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 Incoming links: Past exam of the groups and representations course of the University of Oxford / 2018 / 3 / c

Past exam of the groups and representations course of the University of Oxford / 2018 / 3 / e / Solution 2026-10-25

The spinor basis in c is compatible with the supplied gamma matrices. The rotation generator

r (J)

is anti-Hermitian and commutes with

γ_{0} = - i σ_{1}

. The two boost generators are Hermitian matrices and anticommute with

γ_{0}

. It follows separately for all three generators, and hence for every real

T \in so (1, 2)

, that

r (T)^{†} γ_{0} + γ_{0} r (T) = 0.

For the infinitesimal Lorentz transformation,

δ ψ = r (T) ψ

and

δ ψ^{†} = ψ^{†} r (T)^{†}

. Hence the Dirac adjoint transforms contragrediently:

δ \overline{ψ} = ψ^{†} r (T)^{†} γ_{0} = - ψ^{†} γ_{0} r (T) = - \overline{ψ} r (T) .

The two contributions to the scalar variation cancel:

δ (\overline{ψ} ψ) = (δ \overline{ψ}) ψ + \overline{ψ} δ ψ = 0.

This proves the spinor invariant in three spacetime dimensions has infinitesimal Lorentz invariance. Real transformation parameters matter because the Hermitian adjoint conjugates coefficients; this is not an invariance under arbitrary complexified parameters. For ordinary commuting spinor components, the bilinear is purely imaginary in this mostly-plus convention, and multiplying it by

i

gives a real invariant. Its reality is separate from its invariance.

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