I was playing around with the formula for the volume of an $n$-dimensional sphere, and out of curiosity, I tried plugging in negative values for the dimension $ n $. Surprisingly, the math still works (thanks to analytic continuation), but the behavior gets weird:
Negative dimensions "invert" scaling: A (−1)D "sphere" has volume $∝ 1/$ (shrinking when you enlarge it!).
Dimensions like $ n = -2, -4, \dots $ give zero volume—almost like they’re "collapsed".
This made me wonder:
String theory connection? Could negative dimensions describe compactified dimensions (e.g., the "zero volume" cases matching Planck-scale curled-up dimensions)?
Duality analogy? The inverse scaling reminds me of $T$-duality (where small ↔ large radii swap roles). Is this just a coincidence, or is there a deeper link?
Quantum/statistical mechanics? Could negative dimensions model some kind of "anti-scaling" in renormalization flows?
I’m not asking for rigorous math (I know negative dimensions are exotic!), but rather:
Has this idea been explored before in physics? (e.g., in string theory, QFT, or even condensed matter?)
Is there a plausible physical interpretation? Or is this just a cute math trick with no real-world relevance?
(P.S. I’m aware this might sound speculative — I’m just curious if there’s any prior work or ideas here!)
References:
Standard $n$-sphere volume formula https://en.wikipedia.org/wiki/Volume_of_an_n-ball
String theory compactification (e.g., Polchinski)
