 Solution

ID: past-exam-of-the-groups-and-representations-course-of-the-university-of-oxford/2018/3/c/solution

Choose the following Pauli matrix realization, with its spinor basis fixed for the later Dirac adjoint:

r (K_{1}) = \frac{σ _{3}}{2} = \frac{1}{2} (10 0 - 1), r (K_{2}) = - \frac{σ _{2}}{2} = \frac{1}{2} (0 - i i 0), r (J) = - \frac{i σ _{1}}{2} = \frac{1}{2} (0 - i - i 0) .

[r (J), r (K_{1})] = r (K_{2}), [r (J), r (K_{2})] = - r (K_{1}), [r (K_{1}), r (K_{2})] = - r (J) .

Extending linearly therefore gives a two-dimensional Lie algebra representation. For an irreducible representation, it suffices to rule out an invariant line. Any such line would have to be an eigenline of

r (K_{1})

, whose two eigenvalues are distinct. Those two coordinate lines are interchanged by

r (K_{2})

, so neither is invariant. There is no proper nonzero invariant subspace.

This is a spinorial representation rather than a single-valued two-dimensional representation of the identity component of the specified orthogonal group:

e^{2 π J} = I_{3}, e^{2 π r (J)} = - I_{2} .

The mismatch explicitly exhibits the need for the spin double cover. It causes no obstruction to the requested infinitesimal construction.

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