Einstein and Wigner
- Einstein derived one formula for the energy-momentum relation for massive particles at rest,
massless particles moving with speed of light, and massive particles moving with speed
comparable with that of light. This aspect is well known.
The internal space-time symmetry for a massive particle is that of the three-dimensional rotation group or O(3). This aspect is also well known.
- In his paper of 1939, Wigner noted that the internal space-time
symmetry of a massless has the symmetry of the two-dimensional Euclidean group consisting
of one rotation and two translations in perpindicular directions.
The massless particle has one rotational degree corresponding to helicity, and one unobservable gauage degree of freedom. How can those two translations describe one gauge gauage degree degree of freesom?
With Wigner, in 1987, I published a paper in J. of Math. Pysics which can be summaried with the following figure.
The circle in this figure is for the massive particle at rest with three degrees of freedom. We can consider the plane tangential at the north pole, but there cal also be a cyinder tangential at the equatorial belt. This cylinder has one rotational degree of freedon and one up-down degree of freedom, corresponding to gauge transformation. This is precisely the physical intepretation.
- Furthermore, this sphere can be expanded along the vertical direction to become
a cylider, as shown in this figure.
This procedure is known as the
group contraction procedure formulated by Inonu and Wigner in 1953.
Wightman and Michel were not aware of this contraction procedure. This presumably is the reasoh why they said I was wrong.