A. Zee, Quantum Field Theory in a Nutshell · Problem III.6.2

Field Theory Redux

Physics — QFT

Problem

Check that (7) is consistent with current conservation. [Hint: By translation invariance (we suppress the spin variable) ⟨p′∣Jμ(x)∣p⟩=⟨p′∣Jμ(0)∣p⟩ei(p′−p)x\langle p'| J^\mu(x)|p\rangle = \langle p'| J^\mu(0)|p\rangle e^{i(p'-p)x} and hence ⟨p′∣∂μJμ(x)∣p⟩=i(p′−p)μ⟨p′∣Jμ(0)∣p⟩ei(p′−p)x\langle p'| \partial_\mu J^\mu(x)|p\rangle = i(p' - p)_\mu\langle p'| J^\mu(0)|p\rangle e^{i(p'-p)x} Thus current conservation implies that qμ⟨p′∣Jμ(0)∣p⟩=0q_\mu\langle p'| J^\mu(0)|p\rangle = 0.]

Browse the problem library