Warped Passages - Lisa Randall [222]
III. Mirror Symmetry
T-duality applies when a dimension is rolled up into a circle. But an even weirder symmetry than T-duality is mirror symmetry, which sometimes applies in string theory when six dimensions are rolled up into a Calabi-Yau manifold. Mirror symmetry says that six dimensions can be curled up into two very different Calabi-Yau manifolds, yet the resulting four-dimensional long-distance theory can be the same. The mirror manifold of a given Calabi-Yau manifold could look entirely different: it might have different shape, size, twisting, or even number of holes.* Yet when there exists a mirror to a given Calabi-Yau manifold, the physical theory where six of the dimensions are curled up into either one of the two will be the same. So with mirror manifolds as well, two apparently different geometries give rise to the same predictions. Once again, spacetime has mysterious properties.
IV. Matrix Theory
Matrix theory, a tool for studying string theory, provides still more mysterious clues about dimensions. Superficially, matrix theory looks like a quantum mechanical theory that describes the behavior and interactions of Do-branes (pointlike branes) moving through ten dimensions. But even though the theory doesn’t explicitly contain gravity, the Do-branes act like gravitons. So the theory ends up having gravitational interactions, even though the graviton is superficially absent.
Furthermore, the theory of Do-branes mimics supergravity in eleven dimensions, not ten. That is, the matrix model looks as if it contains supergravity with one more dimension than the original theory seems to describe. This suggestive behavior (along with other mathematical evidence) has led string theorists to believe that matrix theory is equivalent to M-theory, which also contains eleven-dimensional supergravity.
One especially bizarre feature of matrix theory is Edward Witten’s observation that when Do-branes come too close to each other, you can no longer know exactly where they are. As Tom Banks, Willy Fischler, Steve Shenker, and Lenny Susskind—the originators of matrix theory—said in their paper, “Thus for small distances there is no representation of the configuration space in terms of ordinary positions.”* That is, the location of a Do-brane is no longer a meaningful mathematical quantity when you try to define it too precisely.
Although such strange properties make matrix theory very tantalizing to study, it is presently very difficult to use it for computations. The problem is that—like nearly all other theories containing strongly interacting objects—no one has yet found a way to solve many of the most important questions that will help us better to understand what is really going on. Even so, because of the emergence of an extra dimension and the disappearance of dimensions when Do-branes come too close together, matrix theory is one more reason to wonder what dimensions really mean.
What to Think?
Although physicists have mathematically demonstrated these mysterious equivalences between theories with different numbers of dimensions, we are clearly still missing the big picture. Do we know with certainty that these dualities apply, and if so, what they tell us about the nature of space and time? Moreover, no one knows what the best description would be when a dimension is neither very big nor very small (relative to the extraordinarily tiny Planck scale length). Perhaps our notion of spacetime breaks down altogether once we try to describe something so small.
One of the strongest reasons for believing that our spacetime description is inadequate at the Planck scale length is that we don’t know any way, even in theory, to examine such a short distance.