I have often heard that there are more than 4 (3 space and 1 time) dimensions of spacetime. What are the theories that say so, and how many does each predict? Has any experimental evidence been conclusive? And how are these extra-dimensional theories to agree with our daily observations?

# [Physics] Total Number of Dimensions in the Universe?

compactificationspacetimespacetime-dimensionsstring-theory

#### Related Solutions

Actually, let's give this a shot. This isn't evidence **for** extra dimensions (the non-observation of extra dimensions/supersymmetry is one of the big reasons string theory is not accepted universally as true, after all), but this is an argument as to why small extra dimensions are unobservable.

Consider a particle in a box in quantum mechanics of $n$ spatial dimensions. If you do this, then Schrödinger's equation for a pure energy Eigenstate becomes (inside the box):

$$E\psi = - \frac{\hbar^{2}}{2m}\nabla^{2}\psi$$

And where you force $\psi$ to be zero everywhere outside the box, and on the boundary of the box. Using a bunch of PDE machinery involving separation of variables, we find that the unique solution to this equation is a an infinite sum of terms that look like

$$\psi=A\prod_{i=1}^{n}\sin\left(\frac{m_{i}\pi x_{i}}{L_{i}}\right)$$

where all of the $m$ are integers, and the $\Pi$ represents a product with one sine term for each dimension in our space${}^{1}$. Plugging this back into Schrödinger's equation tells us that the energy of this state is

$$E=\frac{\hbar^{2}}{2m}\left(\sum_{i=1}^{n} \frac{m_{i}^{2}}{L_{i}^{2}}\right)$$

Now, let's assume that in the first $d$ dimensions, our box has a large width $L$, while in the last $n-d$ dimensions, our box has a small width $\ell$. Then, we can split this sum into

$$E=\frac{\hbar^{2}}{2m}\left(\sum_{i=1}^{d} \frac{m_{i}^{2}}{L^{2}}+\sum_{i=d+1}^{n} \frac{m_{i}^{2}}{\ell^{2}}\right)$$

So, now we can see what's happening — if $L \gg \ell$, there is a much greater energy cost associated with moving in the more constrained or smaller $n-d$ directions than there is in moving in the less constrained $d$ dimensions — the smallest transitions cost an energy proportional to the inverse square of the size of the dimension. By making these dimensions small enough, we can guarantee that no experiment humans have done has even approached the energy threshold required to induce this transition, meaning that the portion of a particle's wavefunction associated with these extra dimensions is constrained to stay the way they are, making them unobservable.

${}^{1}$So, if $n=2$, a typical state would look something like $\psi=A\sin(\frac{2\pi x}{L_{x}})\sin(\frac{5\pi y}{L_{y}})$

In the mathematics used for physics dimensions are independent mathematical fields on which a variable can be assigned . Independent means that in the algebra used each projected on the other gives zero, is orthogonal. The example is the three dimensions we live in which are assigned orthogonal directions and the field is the real numbers on the axis.

So classical physics uses these three dimensions to model mathematically all observations on macroscopic scales, and in this formulation time is a parameter.

For very large energies it was experimentally found that the mathematics was most efficient by assigning a dimension to time and treating it as the fourth dimension whose mathematical field is an imaginary number, i.e. the real mathematical field multiplied by the square root of minus one. Thus there are four verified space dimensions as far as physics goes.

In the quantum regime, where the size of the items under observation is very small, new theories are proposed in trying to formulate an overall theory. These may have many extra space dimensions and some even time dimensions. String theories as an example. Until they are validated experimentally it is a moot question whether their extra dimensions are observable in our physical reality.

Thus, as far as physics goes, there are four dimensions over all at the moment, three of space and one of time.

## Best Answer

We are not really sure.

Classical theories are based on 4 total: 3 for the observed spatial directions, and 1 for time. However, string theory introduces a whole collection of other possible numbers. Some commonly used ones include 10, 11, or 26. Those extra dimensions are supposed to be "compactified" to explain our inability to observe them.