Srednicki — Quantum Field Theory · Problem 50.5

Canonical Quantization of Scalar Fields — Mark Srednicki

Physics — QFT · spinor-helicity-fierz-identities

Problem

a) Prove the useful identities ⟨p∣γμ∣k]=[k∣γμ∣p⟩,⟨p∣γμ∣k]∗=⟨k∣γμ∣p],⟨p∣γμ∣p]=2pμ,⟨p∣γμ∣k⟩=0,[p∣γμ∣k]=0.(50.38)\begin{aligned} \langle p|\gamma^\mu|k] &= [k|\gamma^\mu|p\rangle , \tag{50.38}\\ \langle p|\gamma^\mu|k]^* &= \langle k|\gamma^\mu|p] , \\ \langle p|\gamma^\mu|p] &= 2p^\mu , \\ \langle p|\gamma^\mu|k\rangle &= 0 , \\ [p|\gamma^\mu|k] &= 0 . \end{aligned} b) Extend the last two identities of part (a): show that the product of an odd number of gamma matrices sandwiched between either ⟨p∣\langle p| and ∣k⟩|k\rangle or [p∣[p| and ∣k]|k] vanishes. Also show that the product of an even number of gamma matrices between either ⟨p∣\langle p| and ∣k]|k] or [p∣[p| and ∣k⟩|k\rangle vanishes. c) Prove the Fierz identities, −12⟨p∣γμ∣q]γμ=∣q]⟨p∣+∣p⟩[q∣,−12[p∣γμ∣q⟩γμ=∣q⟩[p∣+∣p]⟨q∣.(50.43)\begin{aligned} -\tfrac{1}{2}\langle p|\gamma_\mu|q]\gamma^\mu &= |q]\langle p| + |p\rangle[q| , \tag{50.43}\\ -\tfrac{1}{2}[p|\gamma_\mu|q\rangle\gamma^\mu &= |q\rangle[p| + |p]\langle q| . \end{aligned} Now take the matrix element of eq. (50.44) between ⟨r∣\langle r| and ∣s]|s] to get another useful form of the Fierz identity, [p∣γμ∣q⟩ ⟨r∣γμ∣s]=2 [p s]⟨q r⟩.(50.45)[p|\gamma^\mu|q\rangle\,\langle r|\gamma_\mu|s] = 2\,[p\,s]\langle q\,r\rangle . \tag{50.45}

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