Why are extra dimensions necessary? Some theories have more than 4 dimensions of spacetime. But we only observe 4 spacetime dimensions in the real world, cf. e.g. this Phys.SE post.


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*Why are the theories (e.g. string theory) that require more dimensions taken seriously by scientists? 

*Is there any proof that these extra dimensions exist?

*Is there a simple layman's explanation for the need [or strong hint] for extra dimensions?
 A: Let's take your questions in turn


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*Theories that have more dimensions are taken seriousely, because their predictions match the experimental evidence. Of course, the fact that we live in only 4 dimensions constrains such ideas (although the idea of another large dimension is not ruled out by our direct non-observation of it (cf. Flatland, where a 3D Object enters the 2D world), but by the fact that we would have to observe scalar (w.r.t. our 4D Lorentz group) partners of all particles). Nevertheless, theories with compact extra dimensions may explain supersymmetry breaking (if SUSY exists), the fact that we have 3 generations of matter and why the generations have so different masses, or the reason why gravity is weak. One key predictions of compact etra dimensions is a "tower of exitations" of the particles, with mass splittings that depend on the size of the extra dimension. GUTs in extra dimensions predict a rather "small" lifetime of the Proton, so these can be tackled by Hyper-Kamiokande.

*These extra dimensions could be "proven" to exist, if a theory with extra dimensions was able to explain discrepancies within the Standard Model of Particle Physics and/or the Standard Model of Cosmology. Then again, one had to derive a distinct prediction from the theory that can be experimentally falsified. Proving a theory (or a feature thereof) is a hard thing to do, especially if the feature is so general such as an extra dimension.

*There is actually no need or strong hint for extra dimensions. One nice reason would be that one can have a Grand Unified Theory where the symmetry breaking at the high scale occurs through the fact that the extra dimension is compact, rather than having a spontaneous symmetry breaking where a scalar field develops a vacuum expectation value. The spontaneous braking at two different scales (GUT scale and electroweak scale) would then introduce a lot of questions as to the relation of the two scales and the theory would potentially become unstable. More reasons to look at extra dimensions have already been mentioned in 1.

A: 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 $\psi=A\sin(\frac{2\pi x}{L_{x}})\sin(\frac{5\pi y}{L_{y}})$
A: In an abstract sense, a "dimension" is just a component of a state vector.  For example, one might talk about a 10-dimensional phase-space consisting of 3 components for position, 3 for linear momentum, 3 for angular momentum, and 1 for energy.  Or one might have an "event" vector which includes an additional dimension representing time.
There are good reasons to believe that there is no 4th spatial dimension completely analogous to the 3 spatial dimensions that we are familiar with: if there were any way to move perpendicularly to space, then this would be happening all the time as a result of interacting with any object that was already moving in that direction.  For example, consider that a 4-body system (gravitational or electromagnetic) will never stay within a plane once disrupted because it is an unstable equilibrium. Perhaps such a 4th dimension exists, but it would have to have either a different topology, or there would have to be some sort of restorative force which keeps us confined to our hyperplane.  The latter case is illustrated by a pool table -- there is a third dimension perpendicular to the table but the balls are glued to the table because of gravity and the counteracting force is provided by the table itself.  There is an excellent book called Flatland that you can download for free which addresses these issues in an intuitive and accessible way.
A: The short answer is: there is no proof (that is no experimental evidence) so far.
The main reason for considering theories with additional dimensions is that (many) theories that are complicated in 4D can be reformulated in simpler terms as a theory with additional dimensions, which are rolled up in tiny circles (or more generally tiny manifolds) so that we don't experience them as the other "big" dimensions (called "uncompact"). What is meant by "simpler" is that, for example, a theory with only one (vector or tensor) field (think particle) in higher dimensions manifests itself as several fields of different kinds in lower dimensions, and their complicated interactions are described geometrically by the shape of the compactification manifold. In physics people like geometrisation as one can argue that it is more intuitive.
In trying to formulate a theory that describes particle interactions accurately, one is faced with many possibilities and those that can be formulated with extra dimensions are somewhat simpler. So this is often used as a guiding principle for formulating a correct theory - that is a theory which isn't contradicted by experiments. There are several examples that fullfill these requirements. But it might turn out that none of these theories (with extra dimensions) will survive when more experimental data will be gathered and compared to the predictions of these theories.
A: Why superstring theory needs $9+1$ spacetime dimensions? is indeed a very good and fundamental question to ask. Unfortunately, it is very hard to answer this question using only intuitive layman arguments. 
The culprit is the concept of a (quantum mechanical) anomaly. In general, the presence of anomalies would render the quantum version of any classical theory$^{1}$ mathematically inconsistent. 
It turns out that the anomaly cancellation conditions for (quantum) string theory are extremely restrictive. One of their consequences are that flat-spacetime-solutions of (perturbative, quantum) superstring theory must be $9+1$ dimensional.
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$^{1}$ The term classical theory here means a theory where Planck's constant $\hbar=0$ is zero. The classical version of string theory can live in any spacetime dimension.
A: similar meta-physical question came up on skeptics.stackexchange
From a physicists point of view theories can only experimentally falsified. As stated in the link atom lasers might be able to measure directly properties of space. Currently one can only indirectly prove, that the concept of spacetime as a real physical entitiy/object fits measurement data (e.g. gravitational waves from Doppelpulsars predicted by ART). But u cant measure space like using a atomic force microscope to "touch" atoms
It remains currently a meta-physical question whether u assume space is just a describing mathematical concept or a real-world object. Protophysics argues, that geometry is just a set of math. tools to describe universe, spacetime is no object.
A good question might be, can u describe a 4dim spacetime with a more dimensional spacetime consistently, is there some kind of mathematical redundance? Or is the math. concept of dimensionality so unique, that it would force very specific properties of the physics we try to model within it. This is kind of Ockhams Razor principle. An physical explanation should be as simple as possible. Currently these multi-dim theories of everything (TOE) need this additional dimensions to model the physics they try to describe, but they also give out alot of not falsifiable solutions, the more, the more dimensions they use afaik. So this is the borderline between laboratory physics, meta-physics/philosophy. U cant really logically argue here without ad-hoc hypothesis and subjective assumptions. 
A: *

