By Eugene Merzbacher
Offers a scientific and orderly improvement of the total of quantum mechanics when it comes to its purposes to atomic, nuclear, particle, and strong nation physics.
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Additional info for Quantum Mechanics, 2nd Edition
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 Mechanics, 2nd Edition by Eugene Merzbacher