Warped Passages - Lisa Randall [143]
Having too many dimensions was not a novel feature of the superstring. The earlier version of string theory (the one without fermions or supersymmetry) had twenty-six dimensions: one of time and twenty-five of space. But the earlier version of string theory had other problems, like the tachyon. Superstring theory, on the other hand, was sufficiently promising to be worth pursuing.
Even so, string theory was largely ignored until 1984, the year that Green and Schwarz demonstrated a startling feature of the superstring which convinced many other physicists that they were on a promising track. This discovery, together with two other developments that we’ll get to soon, are what put string theory into the mainstream of physics.
Green and Schwarz’s work addressed a phenomenon known as anomalies. As the name suggests, anomalies came as a big surprise when they were first discovered. The first physicists who worked on quantum field theory took for granted that any symmetry of a classical theory would also be preserved by its quantum mechanical extension—the more comprehensive version of the theory that also includes the effects of virtual particles. But that is not always the case. In 1969, Steven Adler, John Bell, and Roman Jackiw showed that even when a classical theory preserves a symmetry, quantum mechanical processes involving virtual particles sometimes violate that symmetry. Such symmetry violations are called anomalies, and the theories that contain anomalies are labeled anomalous.
Anomalies are extremely relevant to the theories of forces. In Chapter 9 we saw that a successful theory of forces requires the existence of an internal symmetry. These symmetries must be exact, or else there’s no way to eliminate the unwanted polarization of the gauge boson, and the theory of forces will then make no sense. The symmetry associated with a force must therefore be anomaly-free—the sum of all symmetry-breaking effects must be zero.
This is a powerful constraint on any quantum theory of forces. For example, we now know that it is one of the most compelling explanations for the existence of both quarks and leptons in the Standard Model. Individually, virtual quarks and leptons would lead to anomalous quantum contributions that would break the symmetries of the Standard Model. However, the sum of the quantum contributions from the quarks and the leptons adds up to zero. This miraculous cancellation is what makes the Standard Model hold together; both leptons and quarks are necessary if the forces of the Standard Model are to make sense.
Anomalies were potentially a problem for string theory, which, after all, includes forces. In 1983, when the theorists Luis Alvarez-Gaume and Edward Witten showed that such anomalies occur not only in quantum field theory but also in string theory, it looked as if this discovery would consign string theory to the annals of interesting but overly far-reaching ideas. String theory didn’t seem as if it would preserve the requisite symmetries. In the skeptical environment created by string theory’s potential for anomalies, Green and Schwarz made quite a splash when they showed that string theory could satisfy the constraints that were needed to avoid anomalies. They computed the quantum contribution to all possible anomalies and showed that for particular forces, anomalies would miraculously add up to zero.
One of the things that made Green and Schwarz’s result so surprising is that string theory allows many worrisome quantum mechanical processes, each of which looks as if it could create symmetry-breaking anomalies. But Green and Schwarz showed that the sum of the quantum mechanical contributions to all these possible symmetry-breaking anomalies in ten-dimensional superstring theory is zero. This meant that the many cancellations that were required in string theory calculations actually occur, and, furthermore, that the cancellations happen in ten dimensions, the number of dimensions that was already known to be special for superstring theory. This discovery was sufficiently miraculous