The Clay Mathematics Institute has offered a prize of US$1,000,000 to the person solving each of seven unsolved problems in contemporary mathematics. One of these seven problems requires proving that the quantum field theory underlying the Standard Model of particle physics, called Yang-Mills theory, satisfies the standard of rigor that characterizes contemporary mathematical physics, i.e. constructive quantum field theory. The winner must also prove that the mass of the smallest particle predicted by the theory be strictly positive, i.e., the theory must have a mass gap.
Contents |
Background
"...one does not yet have a mathematically complete example of a quantum gauge theory in four-dimensional space-time, nor even a precise definition of quantum gauge theory in four dimensions. Will this change in the 21st century? We hope so!" From the Clay Institute's official problem description by Arthur Jaffe and Edward Witten.
Most known and nontrivial (i.e. interacting) quantum field theories in 4 dimensions are effective field theories with a cutoff scale. Since the beta-function is positive for most models, it appears that most such models have a Landau pole as it is not at all clear whether or not they have nontrivial UV fixed points. This means that if such a QFT is well-defined at all scales, as it has to be to satisfy the axioms of axiomatic quantum field theory, it would have to be trivial (i.e. a free field theory).
Quantum Yang-Mills theory with a non-abelian gauge group and no quarks is an exception, because asymptotic freedom characterizes this theory, meaning that it has a trivial UV fixed point. Hence it is the simplest nontrivial constructive QFT in 4 dimensions. (QCD is a more complicated theory because it involves quarks.)
It has already been well proven -- at least at the level of rigour of theoretical physics but not that of mathematical physics -- that the quantum Yang–Mills theory for a non-abelian Lie group exhibits a property known as confinement. This property is covered in more detail in the relevant QCD articles (QCD, color confinement, lattice gauge theory, etc.), although not at the level of rigor of mathematical physics. A consequence of this property is that beyond a certain scale, known as the QCD scale (more properly, the confinement scale, as this theory is devoid of quarks), the color charges are connected by chromodynamic flux tubes leading to a linear potential between the charges. (In string theory, this potential is the product of a string's tension with its length.) Hence free color charge and free gluons cannot exist. In the absence of confinement, we would expect to see massless gluons, but since they are confined, all we see are color-neutral bound states of gluons, called glueballs. If glueballs exist, they are massive, which is why we expect a mass gap.
Results from lattice gauge theory have convinced many that this model exhibits confinement -- as indicated, for example, by an area law for the falloff of the VEV of a Wilson loop. However, these methods and results are not mathematically rigorous.
See also
References
- Arthur Jaffe and Edward Witten "Quantum Yang-Mills theory." Official problem description.
External links
 |
This mathematics-related article is a stub. You can help Wikipedia by expanding it. |
This entry is from Wikipedia, the leading user-contributed encyclopedia. It may not have been reviewed by professional editors (see full disclaimer)
