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1 Quantization of the Dirac Field 2 the Magnetic Moment of the Electron Quantum Field Theory-I Problem Set n. 5 UZH and ETH, HS-2017 Prof. G. Isidori Assistants: M. Bordone, F. Buccioni, A. Karlberg, M. Zoller Due: 26-10-2017 http://www.physik.uzh.ch/en/teaching/PHY551/HS2017.html 1 Quantization of the Dirac field Starting from the decomposition Z d3~p 1 X (x) = as(~p)u(s)(p)e−ipx + bs(~p)yv(s)(p)eipx (1) (2π)3 p 2Ep s and using the anti-commutation relations far(~p); as(~q)yg = fbr(~p); bs(~q)yg = (2π)3δ(3)(~p − ~q) far(~p); as(~q)g = fbr(~p); bs(~q~)g = 0 (2) I. Show that y (3) y y f a(t; ~x); b (t; ~y)g = δ (~x − ~y)δab f a(t; ~x); b(t; ~y)g = f a(t; ~x); b (t; ~y)g = 0 (3) Write the expression of the Hamiltonian, the momentum operator, and the conserved charge Q µ ¯ µ s s y s s s y s (associated to the current J = γ ) in terms of Na (~p) = a (~p) a (~p) and Nb (~p) = a (~p) a (~p). II. Show that 2 [OΓ(x);OΓ(y)] = 0 for (x − y) < 0 (4) ¯ where OΓ(x) = (x)Γ (x) and Γ is a generic combination of Dirac matrices (Γ = γµ; γµγν;:::). Tip: express the commutator in terms of f (x); ¯(y)g and then use the fact that f (x); ¯(y)g = 0 for (x − y)2 < 0 (5) 2 The magnetic moment of the electron The generalization of the Dirac equation in presence of an electromagnetic field is (iD= − m) = (i@= − eA= − m) = 0 (6) I. Show that multiplying from the left by the operator (−i@= + eA= − m) the equation (6) one gets h e i (@ + ieA )(@µ + ieAµ) + σµνF + m2 = 0 (7) µ µ 2 µν As can be noted, this equation differs from a na¨ıve generalization of the Klein-Gordon equation e µν (via the replacement @µ ! Dµ) because of the covariant spin-dependent coupling 2 σ Fµν. 1 II. To understand the physical meaning of the extra term in Eq. (7) it is useful to solve Eq. (6) in the non-relativistic limit for a positive energy state. To this purpose, re-write (~x;t) in terms of two-component spinors, within the standard representation, as U(~x;t) (~x;t) = e−imt (8) V (~x;t) and assume that @tU (@tV ) mU (mV ). Derive the coupled linear equations for U and V , following from Eq. (6), and show that (i@~ − eA~) V (~x;t) ≈ ~σU(~x;t) (9) 2m Using this result, show that the equation of the upper component spinor U(~x;t) is 1 e i@ U(~x;t) = (i@~ − eA~)2 + eΦ − ~σB~ U(~x;t) (10) t 2m 0 2m This is nothing but the Pauli equation, with an intrinsic magnetic moment given by e ~σ e ~µ = − = − S~ (11) Dirac m 2 m where S~ is the spin vector of the spin-1/2 Dirac particle. This is a non-trivial prediction of the Dirac equation, which contradicts the semi-classical expectation. Indeed, a classical particle with charge e, orbiting with an angular momentum L~ , generates a magnetic moment e ~µ = − L~ (12) Classic 2m The Dirac equation predicts an extra factor of 2 (the so-called gyromagnetic ratio) with respect to the na¨ıve replacement L~ ! S~. III. Show how Eq. (11) changes if we include in the Dirac equation the following non-minimal coupling of the electromagnetic field eκ (i@= − eA= − σµνF − m) = 0 (13) 4m µν where κ is an arbitrary coefficient. Does this new equation preserve gauge invariance? 2.
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