 Solution

ID: past-exam-of-the-advanced-quantum-field-theory-for-particle-physics-course-of-the-university-of-oxford/2020/1/ii/solution

γ^{ν} γ^{μ} = 2 g^{νμ} - γ^{μ} γ^{ν}

and

γ^{ν} γ_{ν} = D 1

. Therefore

γ^{ν} γ^{μ} γ_{ν} = 2 g^{νμ} γ_{ν} - γ^{μ} γ^{ν} γ_{ν} = 2 γ^{μ} - D γ^{μ}, γ^{ν} γ^{μ} γ_{ν} = (2 - D) γ^{μ} .

The contraction

g^{ν}_{ν} = D

must be retained until the dimensional regularization calculation is complete; replacing it prematurely by four can change finite terms multiplying a ultraviolet pole.

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