By Ralph Decker Bennett
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Additional info for An Attempt to Test the Quantum Theory of X-Ray Scattering
The Hamiltonian, linear momentum and angular momentum of a scalar ﬁeld are H= 1 2 P = − M μν = d3 x[(∂0 φ)2 + (∇φ)2 + m2 φ2 ] , d3 x∂0 φ∇φ , d3 x(xμ T 0ν − xν T 0μ ) . • The Feynman propagator of a complex ﬁeld is deﬁned by iΔF (x − y) = 0| T (φ(x)φ† (y)) |0 . E) Time ordering is deﬁned by T φ(x)φ† (y) = θ(x0 − y0 )φ(x)φ† (y) + θ(y0 − x0 )φ† (y)φ(x) . 20. 23 present the transformations of a scalar ﬁeld under discrete transformations. 1. Starting from the canonical commutators ˙ [φ(x, t), φ(y, t)] = iδ (3) (x − y) , ˙ ˙ [φ(x, t), φ(y, t)] = [φ(x, t), φ(y, t)] = 0 , derive the following commutation relations for creation and annihilation operators: [a(k), a† (q)] = δ (3) (k − q) , [a(k), a(q)] = [a† (k), a† (q)] = 0 .
This is the Gupta–Bleuler method of quantization. H) Chapter 9. Canonical quantization of the electromagnetic ﬁeld 51 while A0 = 0. I) [π i (t, x), π j (t, y)] = 0 , (3) where π = E and δ⊥ij (x − y) is the transversal delta function given by (3) δ⊥ij (x − y) = 1 (2π)3 d3 keik·(x−y) δij − ki kj k2 . J) [a†λ (k), a†λ (q)] = 0 . • The Feynman propagator for the electromagnetic ﬁeld is given by iDFμν (x − y) = 0| T (Aμ (x)Aν (y)) |0 . 1. G) prove that [Aμ (t, x), A˙ ν (t, y)] = −ig μν δ (3) (x − y) .
10. Prove that ΔR,A |m2 =0 = − 1 θ(±t)δ(x2 ) . 11. If the source ρ is given by ρ(y) = gδ (3) (y), show that φR = g exp(−m|x|) , 4π |x| where φR (x) = − d4 yΔR (x − y)ρ(y). 12. Show that the Green function of the Dirac equation, S(x) has the following form S(x) = (i/ ∂ + m)Δ(x) , where Δ(x) is the Green function of the Klein–Gordon equation with corresponding boundary conditions. 13. B), determine the retarded, advanced, Feynman and Dyson propagators of the Dirac equation. Also, prove that the diﬀerence between any two of them is a solution of the homogenous Dirac equation.