Non-integer value of dimensions?

[jmvizanko]

Given that string theory is built on the idea of one-dimensional entities, which seems much too "nice" given the general fuzziness of interpreting quantum mechanics, would it be possible for a universal theory to be based on a non-integer number of dimensions? I basically know nothing of mathematical formalism, but it was just a thought I had, and I was wondering if it even made any sense or could even be theoretically possible?

 

[MathematicalPhysicist]

Theoretically everything is possible, at the end we need to test our theories with the physical reality.

 

[handrik]

Mathematically, the dimension of a vector space is the number of basis elements it has. This definition doesn't allow non integers

So you'll need a new definition for dimension

 

[Dmitry67]

No, see http://en.wikipedia.org/wiki/Hausdorff_dimension

 

[HallsofIvy]

A vector space must have integer dimension but it is possible to have subsets of Rⁿ with fractional dimension. See http://mathworld.wolfram.com/HausdorffDimension.html

 

[tom.stoer]

There are models of quantum gravity indicating that the so-called spectral dimension of spacetime is (approximately) equal to 4 for large distances, but 2 for small distances. This spectral dimension can be defined by a random walk or the "diffusion" of particles on discrete structures like foam.

Usually the solution to a diffusion problem depends on the dimension of spacetime. But one can turn things round and describe a diffusion process on a discrete structure w/o ever referring to its dimension. Then, instead of using the dimension as input, it can be extracted from certain properties of the diffusion process.

One model which indicates this spectral dimension 2 < dim < 4 is the Causal Dynamical Triangulation approach for quantum gravity. For large distances it seems that an ensemble of particles moves in 3-dim. space, whereas closed to Planck scale it seems that space becomes the a 1-dim. real line

 

[haael]

Fractals aside, the physical reality may be described by something else than a simple vector space. We may live in noncommutative geometry where dimensions may be mixed. Or there may exist "normal" dimensions where vectors have numerical components and "special" dimensions where vectors have operator-valued components. One such theory is supersymmetry, which can be formulated in such a vector space that has several "normal" dimensios and some dimensions where vectors have c-number components.