The solution to this apparent contradiction seems almost obvious once you see it: the open string endpoints must be stuck on some sort of 25 (or 9) dimensional "membrane" while the string interiors (open or closed) can always move in 26 (or 10) dimensions. If more "compact" dimensions are small, the dimensions of the membranes will go down as the string endpoints have fewer and fewer directions in which they can move.
(A note on terminology: Because the word "membrane" usually implies two dimensions, string theorists use the term "brane" for the more general case. In one of the bad puns adored by physicists, a brane that extends in p directions of space is called a "p-brane". And because these particular branes arise when the boundary points of open strings cannot move in certain directions, the are called "Dirichlet branes" after the technical term "Dirichlet boundary conditions" that describes that situation, or "D-branes" for short.)
It turns out that these various dimensional p-branes are not just a mathematical tool but are independent objects in the theory in their own right. It is possible to calculate their masses, charges, and the kinds of vibrations they can carry, all from the equations of string theory that required them to exist in the first place. Their discovery in the early 1990s opened up vast new realms of phenomena in string theory to study, and has been instrumental in almost all of the field's progress in the past decade.
Only after D-branes were understood were the "particle-like" properties of open string endpoints really taken seriously. String endpoints act like particles in a universe defined by the brane (and thus with p space dimensions), and the "photon-like" open strings that we saw even provide forces on the brane like electromagnetism. (Closed strings provide gravity, but as closed strings they can move away from the brane, too.) This picture was the inspiration for "brane world" theories, which suggest that the universe we observe is actually just a 3-brane in a higher dimensional space.
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Copyright © 2004 by Steuard Jensen.