Could negative dimension ever make sense?

After some quick check I found that negative dimensions are not used. But we have negative probability, negative energy etc. So is it so likely that we won't ever use negative dimension(s) ?

Update

I understand there're also dimensions that are not integers e.g. dimension 1½ (?) for fractals or so. Could there also be a dimension such as dimension i (imaginary)?

Answer

The notion of negative dimension has appeared in various places of modern physics. For instance:

1. 
   

   Grassmann-odd variables. Recall that the dimension ${\rm dim}(V)$ of a group representation $\rho: G \to GL(V)$ is given by the trace ${\rm dim}(V)={\rm Tr}(\rho(1))$ of the identity element. For a supergroup, one should use the supertrace, so Grassmann-odd directions can in some sense be viewed as having negative dimension. See also e.g. Ref. 1.

   
   


2. 
   

   K-theory, which is relevant for e.g. string theory and integer quantum Hall effect. Via the Grothendieck group construction for the commutative monoid of vector bundles, it is possible to make sense of how to subtract a vector bundle.

   
   

References:

1. G. Parisi and N. Sourlas, Random Magnetic Fields, Supersymmetry, and Negative Dimensions, Phys. Rev. Lett. 43 (1979) 744.