The Yang-Mills mass gap problem

or how to lure researchers promising a million dollars that nobody will ever win

In the year 2000, the Clay Mathematics Institute asked, in its list of millennium problems, to prove the existence of a non-trivial quantum field theory for every non-Abelian compact simple gauge group in continuous flat space-time. The problem, formulated by Witten and Jaffe, also asks to prove that each such quantum field theory leads to a finite mass gap. A million dollars is promised for the solution.

In the millennium problem, a quantum field theory is defined as a structure realizing the axioms of Streater and Wightman, as well as those of Osterwalder and Schrader. All these axioms are based on perfect locality and continuous quantum fields.

The Yang-Mills millennium problem has two aspects: a physical aspect about nature and a mathematical aspect about axiomatic quantum field theory.

The physical aspect of the Yang-Mills millennium problem is answered by nature. First of all, because of the minimum length, in nature, continuous space-time does not exist. In nature, additional gauge groups do not exist: only the two nuclear interactions are observed to be non-Abelian gauge theories. Experiments and nature thus answer the millennium problem negatively.

In addition, any unified description of nature will answer the physical aspect of the millennium problem negatively. Whatever the unified description may be, it does not comply with the assumptions of spatial continuity and of locality in the Yang-Mills millennium problem. Whatever the unified description may be, it will, by definition, explain why the observed three-dimensionality arises at low energies. Whatever the unified description may be, it will, by definition, explain why only the observed gauge interactions exist. Whatever the unified description may be, it will, by definition, explain why only the observed particle masses and couplings occur.

In summary, if the Yang-Mills millennium problem is restricted to the physical aspect - the existence and mass gap of gauge theories in nature - observations and any unified theory conclude that additional Yang-Mills theories do not exist. The best one can do is to restrict the Yang-Mills millennium problem to proving the existence of a finite mass gap in the case of the gauge group SU(3) of the strong nuclear interaction. The SU(3) case has a positive answer, due to the existence of glueballs - in observations and in the strand tangle model.

The mathematical aspect of the millennium problem also has issues. Continuous space is in contrast with the minimum length and time in nature. Continuous flat space provides no length or time units. All length or time units are tied to Planck’s quantum of action ℏ. Therefore, continuous space without length or time units is in contrast with quantum field theory. But there is more.

In mathematics, one can choose to ignore nature and physics and assume both continuity and quantum field theory at the same time. This leads to a further issue. The statement of the millennium problem defines a quantum field theory as a structure realizing specific axioms. The axioms assume that the quantum of action ℏ and the speed of light c are finite. Now, exact space continuity implies that the Planck length is zero. This implies that the gravitational constant G must vanish. A vanishing G is in contrast to observations. A vanishing G also implies that mass units cannot be defined, that mass values cannot be measured, and thus that mass cannot be defined. However, the Yang-Mills millennium problem explicitly asks for a mass value.

In mathematics, one can choose to ignore nature and physics and assume continuity, quantum field theory, and non-vanishing G. However, if one insists that G is finite after all, then one has a non-vanishing minimum length. The minimum length implies that there are no points, no sets, and no axioms (as shown in the reference cited at the end). Quantum field theory and finite G, taken together, thus imply that there is no axiomatic quantum field theory. However, the Yang-Mills millennium problem asks for such a theory.

In mathematics, one can choose to ignore nature, and physics further and assume both that continuity and axiomatic quantum field theory are valid at the same time. The two assumptions, taken together, imply the description of particles as exact point particles. This purely mathematical result contradicts general relativity: no object can be smaller than the Schwarzschild radius determined by its mass. Thus, the concepts of ‘particle’ and ‘quantum field’ assumed in the millennium problem contradict general relativity and have no relation to actual particles in nature or actual quantum fields. The two mathematical assumptions have a further consequence. Different elementary particles differ at least at the Planck scale. Only Planck scale differences explain the existence of different types of quarks and and leptons. Eliminating the Planck scale by assuming point particles implies that the existence of different elementary particles cannot be derived but must be assumed. In particular, assuming point particles also prevents deducing whether additional elementary gauge bosons exist. Therefore, it is impossible to derive whether additional gauge interactions with additional gauge groups exist. It is equally impossible to derive whether additional particles exist, such as generalizations of glueballs for additional gauge interactions. Thus, it is impossible to deduce the existence of any additional mass gap. However, the Yang-Mills millennium problem asks for additional gauge theories and for additional mass gaps.

In short, the speed limit c, the quantum of action ℏ and the maximum force c⁴/4G imply a minimum length, the lack of continuity of space, and the lack of perfect locality. The Planck limits lead to the emergence of wave functions, quantum field theory, gauge theories, elementary particles, measurement units, mass, physical observables and the three dimensions. In contrast, the Yang-Mills millennium problem starts with different assumptions: continuity of space, perfect three-dimensionality, perfect locality, point-like particles, axiomatic quantum field theory, predetermined gauge bosons, and predetermined mass generation mechanisms. Because of the logical contradictions among the assumptions, a mathematical proof of the existence of additional Yang-Mills theories and of their mass gaps is impossible.

The only known explanation for the gauge theories and their gauge groups, the strand tangle model, confirms the lack of any Yang-Mills theory beyond SU(3) and deduces the existence of a finite mass gap for SU(3). Strands exclude every imaginable alternative to the standard model with massive Dirac neutrinos: because of the existence of just three Reidemeister moves and of the minimum length, additional gauge groups and additional gauge particles are impossible. Strands clarify the relation between the three dimensions of space, the three possible gauge groups and the three fermion generations. In particular, strands imply that the emergence of three-dimensional space and the lack of higher gauge groups are two sides of the same coin. Strands thus answer the physical challenge that motivated the millennium problem negatively, as expected from a unified description.

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This is a slightly edited version of Appendix F in the preprint https://www.researchgate.net/publication/361866270. The references to the literature are found there.