What experiment could determine whether our universe is fractal?
Your question can't be answered. Fractal isn't a well-defined property. Fractal vaguely implies self-symmetry, but this isn't good enough to give a yes or no answer.
From what I understand of fractional dimensions, they are such that there is not complete freedom of movement in them (i.e. their freedom of movement is a fraction of 1). This means there are discrete positions, not a continuum.
There should in principle be a test that could show that an object could not move continuously through that dimension, but would move in quantized jumps.
You mean like quantum states for example? Oops sorry, just re read and see you said quantized jumps that's what I was thinking of. So since energy is quantized then is that evidence that there are fractional dimensions?
This thread has discussion of an actual model where reality turns fractal at the planckscale. May help you to visualise the issues.
1] I'm just hypothesizing. None of this comes close to constituting evidence.
2] My supposition is that a fractal universe would require a quantized spatial dimension. Don't see how energy being quantized would apply.
Take a really drunken guy, suppose he can walk in all directions possible, and measure how long it takes for the guy to drift away from a starting point. That is related to the number of dimensions. Some people even call it "spectral dimension".
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