How to define *dynamical* dimensions?

I'm considering a simple toy model. The spacetime is flat with $d$ space dimensions. Using cartesian coordinates, the spacetime metric is Minkowskian : $$\tag{1} ds^2 = dt^2 - dx_1^2 - dx_2^2 - dx_3^2 - \ldots - dx_d^2.$$ A massless scalar field $\Phi$ is propagating in that spacetime according to the following PDE : $$\tag{2} \frac{\partial^2 \Phi}{\partial \, t^2} - \frac{\partial^2 \Phi}{\partial \, x_1^2} - \frac{\partial^2 \Phi}{\partial \, x_2^2} - \frac{\partial^2 \Phi}{\partial \, x_3^2} - \ldots - \frac{\partial^2 \Phi}{\partial \, x_d^2} = 0.$$ Of course, $d \ge 1$ is just any integer. But what if it becomes a dynamical variable, dependant on the value of $\Phi$? I'm not considering an extension to real values ; $d$ would still be an integer, but that could vary from place to place in spacetime : $d = 1$ in some patch of spacetime while $d = 2$ or $d = 3$ in other parts of spacetime. Matching the patches boundaries could be done a bit like a thickless string ($d = 1$) smoothly getting a thickness (cylindrical surface ; $d = 2$), then getting a bulk ($d = 3$), etc, but I'm not sure this is actually making any sense.

How could we make sense of this idea of a dynamical spacetime dimension, and define the dynamics of $d$ now considered as a kind of a new scalar field? (constrainted on integers ?).

Introducing some variable scale factors, I was thinking about an infinite metric like this : $$\tag{3} ds^2 = dt^2 - a_1^2(t, x)\, dx_1^2 - a_2^2(t, x)\, dx_2^2 - a_3^2(t, x)\, dx_3^2 - a_4^2(t, x)\, dx_4^2 - \ldots,$$ where $a_i(t, x)$ are new dynamical variables and $i = 1, 2, 3, \dots, \infty$. In a one dimensional space ($d = 1$), $a_i = 0$ for all $i = 2, 3, \dots, \infty$. If space is two dimensional ($d = 2$), then $a_i = 0$ for all $i = 3, 4, \dots, \infty$, etc.

I'm "visualizing" these dynamical dimensions as an infinity of closed loops ; $0 \le x_i < 2 \pi \ell$, all fluctuating at a tiny scale ($a_i$ randomly pulsating and oscillating, such that $a_i \approx 0$ for most $i > 1$), then suddenly blowing up if the scalar field $\Phi$ gets a special high value, locally or globally. When the spacetime is getting (or losing) some new (old) local space dimension, it would define a dimensional phase transition.

How could we obtain the Einstein (or FLRW) equations for such an infinite metric like (3) above, especially if $a_i$ depends on $t$ only? Was this idea already considered before?

• Your idea intrigues me, but I think you are making things less elegant and that's not a good sign. Personally, the only case where I see the dimensionality of spacetime as a variable is in string theory where analyzing the movement of the string in the D-dimensional flat spacetime is equivalent to considering D free scalar fields in a 2-dimensional (target) spacetime. Fixing the gauge freedom then yields the critical number of such scalar fields, i.e. the dimensionality D. The bad news are that this isn't a place to share your personal ideas and future research but I hope this helps you a bit! – Panos C. Jul 8 '18 at 14:42
• AFAIK, string theory uses a finite number of dimensions ($d = 10, 11$ or $26$ ?). The idea I'd like to explore is an infinite number of variables (i.e. dynamical) dimensions. This idea appears to be very appealing to me, as a "natural" extension of classical General Relativity. I'm intrigued by the integer nature of spacetime dimensions , and I "see" a kind of analogy with the radial quantum number of the atomic (hydrogen) spectra. Energy $E_n$ is higher for large values of $n = 1, 2, 3, \dots, \infty$. In our spacetime ; $d = 3$, the number of dimensions is pretty "low energy". – Cham Jul 8 '18 at 14:52
• General Relativity works just fine in any number of dimensions, but it has some very special properties in D=4 that make the Universe... habitable. In any case, you can always work in D=p+q dimensions (p spatial and q temporal dimensions) with the aspects of General Relativity still working perfectly fine. Having an infinite number of dimensions yields a lot of zeros and infinities. Take for example the Newtonian gravitational potential. It would be identically zero in an infinite number of dimensions and gravity wouldn't matter anymore. – Panos C. Jul 8 '18 at 14:56
• @PanosC., usual GR assumes that the spacetime dimensions are fixed, i.e non-dynamical. Why a fixed $d$ in the most general "universe" ? Where the value of $d$ comes from ? I don't believe that there are any true universal constant in Nature, only interelated dynamical quantities. Of course, this is pure speculation, but then this is still physics. I'm wondering if there are any studies/theories/publications about an infinite number of dynamical dimensions. – Cham Jul 8 '18 at 15:00

With that said, if all you care about is field theories, why not instead try to mock up something where the space is higher dimensional but the field simply has regions where it does not vary with $x_1$?
• From the metric (3) in my question, I already consider an higher manifold (of infinite dimensions), on which the scale factors $a_i$ could vary. That manifold is a kind of "ancilla", or a trick to realize the dynamical dimensions. – Cham Jul 8 '18 at 16:05