Why are there 4 Dimensions and 4 Fundamental Forces?

Question

Is it a coincidence that there are four fundamental forces and four spacetime dimensions ? Does a universe with three spacetime dimension contain four fundamental forces? Can magnetism be realized in three dimensional spacetime?

Edit 1: Thanks alot for the answer. I'm a first year undergrad student so this may seem naive. If there is a symmetry that mixes spacetime symmetries and internal symmetries of the theory (e.g. gauge symmetry) then there might be ways that spacetime geometry have effects on the particle spectrum and interactions of the theory so you can describe things in geometrical manner . The mechanisms that lead to the structure of spacetime may explain why there are four forces.

Answer 1

Well, I will not join the chorus in agreeing there are four fundamental forces.

Our everyday world depends on two forces, gravity and electromagnetism, and in this sense they would be fundamental to our everyday world.

When we started the scattering experiments we discovered another two forces, strong and weak. The higher we go in energies the more forces may be discovered, because forces are exchanged particles in Feynman diagrams and none would be more fundamental than another one, imo. There are the grand unified theories for example, where there are a lot of carriers of force similar to a photon or a gluon.

If/when string theory becomes evident, even the four dimensions will no longer be true, as string require eleven dimensions.

Thus yes, it is a coincidence that at this point in time we know four dimensions and have two forces well explored, another two under examination at the LHC and an unknown number N for the future.

Answer 2

I think the simple answer is "yes, it's a coincidence", but as Manishearth noted, it's really not a bad question and no worse than some very serious speculations I've seen in physics about other odd number coincidences. The history of physics is in fact full of speculations about whether "same" numbers mean something or not.

Here's the main reason why the 4-to-4 is a coincidence: the two groups of items have completely different structure. Spacetime has three isotropic (identical, interchangeable, and "rotatable" -- hmm, is that really a word?) dimensions of space plus one of time, while the four forces are all different in some really interesting and unique ways.

And one other note: You could argue that magnetism is already the most "3D" of the apparent forms of electromagnetism that you encounter in everyday life. (Please note that electric and magnetic are really different view of a single unified force, so the kinds of distinctions I'm making here only apply in a limited context.) Magnetism is the most 3D because its relativistic representation requires only components of the form $\{xy, yz, zx\}$. As you can see, those components only reference the spatial dimensions xyz. Electric charge in contrast requires a slightly more complicated set $\{tx, ty, tz\}$ of components, making it 4D due to the addition of t (time). Thus the seemingly exact symmetry of static electric and magnetic fields in 3D space is a bit of an illusion. (See Feynman's Lectures on Physics Vol II if you are want to know what all that actually means.)

Addendum by Terry Bollinger on 2012-03-08.20:15 EST (Thu)

PhysicsGuy: In your addendum, I think you last sentence best captured your postulate: "The mechanisms that lead to the structure of spacetime may explain why there are four forces."

That could be a tough one to answer. I'll point out an interesting issue though: You could argue that the electromagnetic force all by itself does a pretty good job of defining the need for an xyz+t space, without the need for any other forces. Abstracted down to simplest topological forms, B loops in three orientations define xyz, while E adds in time. E does that in a peculiar way: It has the minimal abstraction symmetry of a hollow sphere (a 2-sphere) whose poles have been rotated onto the t axis. That leaves only the equator of the sphere intersecting, ring-like, with 3D space. The equator of the E sphere then looks just like a B loop from our 3D perspective, resulting in the apparent (but broken) symmetry of B and E in at low speeds. It's a pretty tight structure overall, and not one that is easily generalizable to anything other than a 3+1 space. That's one reason why Maxwell wrote his original equations using quaternions.

(It was Heaviside, not Maxwell, who gave us the four modern non-quaternion Maxwell's equations. Heaviside's equations are also hugely reduced in number, from I think about 18 down to 4. Heaviside himself insisted that these dramatically transformed equations still be called Maxwell's equations, however.)

Answer 3

From the little I know, the number of dimensions is just based on prior experience (i.e experiments). There is no reason not to work in higher dimensions. What I find more interesting is that we choose the Poincare Group to describe nature. I don't know why but it works. It seems that the representation of this group in 4D seems to answer many questions. In my view, the choice of the Group and the representation of its generators is more interesting.

As for the fundamental interactions, I think Group theoretic analysis provides the answer one way or the other. I think that 4 is not a theoretical limit but it is a limit borne out by current experiments.