By F.J. Dyson (lecture notes), Michael J. Moravcsik (editor)
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Extra resources for Advanced Quantum Mechanics, 2nd Edition
It is a very bad method, it is complicated; and it is not at all obvious or even easy to prove that the theory so made is relativistic, because the whole Hamiltonian approach is non-covariant. Just recently we learnt a much better way of doing it, which I shall now expound in these lectures. 16 35 FIELD THEORY References: R. P. Feynman, Rev. Mod. Phys. 20 (1948) 367 Phys. Rev. 80 (1950) 440 J. Schwinger, Phys. Rev. 82 (1951) 914 It is relativistic all the way, and it is much simpler than the old methods.
Solve the Schr¨odinger equation in the Born approximation a) By stationary perturbation theory b) By time-dependent perturbation theory. Show that the results agree, with a transition probability per unit time given by w = (2π/ )ρE |VBA |2 . Evaluate the cross section in the case V = −Ze2 /r, averaging spin over initial state and summing over final state. c) Repeat the calculation with particles obeying the Klein-Gordon equation, leaving out the V 2 term, by either method. Compare the angular distribution in the two cases.
Let the H H surfaces σ1 and σ2 be varied so that the point xµ moves to xµ + δxµ . And let the function L also be varied so that it is replaced by L + δL where δL is any expression involving the φα and φα µ . Under this triple variation (174) gives i i δIH (Ω) exp IH (Ω) (176) δ φ′1α , σ1 φ′2α , σ2 = N H Using (175) this may be written δ φ′1α , σ1 φ′2α , σ2 = i φ′1α , σ1 δI(Ω) φ′2α , σ2 . (177) Here δI(Ω) is the operator obtained by making the three variations on the operator I(Ω). Formally δI(Ω) is the same as the variation obtained in the classical theory, δI(Ω) = 1 c δL + Ω α,µ ∂ ∂L ∂L − ∂φα ∂xµ ∂φα µ δφα + σ1 − d 4x πα δφα + σ2 α,µ 1 nµ L − φα µ πα δxµ c dσ (178) 37 FIELD THEORY Only now everything on the RHS of (178) is an operator.