Yang-Mills Theory
In the last half of the twentieth century, physicists succeeded in unifying three of the four fundamental forces of nature: electromagnetism, the "weak force" responsible for radioactive decay, and the "strong force" responsible for holding the nuclei of atoms together. The mathematical foundation underlying these advances is called Yang-Mills theory, after Chen Ning Yang and Robert Mills, the physicists who introduced it in a short paper in 1954. At present, it is the dominant approach to quantum field theory.
In quantum field theory, the classical distinction between particles and waves breaks down. A light wave, for example, can be thought of either as a wave or as a particle (a photon). A bowling ball, to take another example, is a superposition or "tensor product" of all the waves making up the neutrons, protons and electrons in its constituent atoms.
Instead of having a particular location in space, a particle/wave is described by its wave function psi(x, t). This function is sometimes described as giving the probability that the particle is observed at position x at time t. In fact, the story is a bit more complicated than that. For the purposes of electromagnetic theory, the value of psi(x, t) at each point and time is a complex number, and it is the magnitude squared, |psi(x, t)| 2, that actually represents the probability of observing the particle there. This subtlety takes on major importance below.
Having abandoned the idea that a particle is located at a single point in space, we also have to abandon the idea that it moves along a single path through spacetime. In fact, its evolution is a complicated "sum over histories," some of which are more probable than others. The probability of a particular history is given by evaluating a certain integral, called the "action"; in rough agreement with the principle of least action (Maupertuis' principle) from classical physics, histories with more action have a much smaller probability, and the history with the least action is the most likely.
For mathematicians, the question of how to define this "sum over histories" correctly is still a thorny and unsolved problem. For physicists, though, the most important question is what function to put inside the action integral. Different choices for this function, called the Lagrangian, lead to different physical theories.
Yang and Mills realized that the possible choices for a Lagrangian are strongly constrained by physical symmetries. In electromagnetic theory, all physical observables (such as a particle's electric charge) are unchanged when psi(x, t) is multiplied by a complex number of magnitude 1. This property of the electric field is called "gauge invariance." Invariance under multiplication by unit complex numbers is an especially simple kind of symmetry, because these numbers form an Abelian group, called U(1). In their 1954 paper, Yang and Mills proposed that the weak and strong nuclear forces could be described by action functionals with more complicated, non-Abelian symmetry groups.
Yang and Mills' idea was not accepted immediately—in fact, Yang won his Nobel Prize for a completely different discovery--but in time, it turned out to be just what physicists needed. Steven Weinberg, Abdus Salam, and Sheldon Glashow used an SU(2) × U(1) - invariant Lagrangian to unite electromagnetism and the weak force. Murray Gell-Mann found an SU(3)-invariant Lagrangian that described the strong nuclear force and predicted the existence of quarks. All of these physicists won Nobel Prizes for their work.
In many ways, the status of Yang-Mills theory today resembles the status of calculus in the 1700s. When Newton developed calculus in the late 1600s, his main motivation was not mathematical but physical: He wanted to understand the motion of planets subject to his inverse-square law of gravitation. Calculus was a powerful calculational device (thus its name) that suited this purpose, as well as many others. But its mathematical soundness was still very much open to debate for over a century. What were the infinitesimals or "fluxions" that Newton wrote of? How could you add up an infinite collection of infinitesimals and obtain a sensible, finite answer? These questions were finally answered by mathematicians like Cauchy and Riemann in the 1800s, and their work transformed calculus from a computational tool into the mathematical disciplines of real analysis and complex analysis.
Like calculus in the 1700s, Yang-Mills theory works, and works brilliantly. But, at present, it makes no logical sense. No one has yet proved the existence of a four-dimensional spacetime (such as our own) with well-defined solutions to the quantum Yang-Mills equations. At best, they have proved the existence of solutions in certain two- and three-dimensional "toy universes." For this reason, in 2000 the Clay Mathematics Institute named the proof of the existence of Yang-Mills fields as one of its seven Millennium Prize Problems. Anyone who solves the problem, and whose solution is generally accepted by the mathematical community, will receive a prize of one million dollars.
The Millennium Prize Problem comes with a second part, the "mass gap," which is also of great relevance to physicists. At present, there is no good explanation for the fact that the strong nuclear force is "local"--that is, we do not observe it at distances larger than the size of an atomic nucleus. One way to prove locality is to show that there is a minimum nonzero mass Delta for any solution to the quantum Yang-Mills equations. As a minimum requirement for a successful solution to the Yang-Mills existence problem, the Clay Mathematics Institute specified that the solution should explain this "mass gap" between the zero mass of the vacuum and the minimum mass Delta. There are other important physical phenomena that the solution should explain, such as "quark confinement" (the fact that quarks are never seen in isolation), but they are not required to win the million-dollar prize. According to mathematical physicists Arthur Jaffe and Edward Witten, "A solution of the existence and mass gap problem... would be a turning point in the mathematical understanding of quantum field theory."
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