Further Contents of E = mc²

Einstein lived in this house (Bern, Switzerland) from 1903 to 1905. During this period, he formulated his special theory of 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.

Einstein's Genealogy Massive/Slow between Massless/Fast

Energy Momentum E=p2/2m Einstein's E=(m2 + p2)1/2 E=p

Spin, Gauge, Helicity S3 S1 S2 Wigner's Little Group S3 Gauge Trans.

Click here for a photo. Build YOUR OWN HOUSE!

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.