Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 301 / 2 / b

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- Solution b

Solution

Write every displayed block identity as

I_{2}

. Direct block multiplication in the Dirac representation of the gamma matrices gives

(γ^{0})^{2} = I_{4}, γ^{0} γ^{i} = (0 σ^{i} σ^{i} 0), γ^{i} γ^{0} = (0 - σ^{i} - σ^{i} 0) .

Thus their mixed anticommutator is zero. For two spatial indices,

γ^{i} γ^{j} = - (σ^{i} σ^{j} 0 0 σ^{i} σ^{j}) .

Using the Pauli matrix multiplication law gives

{γ^{0}, γ^{0}} = 2 I_{4}, {γ^{0}, γ^{i}} = 0, {γ^{i}, γ^{j}} = - 2 δ^{ij} I_{4} .

These exhaust all time-time, time-space and space-space cases of the Clifford algebra. No special choice of individual Pauli matrices beyond their stated anticommutator is needed.

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