Further Contents of E = mc2Let us look at the energy-momentum relations for massive and massless particles. They are E = p2/2m and E = cp respectively. As you know, it was Einstein who unified these relations with his special relativity.
As for the internal space-time symmetries, massive particles have
the O(3)-like symmetry which produces the concept of spin.
Massless particles have the symmetry which is like the two-dimensional
Euclidean group. Wigner made these observations in his 1939 paper
on "inhomogeneous Lorentz group" where introduces the little group as
the subgroup of the Lorentz group whose transformations leave the
four-momentum of a given particle invariant.
If the massive particle is at rest, its O(3)-like symmetry leads to the concept of spin. For the massless particle, the rotational degree of freedom of the E(2)-like little group corresponds to the helicity. The translational degrees of freedom have been shown to correspond to gauge transformations. Indeed, massive and massless particles appear to have two distinct internal space-time symmetries. Can they be unified as Einstein did for the energy-momentum relation?
In order to answer this question, let us boost a particle with spin. Its longitudinal component remains invariant and is called the helicity. How about its transverse components? They become contracted to the translational-like degrees of freedom in the E(2)-like little group, which physically are gauge degrees of freedom. Together with D. Han and D. Son, I published a series of papers in the 1980s establishing that Wigner's little group together with group contraction techniques unifies the internal space-time symmetries of massive and massless particles.
You are then invited to build your own house. How are you going to build your own house? With what? In order to build a house consistent with it neighborhood is to find a Lorentz-covariant entity which takes different forms for slow and fast particles.
portrait by Bulent Atalay (1978)