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Additional resources for Introduction to QFT (CMU lecture notes draft)
Example text
Xn − 1) . (x1 A1 + . . 2. A) holds. 3. F). 4. Regularize the integral I= d4 k using Pauli–Villars regularization. 1 1 , k 2 (k + p)2 − m2 Chapter 11. 5. Compute kα kβ kµ kν kρ kσ . (k 2 )n Also, find the divergent part of the previous integral for n = 5. Apply the dimensional regularization. 6. Consider the interacting theory of two scalar fields φ and χ: L= 1 1 1 1 (∂φ)2 − m2 φ2 + (∂χ)2 − M 2 χ2 − gφ2 χ . 2 2 2 2 (a) Find the self–energy of the χ particle, −iΠ(p2 ). (b) Calculate the decay rate of the χ particle into two φ particles.
H) • Cutkosky rule for computing discontinuity of any Feynman diagram contains the following steps: 1. Cut through the diagram in all possible ways such that the cut propagators can be put on–shell. 2. For each cut, make the replacement p2 1 → (−2iπ)δ (4) (p2 − m2 )θ(p0 ) . − m2 3. Sum the contributions of all possible cuts. 1. Prove the following formula (the Feynman parametrization) 1 = (n − 1)! A1 . . An 1 1 ... 0 dx1 . . dxn 0 δ(x1 + . . + xn − 1) . (x1 A1 + . . 2. A) holds. 3. F). 4. Regularize the integral I= d4 k using Pauli–Villars regularization.
G. : c(q)c† (p) := −c† (p)c(q) , : c(q)c(k)c† (p) := c† (p)c(q)c(k) . • The Hamiltonian, momentum and angular moment of the Dirac field are: H= ¯ + m]ψ , d3 xψ[−iγ∇ P= −i Mµν = d3 xψ † ∇ψ , 1 d3 xψ † (i(xµ ∂ν − xν ∂µ ) + σµν )ψ . 2 • The Feynman propagator is given by ¯ |0 . F) Time ordering is defined by ¯ ¯ ¯ − θ(y0 − x0 )ψ(y)ψ(x) . G) U (Λ)ψ(x)U −1 (Λ) = S −1 (Λ)ψ(Λx) . Here U (Λ) is a unitary operator in spinor representation which generates the Lorentz transformation. H) where P is the appropriate unitary operator.