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The states c† (p, r) |0 , and d† (p, r) |0 are the electron and positron one–particle states, respectively with defined momentum and polarization. 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) .

D), V d3 p/(2π)3 is the volume element of phase space. • Feynman rules for QED: ◦ Vertex: = ieγ µ ◦ Photon and lepton propagators: =− iDF µν = iSF (p) = = igµν , k2 + i i . e. e. positron): = v¯(p, s) final = εµ (k, λ) initial c) photons: = ε∗µ (k, λ) final ◦ Spinor factor are written from the left to the right along each of the fermionic lines. The order of writing is important, because it is a question of matrix multiplication of the corresponding factors. ◦ For all loops with momentum k, we must integrate over the momentum: d4 k/(2π)4 .

Let (a) Fµ = ∂µ φ , (b) S = d4 x 12 (∂µ φ)2 − V (φ) , be functionals. Calculate the functional derivatives 2 δ S δφ(x)δφ(y) δFµ δφ in the first case, and in the second case. 2. Find the Euler–Lagrange equations for the following Lagrangian densities: (a) L = −(∂µ Aν )(∂ν Aµ ) + 12 m2 Aµ Aµ + λ2 (∂µ Aµ )2 , (b) L = − 41 Fµν F µν + 12 m2 Aµ Aµ , where Fµν = ∂µ Aν − ∂ν Aµ , (c) L = 12 (∂µ φ)(∂ µ φ) − 12 m2 φ2 − 14 λφ4 , (d) L = (∂µ φ − ieAµ φ)(∂ µ φ∗ + ieAµ φ∗ ) − m2 φ∗ φ − 14 Fµν F µν , ¯ 5 ψφ . 3.

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A Statistical Quantum Theory of Regular Reflection and Refraction by Cox R.T., Hubbard J.C.

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