity to zero.