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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)?

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1: What do you mean by negative dimension? Are you refering the the dimensionality of spaces (point -> 0D, line -> 1D, plane -> 2D, etc.)? I can't imagine a definition of negative dimensions in this sense which makes any sense, which is not to say it's impossible. It would have to refer to some truly bizarre mathematical object, however. You might want to ask this on the maths stackexchange. 2: Negative probability? Probabilities are only defined on the interval $0\leq p \leq 1$. Maybe you mean probability amplitudes which are not the same. –  Michael Brown Jan 25 '13 at 14:18
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See Penrose, "Applications of negative dimensional tensors," 1971. –  Ben Crowell Mar 25 '13 at 0:25
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4 Answers

up vote 10 down vote accepted

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.
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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.

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Anything that has a direction and a magnitude can be a dimension.

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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".

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There are various ways of defining the dimension of a topological space, e.g., Lebesgue covering dimension and inductive dimension. But I don't think any of these can give negative results. –  Ben Crowell Mar 25 '13 at 4:06
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