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 Incoming links: Past exam of the quantum field theory course of the University of Oxford / 2024 / 1 / c

Choose the zero of energy so that the translation-invariant vacuum state has

P^{μ} ∣Ω ⟩ = 0

m_{ψ}

p^{μ} = (E_{p}, p), E_{p} = m_{ψ}^{2} + ∣ p ∣^{2}, p^{2} = m_{ψ}^{2} .

⟨ Ω∣ ϕ (x) ∣ ψ, p ⟩ = ⟨ Ω∣ e^{i P \cdot x} ϕ (0) e^{- i P \cdot x} ∣ ψ, p ⟩ = e^{- i p \cdot x} ⟨ Ω∣ ϕ (0) ∣ ψ, p ⟩ .

Here

p \cdot x = E_{p} t - p \cdot x

. A different additive vacuum energy must be subtracted consistently from both energy eigenvalues before identifying the physical

p^{0}