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If we live in more than three spatial dimensions, is it not right to conclude that all matter observable to us shares almost the same coordinates of extra dimensions. Or is it just that ordinary interactions (like electromagnetism) are simply independent of these extra dimensions?

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In some extra-dimensional models, such as brane cosmology, the fields (except gravity) are indeed confined to a lower-dimensional surface, which is sort of like "sharing almost the same coordinates in the extra dimensions".

In Kaluza-Klein theory with compact extra dimensions, the fields are basically spread evenly across the entire size of the extra dimensions, so you can't probe them directly by, say, sending wave packets around them. (Maybe in a sufficiently high-energy experiment you could—this would mean exciting higher "Kaluza-Klein modes". Some people thought the LHC might find such modes, but so far it hasn't.)

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