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One of the most used methods in unification of theories is the use of higher dimensions. How does it actually work? If these dimensions are extremely small curled up, how does it affect the universe. There many theories giving explanation through mathematics. But what is the physical reality? Is it possible that these dimensions are not curled up but actually include stuff like multiple universes?

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If you're after an intuitive understanding then I think the original Kaluza-Klein theory is a good place to start. NB no-one believes the Kaluza-Klein theory to be a realistic description of the universe, but it's a good illustration of how extra dimensions may influence physics. See http://en.wikipedia.org/wiki/Kaluza%E2%80%93Klein_theory for a good article on KK theory, but be warned that this is a bit mathematical. Some Googling may find more accessible articles.

Anyhow, with KK theory any point in spacetime has an extra degree of freedom. You specify a point by it's 4D co-ordinates (t, x, y, z) but there is a fifth co-ordinate that describes it's position on the curled up fifth dimension, and the point could take any position on this fifth dimension. You can describe electromagnetism by this position in the fifth dimension. In effect you're taking advantage of the extra dimension/extra degree of freedom to describe another field. More precisely, the freedom to move anywhere in the fifth dimension creates a local gauge symmetry called U(1), and including this symmetry in the Lagrangian requires electromagnetism to exist.

As for physical reality, there is currently no experimental evidence that extra dimensions exist, but then there is no experimental evidence that they don't. String theory requires extra dimensions, but then Professor S. James Gates Jr in his video series on string theory (called "The DNA of Reality" I think) claims it can be formulated without extra dimensions, though he does admit this makes it harder and it's a minority view.

There is a sense that the extra dimensions may contain "multiple universes" provided you're allowed rather specific definitions of what constitutes "other universes". have a look at http://en.wikipedia.org/wiki/Brane_world for an article on this. However this probably isn't what you mean by "multiple universes".

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the string theory is perhaps the best contender for an actual unified theory dreamed by einstein in his last years of his life. The stuff you are talking about is multiple universes in a single dimension is a idea i think you mistook.According to the theory, the multiverse contains all the universes you are talking about and these universes could actually contain any number of dimensions and there physical states might be different as well as the laws of physics which govern them. the string theory or m-theory would be mathematically inconsistent and hence only adding dimensions to the whole would make it consistent and actually an idea to think about.

As for "physical reality", we will not know if these assumptions made during the last 30 years are actually true, at least not in our lifetime.Physicists all over the world are calculating and researching about the various applications of the string theory and it's eventual "proof" itself. It may be possible for these dimensions to not be curled up as much research is going on.As far as i know, physicists are actually thinking of universes containing dimensions and not dimensions containing universes. For more information, you may download brian greene's THE ELEGANT UNIVERSE which is his attempt to explain all the advancements in modern physics through a simple,non-mathematical way which can be understood by all.

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