By Srednicki M.
This can be a draft model of half III of a three-part introductory textbook onquantum box thought.
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Extra resources for Quantum field theory: spin one
Example text
The vertex factors associated with the last two terms are −2ie2 gµν and −iλ. To get the vertex factor for the first term, we note that if |k is an incoming selectron state, then 0|ϕ(x)|k = eikx and 0|ϕ† (x)|k = 0; and if k ′ | is an outgoing selectron ′ state, then k ′ |ϕ† (x)|0 = e−ik x and k ′ |ϕ(x)|k = 0. Therefore, in free field theory, ′ k ′ |(∂µ ϕ† )ϕ|k = −ikµ′ e−i(k −k)x , ′ k ′ |ϕ† ∂µ ϕ|k = +ikµ e−i(k −k)x . (191) (192) This implies that the vertex factor for the first term in eq. (190) is given by i(ie)[(−ikµ′ ) − (ikµ )] = ie(k + k ′ )µ .
Using results in section 50 we find 0 √ 1 , |k] = 2ω 0 0 |k = √ 0 0 . 2ω 1 (148) 0 For any value of q, the twistor q| takes the form q| = (0, 0, α, β) , (149) where α and β are complex numbers. Plugging eqs. (148) and (149) into eq. (145), and using 0 σµ µ γ = (150) σ ¯µ 0 along with σ µ = (I, σ) and σ ¯ µ = (I, −σ), we find that we reproduce eq. (147) √ with eiφ = 1 and C = −β/( 2αω). There is now no need to check eq. (146), because εµ− (k) = −[εµ+ (k)]∗ , as can be seen by using q k ∗ = −[q k] along with another result from section 50, q|γ µ |k]∗ = k|γ µ |q].
With a factor of e, this current should be identified as the electromagnetic current. We have not previously contemplated the notion that the electromagnetic current could involve the gauge field itself, but in scalar electrodynamics this arises naturally, and is essential for gauge invariance. It also poses no special problem in the quantum theory. We will make the same assumption that we did for spinor electrodynamics: namely, that the correct procedure is to omit integration over the component of A˜µ (k) that is parallel to kµ , on the grounds that this integration is redundant.
Quantum field theory: spin one by Srednicki M.
by Daniel
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