Practical task for exam in QFT(量子场论复习).pdfVIP

1 1. Consider free scalar field ϕ(x) = ϕ (x) + ϕ− (x), with ϕ− being positive(negative)-energy part of field operator: ⟨Ω|ϕ− = ϕ |Ω⟩ = 0, and |Ω⟩ is vacuum state. - Show that [ − ] − ϕ (x), ϕ (y) = ⟨Ω|ϕ (x)ϕ (y)|Ω⟩I = ⟨Ω|ϕ(x)ϕ(y)|Ω⟩I, where I is unity operator. - Show that ⟨Ω|Φ(x)Φ(y)|Ω⟩ = ⟨Ω|ϕ (x)ϕ (x)ϕ− (y)ϕ− (y)|Ω⟩, where Φ(x) =:ϕ (x): (normal ordered). - Use the commutation relation and derive ⟨Ω|T (Φ(x)Φ(y)) |Ω⟩ = 2 (⟨Ω|T (ϕ(x)ϕ(y)) |Ω⟩) , (1) ( ) * Compare the previous result (1) to ⟨Ω|T ϕ (x)ϕ (y) |Ω⟩ given by application of Wick’s theorem. Discuss the discrepancy. 2. Electron-electron scattering is called Møller scattering. Consider the Møller scattering e− p e− p′ → − k e− (k′ ) at the leading order. In particular: ( ) ( ) e ( ) - Draw Feynman diagrams for the leading(tree) order amplitude and write down their expres- sions. - Calculate the scattering probability (squared amplitude) for unpolarized final and initial states. (for simplicity set mass of electron to 0) - Express the scattering probability via the Mandelstam variables, and via the scattering angle in the center-mass frame θ. * Explain the origin of singularity at θ = 0. 2 3. Vacuum expectation of odd number of electromagnetic currents is zero. P