What is the mass gap?
In a relativistic theory, whenever there is a state with definite 4-momentum p, there is also one with definite momentum p' = Lambda p obtained by applying a Lorentz transform Lambda. The orbit of 4-momenta obtained in this way forms a hyperboloid in the future cone (because of causality), characterized by a mass m &ge 0. Its equations are
The only state with zero momentum is the ground state, usually called
the vacuum. If the values of p^2 for the realizable nonzero p is
bounded below by a positive number, the theory is said to have a mass
gap. The largest value of m>0 for which m^2 is such a lower bound
defines the precise value of the mass gap. Usually there is a state
for which p^2=m^2; this is then interpreted as the state of a
single 'dressed' particle.
In general, the mass spectrum consist of a discrete and a continuous part. The discrete part of the spectrum corresponds to bound states, the continuous part to scattering states.
The continuous spectrum starts when there is the possiblity of scattering. which means that the energy is large enough that two asymptotically independent systems can exist. Given a state of mass m, one expects to have states with two almost independent systems of mass m and an arbitrary relative momentum, giving a continuous spectrum of scattering states with all possible squared momenta exceeding (2m)^2, as a simple calculation reveals: If p is the sum of two timelike vectors p1,p2 of mass m then
If there is no mass gap, one expects massless dressed particles
to be present. This corresponds to the limiting case m --> 0 of the
Arnold Neumaier (Arnold.Neumaier@univie.ac.at) A theoretical physics FAQ