Lectures on Quantum Field Theory. Ashok Das

      (Alternatively, we can say that Π₁|p, h〉 = 0 = Π₂|p, h〉 and this is the reason for the earlier assertion.) This corresponds to the one dimensional representation of E₂ known as the “degenerate” representation. Clearly, such a state would correspond to the highest or the lowest helicity state. Furthermore, if our theory is also invariant under parity (or space reflection), the space of physical states would also include the state with the opposite helicity (recall that helicity changes sign under space reflection, see (3.148)). As a result, massless theories with nontrivial spin that are parity invariant would have two dimensional representations corresponding to the highest and the lowest helicity states, independent of the spin of the particle. On the other hand, if the theory is not parity invariant, the dimensionality of the representation will be one dimensional, as we have seen explicitly in the case of massless fermion theories describing neutrinos.

      Incidentally, the fact that the massless representations have to be one dimensional, in general, can be seen in a heuristic way as follows. Let us consider spin as arising from a circular motion. Then, it is clear that since a massless particle moves at the speed of light, the only consistent circular motion a massless particle can have, is in a plane perpendicular to the direction of motion (otherwise, some component of the velocity would exceed the speed of light). In other words, in such a case, spin can only be either parallel or anti-parallel to the direction of motion leading to the one dimensional nature of the representation. However, if parity (space reflection) is a symmetry of the system, then we must have states corresponding to both the circular motions leading to the two dimensional representation.

4.4References

      1.V. Bargmann, Irreducible unitary representations of the Lorentz group, Annals of Mathematics 48, 568 (1947).

      2.A. Das and S. Okubo, Lie Groups and Lie Algebras for Physicists, Hindustan Publishing, India and World Scientific Publishing, Singapore (2014).

      3.E. Wigner, On unitary representations of the inhomogeneous Lorentz group, Annals of Mathematics 40, 149 (1939).

      4.E. Wigner, Group Theory and its Applications to the Quantum Mechanics of Atomic Spectra, Academic Press, New York (1959).

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      ¹See, for example, Quantum Mechanics: A Modern Introduction, A. Das and A. C. Melissinos (Gordon and Breach), page 289 or Lectures on Quantum Mechanics, A. Das (Hindustan Book Agency, New Delhi), page 182 (note there is a typo in the sign of the 23 element for L₂ in this reference).