Most of the several dozen "elementary'' particles listed in the tables published annually by the Particle Data Group at the Lawrence Berkeley Laboratory are unstable: if produced, they decay according to an exponential decay law familiar from radioactive decay. Their mean lifetimes vary over a wide range. For instance, the mean lifetime of the first excited state of the proton, the so-called Δ particle, is about 10-23 seconds, that of the muon is a few microseconds, and that of the proton (if it is unstable at all) is comparable to the known lifetime of the universe (about 10 billion years.)
Typically, a short-lived unstable particle can be produced if one scatters its decay products on each other. For instance, the excited states of the proton can be produced by scattering a beam of π mesons (pions) or photons on a hydrogen (proton) target. In such an experiment, the incident particle (the pion or photon) spends a time comparable to the mean lifetime of the excited state close to the proton. This leads to an enhancement of the scattering probability, measured by the scattering cross section. The cross section has a peak at a center-of-mass energy of the incoming particles nearly equal to the rest energy of the unstable particle produced. The peak is of a finite width: it is inversely proportional to the mean lifetime of the unstable particle or resonance, corresponding to the uncertainty relation between time and frequency.
Unlike other uncertainty relations known in quantum mechanics, the uncertainty relation between frequency and time exists in classical physics as well. For instance, if one wants to measure the frequency of a pendulum, one has to measure it over several periods. The error of the frequency measurement is inversely proportional to the time spent on the measurement. This purely classical result translates into quantum theory by the use of Planck's relation: E = hν , where h is Planck's constant and ν is the frequency of oscillation.
In a situation like that just described, one speaks about a resonance in the cross section. The shape of the cross section at energies close to the energy of the unstable particle excited is well described by the Breit-Wigner formula. This expression for the scattering cross section is identical to the response of a damped, forced harmonic oscillator at forcing frequencies close to the eigenfrequency of the oscillator, hence the name.
In general, the existence of resonances in a scattering cross section indicates that the target particle is a composite one, just as in atomic or nuclear physics. (For instance, the scattering of a beam of photons on an atomic target excites the various electronic levels and it leads to peaks in the scattering cross section.) In the example quoted, the proton is now known to be composed of quarks and gluons. String models may be different in this respect from other physical theories. In such models, resonances result from a novel concept of space-time rather than from the composite nature of elementary particles.
Green, M. B.; Schwarz, J. H.; and Witten, E. Superstring Theory, Vol. 1 (Cambridge University Press, Cambridge, UK,1987).
Griffiths, D. J. Introduction to Elementary Particles (Harper and Row Publishers, New York, 1987).
Particle Data Group. <http://pdg.lbl.gov>.