Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 50 / 1 / b

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- Solution b

Solution

Apply

i γ^{ν} \partial_{ν} + m

to the Dirac equation. Since partial derivatives commute, only the symmetric part of the gamma matrix product contributes:

0 = (i γ^{ν} \partial_{ν} + m) (i γ^{μ} \partial_{μ} - m) ψ = - (\frac{1}{2} {γ^{ν}, γ^{μ}} \partial_{ν} \partial_{μ} + m^{2}) ψ = - (□ + m^{2}) ψ .

Here

□ = \partial_{t}^{2} - \nabla^{2}

in the signature fixed above. The remaining operator is scalar in spinor indices, giving

(□ + m^{2}) ψ_{α} = 0 for each of the four components .

This is the Klein-Gordon equation for each component, although the first-order Dirac equation additionally couples them.

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