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) ?
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)?
The notion of negative dimension has appeared in various places of modern physics. For instance:
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.
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:
Dimension of a (finite dimensional) vector space is defined as the cardinality of a basis for the vector space. Since the cardinality cannot be negative, negative dimension for vector spaces is meaningless. The same holds for manifolds, because they are locally defeomorphic to vector spaces. However, if you consider dimension as the value of some sort of integration which, in vector space case, coincides with the above definition, then a negative dimension is possible (for example, you can use all types of measures for integration, negative, complex, etc). But it is certainly a misuse of the word "dimension".
There are algebraic stacks, which generalize algebraic varieties, and differentiable stacks, which generalize smooth manifolds. Each variety or manifold can be considered as a stack, and its dimension as stack is the same as its dimension as variety/manifold. But there are many stacks which don't correspond to varieties/manifolds, and some of these have negative dimension.
Specifically, if $V$ is a variety/manifold, and $G$ is an algebraic/Lie group acting on $V$, then we can form the quotient stack $[V/G]$, and we have $$\dim(V/G)=\dim(V)-\dim(G)$$ which may well be negative.
The infinite lattice is a fractal of negative dimension: if you scale the infinite lattice on a line 2x, it becomes 2x less dense, thus 2 scaled lattices compose one non-scaled. If you take a lattice or on a plane, scaling 2x makes it 4x less dense so that 4 scaled lattices compose one non-scaled, etc.
One way to look at it is that in any space, the magnitude of any volume element changes in proportion to the magnitude of a length inside that volume element, raised to the power of the dimension of the space containing the volume element. Alternatively, $\log(v)$ is proportional to $d \log(l)$; where $v$ is the magnitude of the volume element, $l$ is the magnitude of the length considered within the volume element, and d is the dimension of the space containing the volume element. Thus we have $d = C \log(v)/\log(l)$, where $C$ is some arbitrary constant.
Given that kind of definition of dimension it is possible to contemplate fractional dimension spaces, but I don't know what to make of the idea of a negative dimensional space.
Using modules (e.g. vector spaces), or abelian groups, or any object which has a notation of rank / dimension, there is yet another concept of negative dimension, not in the original spaces but in their Homology.
Take a "mixed complex" of both traditional (forward) and reverse sequences of chain maps. A sequence $M_0 \xrightarrow{d_0} M_1 \xrightarrow{d_1} M_2$ is called "reverse" if instead of $\operatorname{im}d_0 \subset \ker d_1$ we have $\ker d_1 \subset \operatorname{im} d_0$. Therefore a reverse exact sequence is the same thing as an exact sequence, which is like saying $-0 = 0$ when speaking of the dimensions of the $n$th homology group.
If the homology groups have a concept of dimension, we define the strict reverse homology groups to have negative dimension.
You can have complexes with mixed forward and reverse homology. Any sequence of surjections and any sequence of injections, while they may not have interesting or even defined forward homologies, they will always have reverse homology. So there is an application. That is, you can usually apply it when traditional homology is not defined, or in other words where the sequence maps $d$ are not such that $d^2 = 0$.
