Question

How to define *dynamical* dimensions?

Answer 1

I'm considering a simple toy model. The spacetime is flat with dd space dimensions. Using cartesian coordinates, the spacetime metric is Minkowskian :

ds2=dt2−dx21−dx22−dx23−…−dx2d.(1)(1)ds2=dt2−dx12−dx22−dx32−…−dxd2.

A massless scalar field ΦΦ is propagating in that spacetime according to the following PDE :

∂2Φ∂t2−∂2Φ∂x21−∂2Φ∂x22−∂2Φ∂x23−…−∂2Φ∂x2d=0.(2)(2)∂2Φ∂t2−∂2Φ∂x12−∂2Φ∂x22−∂2Φ∂x32−…−∂2Φ∂xd2=0.

Of course, d≥1d≥1 is just any integer. But what if it becomes a dynamical variable, dependant on the value of ΦΦ? I'm not considering an extension to real values ; ddwould still be an integer, but that could vary from place to place in spacetime : d=1d=1 in some patch of spacetime while d=2d=2 or d=3d=3 in other parts of spacetime. Matching the patches boundaries could be done a bit like a thickless string (d=1d=1) smoothly getting a thickness (cylindrical surface ; d=2d=2), then getting a bulk (d=3d=3), etc, but I'm not sure this is actually making any sense.