Erratum

The subscript $0$ on $𝜙 0$ seems to appear without explanation. Presumably, it is meant to indicate that the field under consideration is a free field. However, the subscript is not used anywhere from Eq. (2.82) through Eq. (2.85), so every $𝜙$ in this passage should be understood as $𝜙 0$ .

Notes

Understanding the “low-energy/large-mass” limit

⃗ 𝑝 \leftrightarrow - 𝑖 ⃗ \nabla \Rightarrow \sqrt 𝑚 2 - ⃗ \nabla 2 \leftrightarrow \sqrt 𝑚 2 + ⃗ 𝑝 2

\sqrt 𝑚 2 - ⃗ 𝑝 2 = 𝑚 \sqrt \sqrt \sqrt \sqrt \sqrt ⎷ 1 - (∣ ⃗ 𝑝 ∣ 𝑚) 2

Let $𝛿 = (∣ ⃗ 𝑝 ∣ 𝑚) 2$ . In the low-velocity limit,

𝛿 = (∣ ⃗ 𝑝 ∣ 𝑚) 2 = (∣ ⃗ 𝑣 ∣ 𝑐) 2 \approx 0

\sqrt 𝑚 2 + ⃗ 𝑝 2 = 𝑚 \sqrt 1 + 𝛿 = 𝑚 (1 + 12 𝛿 + O (𝛿 2)) \approx 𝑚 + 𝑚 2 𝛿 = 𝑚 + ⃗ 𝑝 2 2 𝑚

while

𝑚 + ⃗ 𝑝 2 2 𝑚 \leftrightarrow 𝑚 - ⃗ \nabla 2 2 𝑚

Therefore, the “low-energy/large-mass” limit here is precisely the nonrelativistic limit $∣ ⃗ 𝑣 ∣ ≪ 𝑐$ .

Schwartz 2.3.2

Erratum

Notes

Understanding the “low-energy/large-mass” limit