The known elementary particles range in mass from about 0.0004eV for the lightest neutrino to around 170 billion eV = 1.7 x 10^11 for the heaviest quark, the top. This is a ratio of about 400,000,000,000,000 to 1.

On the other hand, the energy scale available from Einstein’s theory of gravitation (which relates mass to energy) suggests that the natural mass for a typical particle should be the Planck mass, about 2.43 × 10^27 eV, 11 orders of magnitude larger than even the top quark and 25 orders of magnitude larger than the lightest neutrino mass.

From the point of view of the Planck energy, all particles known to man have mass very close to zero. Let’s write the Hamiltonian for the system as a first order Hamiltonian , in which the energies (and therefore the masses) of all our usual particles are zero, plus a perturbation , which will provide a correction to the zero energies. For the full Hamiltonian we have:

where is a small number.

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