Question: For a massless particle, there is no rest frame. One can fix the axis

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Question:

For a massless particle, there is no rest frame. One can fix the axis of the Minkowski coordinates so that p = (1,0,0,1). Then one can check that the little group is E(2), which is the isometry group of 21) Euclidean space: rotation and translations. Assuming that the internal space of the particle is finite dimensional, it has to be an eigenstate of both translation operators in E(2) with the same eigenvalue 0. The eigenvalue A for these states is quantized lambda = 0, plusminus 1/2, plusminus 1, plusminus 3/2,.., for the rotation of E(2). Prove that if the eigenvalues of the 2 translations of E(2) are not both zeros for all states in a representation, the representation is infinite dimensional.

Explanation / Answer

here the spectrum of a bounded operator is non empty,hence it is a infinite dimensional

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