Section 5: Extra Dimensions and Particle Physics
The Standard Model of particle physics described in Units 1 and 2 is very successful, but leaves a set of lingering questions. The list of forces, for instance, seems somewhat arbitrary: Why do we have gravity, electromagnetism, and the two nuclear forces instead of some other cocktail of forces? Could they all be different aspects of a single unified force that emerges at higher energy or shorter distance scales? And why do three copies of each of the types of matter particles exist—not just an electron but also a muon and a tau? Not just an up quark, but also a charm quark and a top quark? And how do we derive the charges and masses of this whole zoo of particles? We don't know the answers yet, but one promising and wide class of theories posits that some or all of these mysteries are tied to the geometry or topology of extra spatial dimensions.
Perhaps the first attempt to explain properties of the fundamental interactions through extra dimensions was that of Theodor Kaluza and Oskar Klein. In 1926, soon after Einstein proposed his theory of general relativity, they realized that a unified theory of gravity and electromagnetism could exist in a world with 4+1 spacetime dimensions. The fifth dimension could be curled up on a circle of radius R so small that nobody had observed it.
Figure 13: Theodor Kaluza (left) and Oskar Klein (right) made a remarkable theoretical description of gravity in a fifth dimension.
Source: © Left: University of Göttingen, Right: Stanley Deser. More info
In the 5D world, there are only gravitons, the force carriers of the gravitational field. But, as we saw in the previous section, a single kind of particle in higher dimensions can give rise to many in the lower dimension. It turns out that the 5D graviton would give rise, after reduction to 4D on a circle, to a particle with very similar properties to the photon, in addition to a candidate 4D graviton. There would also be a whole tower of other particles, as in the previous section, but they would be quite massive if the circle is small, and can be ignored as particles that would not yet have been discovered by experiment.
This is a wonderful idea. However, as a unified theory, it is a failure. In addition to the photon, it predicts additional particles that have no counterpart in the known fundamental interactions. It also fails to account for the strong and weak nuclear forces, discovered well after Kaluza and Klein published their papers. Nevertheless, modern generalizations of this basic paradigm, with a few twists, can both account for the full set of fundamental interactions and give enormous masses to the unwanted additional particles, explaining their absence in low-energy experiments.
Particle generations and topology
Figure 14: The Standard Model of particle physics.
Source: © Wikimedia Commons, Creative Commons 3.0 Unported License. Author: MissMJ, 27 June 2006. More info
One of the most obvious hints of substructure in the Standard Model is the presence of three generations of particles with the same quantum numbers under all the basic interactions. This is what gives the Standard Model the periodic table-like structure we saw in Unit 1. This kind of structure sometimes has a satisfying and elegant derivation in models based on extra dimensions coming from the geometry or topology of space itself. For instance, in string theories, the basic elementary particles arise as the lowest energy states, or ground states, of the fundamental string. The different possible string ground states, when six of the 10 dimensions are compactified, can be classified by their topology.
Because it is difficult for us to imagine six dimensions, we'll think about a simpler example: two extra dimensions compactified on a two-dimensional surface. Mathematicians classified the possible topologies of such compact, smooth two-dimensional surfaces in the 19th century. The only possibilities are so-called "Riemann surfaces of genus g," labeled by a single integer that counts the number of "holes" in the surface. Thus, a beach ball has a surface of genus 0; a donut's surface has genus 1, as does a coffee mug's; and one can obtain genus g surfaces by smoothly gluing together the surfaces of g donuts.
Figure 15: These objects are Riemann surfaces with genus 0, 1, and 2.
Source: © Left: Wikimedia Commons, Public Domain, Author: Norvy, 27 July 2006; Center: Wikimedia Commons, Public Domain, Author: Tijuana Brass, 14 December 2007; Right: Wikimedia Commons, Public Domain. Author, NickGorton, 22 August 2005. More info
To understand how topology is related to the classification of particles, let's consider a toy model as we did in the previous section. Let's think about a 6D string theory, in which two of the dimensions are compactified. To understand what particles a 4D observer will see, we can think about how to wind strings around the compactified extra dimensions. The answer depends on the topology of the two-dimensional surface. For instance, if it is a torus, we can wrap a string around the circular cross-section of the donut. We could also wind the string through the donut hole. In fact, arbitrary combinations of wrapping the hole N1 times and the cross-section N2 times live in distinct topological classes. Thus, in string theory on the torus, one obtains two basic stable "winding modes" that derive from wrapping the string in those two ways. These will give us two distinct classes of particles.
Figure 16: Strings can wind around a double torus in many distinct ways.
Similarly, a Riemann surface of genus g would permit 2g different basic stable string states. In this way, one could explain the replication of states of one type—by, say, having all strings that wrap a circular cross-section in any of the g different handles share the same physical properties. Then, the replication of generations could be tied in a fundamental way to the topology of spacetime; there would, for example, be three such states in a genus 3 surface, mirroring the reality of the Standard Model.
Semi-realistic models of particle physics actually exist that derive the number of generations from specfic string compactifications on six-dimensional manifolds in a way that is very similar to our toy discussion in spirit. The mathematical details of real constructions are often considerably more involved. However, the basic theme—that one may explain some of the parameters of particle theory through topology—is certainly shared.