Examples From Classical Theories
An example of mass gap arising for massless theories, already at the classical level, can be seen in spontaneous breaking of symmetry or Higgs mechanism. In the former case, one has to cope with the appearance of massless excitations, Goldstone bosons, that are removed in the latter case due to gauge freedom. Quantization preserves this property.
A quartic massless scalar field theory develops a mass gap already at classical level. Let us consider the equation
This equation has the exact solution
-- where and are integration constants, and sn is a Jacobi elliptic function -- provided
At the classical level, a mass gap appears while, at quantum level, one has a tower of excitations and this property of the theory is preserved after quantization in the limit of momenta going to zero.
While lattice computations have suggested that Yang-Mills theory indeed has a mass gap and a tower of excitations, a theoretical proof is still missing. This is one of the Clay Institute Millennium problems and it remains an open problem. Such states for Yang-Mills theory should be physical states, named glueballs, and should be observable in the laboratory.
Read more about this topic: Mass Gap
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