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associated with a symmetry. Without such symmetries, too many oscillation modes of the gauge bosons—the particles that communicate those forces—would be predicted to be present by the theory that quantum mechanics and special relativity tells us describes them. In the theory without symmetries, theoretical calculations would generate nonsensical predictions, such as probabilities for high-energy interactions greater than one for the spurious oscillation modes. In any accurate description of nature, such unphysical particles—particles that don’t actually exist because they oscillate in the wrong direction—clearly need to be eliminated.

In this context, symmetries act like spam filters, or quality control constraints. Quality requirements might specify keeping only those cars that are symmetrically balanced, for instance, so that the cars that make it out of the factory all behave as expected. Symmetries in any theory of forces also screen out the badly behaved elements. That’s because interactions among the undesirable, unphysical particles don’t respect the symmetries, whereas those particles that interact in a way that preserves the necessary symmetries oscillate as they should. Symmetries thereby guarantee that theoretical predictions involve only the physical particles and therefore make sense and agree with experiments.

Symmetries therefore permit an elegant formulation of a theory of forces. Rather than eliminate unphysical modes in each calculation one by one, symmetries eliminate all the unphysical particles with one fell swoop. Any theory with symmetric interactions involves only the physical oscillation modes whose behavior we want to describe.

This works perfectly in any theory of forces involving zero mass force carriers, such as electromagnetism or the strong nuclear force. In symmetric theories, predictions for their high-energy interactions all make sense and only physical modes—modes that exist in nature—get included. For massless gauge bosons, the problem with high-energy interactions is relatively straightforward to solve, since appropriate symmetry constraints remove any unphysical, badly behaved modes from the theory.

Symmetries thereby solve two problems: unphysical modes are eliminated, and the bad high-energy predictions that would accompany them are as well. However, a gauge boson with nonzero mass has an additional physical—existent in nature—mode of oscillation. The gauge bosons that communicate the weak nuclear force fall into this category. Symmetries would eliminate too many of their oscillation modes. Without some new ingredient, weak boson masses cannot respect the Standard Model symmetries. For gauge bosons with nonzero mass, we have no choice but to keep a badly behaved mode—and that means the solution to the bad high-energy behavior is not so simple. Nonetheless something is still required for the theory to generate sensible high-energy interactions.

Moreover, none of the elementary particles in the Standard Model without a Higgs can have a nonzero mass that respects the symmetries of the most naive theory of forces. With the symmetries associated with forces present, quarks and leptons in the Higgsless Standard Model would not have nonzero masses either. The reason appears to be unrelated to the logic about gauge bosons, but it can also be traced to symmetries.

In Chapter 14, we presented a table that included both left-and right-handed fermions—particles that get paired in the presence of nonzero masses. When quark or lepton masses are nonzero, they introduce interactions that convert left-handed fermions to right-handed fermions. But for left-handed and right-handed fermions to be interconvertible, they would both have to experience the same forces. Yet experiments demonstrate that the weak force acts differently on left-handed fermions than on the right-handed fermions that massive quarks or leptons could turn into. This violation of parity symmetry, which if preserved would treat left and right as equivalent for the laws of physics, is startling to everyone when they first learn