Cylindrical group and massless particles

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Abstract
It is shown that the representation of the E(2)‐like little group for photons can be reduced to the coordinate transformation matrix of the cylindrical group, which describes movement of a point on a cylindrical surface. The cylindrical group is isomorphic to the two‐dimensional Euclidean group. As in the case of E(2), the cylindrical group can be regarded as a contraction of the three‐dimensional rotation group. It is pointed out that the E(2)‐like little group is the Lorentz‐boosted O(3)‐like little group for massive particles in the infinite‐momentum/zero‐mass limit. This limiting process is shown to be identical to that of the contraction of O(3) to the cylindrical group. Gauge transformations for free massless particles can thus be regarded as Lorentz‐boosted rotations.