*Because of the hope of quantising gravity and unifying it with other forces. The first attempt to unify forces with small extra dimensions was Kaluza–Klein theory, which added a single dimension to unify gravity with electromagnetism. String theory can also include the strong and weak nuclear interactions.

*There's nothing empirical yet, but there are numerous proposals for how we could get it, depending on these dimensions' size. There is a nice theoretical benefit of them: they fold into a manifold, and the shape and size of the holes in that manifold respectively determine the laws of physics and the constants in them, so in principle string theory can explain all of physics just in terms of the shape these dimensions take (which theorists usually expect to be a Calabi-Yau manifold). Sadly, we're a long way from knowing what shape it is.

*This page can be summarised as, "there's no easy explanation, but I'll make several attempts at half-easy ones". Early string theories only considered bosons, and required $26$ dimensions; later string theories are supersymmetric to incorporate fermions, and require $10$. The above link concentrates on reasons $26$ is special in the first case; slight variations on the same arguments explain $10$ in the second case.

A: 3+1 only describes the empty 'stage' where things happen.
An empty stage is insufficient to describe the world.
As an answer the above two lines appears to be enough, but I will say a little more.
It seems so difficult to prove, as to disprove, your radical approach. 
If anyone could prove that 3+1 is sufficient this question was not here. I think that you are presuming that 3+1 is all the story. Presuming is no good.
The objective of physics is to explore the better representation of the world and sometimes it is needed to 'complicate' a little to see clearer. 
Kaluza-Klein theory has one more dimension: charge. This has to be seen as a property inherent to the 'actors' (ultimately field), and to me this kind of representation makes sense.
How sure we are that 3+1+1 is not enough? I think that this path has more to explore before I can consider going to higher dimensions, but this is no more than a personal feeling.
As an example: Electromagnetism happens in the 3+1 world (and the set of 'real' numbers is enough: real amplitudes, real freqs, real phases, etc..) but the mathematical representation is much more convenient when we use 'complex' numbers instead of 'reals'.
I'm not talking in defense of any 'strangy' theory, but it seems unwise to be so radically simplistic and say 3+1 is sufficient. 
A: Your line of arguments is good.
Language is very confused thing.
That's why ->
There are many different aspects of what you call Dimension both in physics and mathematics.
In mathematics dimension is defined as a part of Descarte's method of coordinates.
In physics on the other hand dimension is defined as ability to TRAVEL.
So in my room i can go back and forth but in time i can go only forth, hence strictly speaking time is not a physical dimension EITHER. But! In mathematics since time can be used as a parameter it can easily become a dimension.
So. In theoretical physics we are in 3+1 dimensions, in mathematics it can be anything, and in real life only 3.
Discovering new dimension is very tricky thing - first you need a GOOD theory which will allow you to build an experiment which will show that something CAN travel in 4th or 5th dimension.
A: Extra dimensions are not "necessary" as the question phrases it, but rather can a higher dimensional theory better explain some of the unresolved anomalous cosmologocal observations, such as accelerating expansion and anomalous galactic rotation, without resorting to the invention of magical mystery matter?
For example, if time, like space, was 3 dimensional (t1 t2.t3) but we were only aware of its magnitude t = root(t1^2 ÷ t2^2 + t3^2)  we could still write physical laws in only (x.y.z.t). However, the GR  solutions in 6D (x.y.z.t1.t2.t3) would be different and possibly suggest testable features of a spacetime having more time dimensions.
