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 ψ(x) = as(~p)u(s)(p)e−ipx + bs(~p)†v(s)(p)eipx (1) (2π)3 p 2Ep s and using the anti-commutation relations

{ar(~p), as(~q)†} = {br(~p), bs(~q)†} = (2π)3δ(3)(~p − ~q) {ar(~p), as(~q)} = {br(~p), bs(~q~)} = 0 (2)

I. Show that

† (3) † † {ψa(t, ~x), ψb (t, ~y)} = δ (~x − ~y)δab {ψa(t, ~x), ψb(t, ~y)} = {ψa(t, ~x), ψb (t, ~y)} = 0 (3) Write the expression of the Hamiltonian, the momentum operator, and the conserved charge Q µ ¯ µ s s † s s s † s (associated to the current J = ψγ ψ) 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) = ψ(x)Γψ(x) and Γ is a generic combination of Dirac matrices (Γ = γµ, γµγν,...). Tip: express the commutator in terms of {ψ(x), ψ¯(y)} and then use the fact that

{ψ(x), ψ¯(y)} = 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)ψ = (i∂/ − eA/ − m)ψ = 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 ψ = 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 ψ(~x,t) in terms of two-component spinors, within the standard representation, as

 U(~x,t)  ψ(~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)ψ = 0 (13) 4m µν where κ is an arbitrary coefficient. Does this new equation preserve gauge invariance?

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