Warped Passages - Lisa Randall [220]
Where do I expect physics to go from here? There are too many possibilities to enumerate. But let me describe a few intriguing observations that suggest that more important theoretical surprises lie in store—ones that might come closer to resolution some time soon. These mysteries center on a question that at this point might sound shocking, namely:
What Are Dimensions, Anyway?
How can I ask such a question this late in the game? I’ve already spent much of this book discussing the meaning of dimensions and some of the potential implications of proposed extra-dimensional worlds. But now that I’ve told you what we understand about dimensions, allow me to return briefly to this question.
What does the number of dimensions really mean? We know that the number of dimensions is defined as the number of quantities that you need to locate a point in space. But I also presented examples in Chapters 15 and 16 showing that ten-dimensional theories sometimes have the same physical consequences as eleven-dimensional theories.
Such duality suggests that our notion of dimension isn’t quite as firm as it looks—there’s a plasticity in the definition that eludes the conventional terminology. Dual descriptions of a single theory tell us that no single formulation is necessarily the best one. The formulation and even the number of dimensions in the best description might depend on the strength of the string coupling, for example. Because no single theory is always the best description, the question of the number of dimensions doesn’t always have a simple answer. This ambiguity in the meaning of dimensions and the apparent emergence of an additional dimension in strongly interacting theories are among the most important theoretical physics observations of the last decade. Let me now list a few more intriguing recent theoretical discoveries that indicate that the notion of dimension is somewhat fuzzier than we’d maybe like to believe.
I. Warped Geometry and Duality
In Chapters 20 and 22, I explained some of the consequences of the warped spacetime geometry that Raman Sundrum and I developed. In that geometry, the masses and sizes of objects depend on location along a fifth dimension and, furthermore, gravity is localized in the vicinity of a brane. But there is still one more amazing feature of this warped spacetime, known technically as anti de Sitter space, that I have yet to tell you about—one that leads to further questions about dimensionality.
The remaining remarkable feature of anti de Sitter space is the existence of a dual four-dimensional theory. Theoretical clues tell us that everything that happens in five-dimensional anti de Sitter space can be described using a dual four-dimensional framework in which there are extremely strong forces that have special properties. According to this mysterious duality, everything in the five-dimensional theory has an analog in the four-dimensional theory. And vice versa.
Although mathematical reasoning tells us that a five-dimensional theory in anti de Sitter space is equivalent to a four-dimensional one, we don’t always know the precise particle content of that four-dimensional dual theory. However, Juan Maldacena, an Argentinian-born string theorist now at the Institute for Advanced Study in Princeton, triggered a string theory frenzy in 1997 when he derived an explicit example of a similar duality in string theory. He realized that a version of string theory with a large number of overlapping D-branes on which strings interact strongly can be described either with a four-dimensional quantum field theory or with a ten-dimensional gravitational