PHY–396 K. Problem set #4. Due October 2, 2012. 1. Consider an O ...

PHY–396 K. Problem set #4. Due October 2, 2012.1. Consider an O(N) symmetric Lagrangian for N interacting real scalar fields,L = 1 2N∑ ( ) 2 m 2∂µ Φ a −a=12(N∑Φ 2 a − λ N 2∑Φ 224a). (1)a=1a=1By the Noether theorem, the continuous SO(N) subgroup of (N) symmetry gives rise to12N(N − 1) conserved currentsJ µ ab (x) = −Jµ ba (x) = Φ a(x) ∂ µ Φ b (x) − Φ b (x) ∂ µ Φ a (x). (2)In the quantum field theory, these currents become operatorsĴ ab (x, t) = −Ĵ ba (x, t) = −ˆΦ a (x, t)∇ˆΦ b (x, t) + ˆΦ b (x, t)∇ˆΦ a (x, t),Ĵ 0 ab (x, t) = −Ĵ0 ba (x, t) ˆΦ a (x, t)ˆΠ b (x, t) − ˆΦ b (x, t)ˆΠ a (x, t).(3)This problem is about the net charge operators∫∫ˆQ ab (t) = − ˆQ ba (t) = d 3 x Ĵ0 (x) =d 3 x(ˆΦa (x, t)ˆΠ b (x, t) − ˆΦ)b (x, t)ˆΠ a (x, t) . (4)(a) Write down the equal-time commutation relations for the quantum ˆΦ a and ˆΠ a fields.Also, write down the Hamiltonian operator for the interacting fields.(b) Show that[ˆQab (t), ˆΦ c (x, same t)][ˆQab (t), ˆΠ c (x, same t)]= iδ bc ˆΦa (x, t) − iδ ac ˆΦb (x, t),= iδ bc ˆΠa (x, t) − iδ ac ˆΠb (x, t),(5)(c) Show that the all the ˆQ ab commute with the Hamiltonian operator Ĥ. In the Heisenbergpicture, this makes all the charge operators ˆQ ab time independent.1

(d) Verify that the ˆQ ab obey commutation relations of the SO(N) generators,[ˆQab , ˆQ cd]= −iδ [c[b[ ˆQa]d] ≡ −iδ bc ˆQad + iδ ac ˆQbd + iδ bd ˆQac − iδ ad ˆQbc . (6)(e) In the Schrödinger picture ˆΦ a (x) and ˆΠ a (x) can be expanded into creation and annihilationoperators as if they were free fields. Show that in terms of creation andannihilation operators, the charges (4) becomeˆQ ab= ∑ p()−iâ † p,aâ p,b+ iâ † p,bâp,b . (7)Finally, for N = 2 the SO(2) symmetry is the phase symmetry of one complex fieldΦ = (Φ 1 + iΦ 2 )/ √ 2 and its conjugate Φ ∗ = (Φ 1 − iΦ 2 )/ √ 2. In the Fock space, they giverise to particles and anti-particles of opposite charges.(f) Show that for N = 2ˆQ 21 = − ˆQ 12 = ˆN particles − ˆN antiparticles = ∑ p(â † pâ p − ˆb † pˆb p)(8)whereâ p = âp,1 + iâ p,2√2are particle annihilation operators,ˆbp = âp,1 − iâ p,2√2â † p = ↠p,1 − i↠p,2√2are antiparticle annihilation operators,are particle creation operators,(9)ˆb† p = ↠p,1 + i↠p,2√2are antiparticle creation operators.2

2. Now consider a massive relativistic vector field A µ (x) with the Lagrangian densityL = − 1 4 F µνF µν + 1 2 m2 A µ A µ − A µ J µ (10)(in ¯h = c = 1 units) where the current J µ (x) is a fixed source for the A µ (x) field. Becauseof the mass term, the Lagrangian (10) is not gauge invariant. However, we assume thatthe current J µ (x) is conserved, ∂ µ J µ (x) = 0.In an earlier homework (set 1, problem 1) we have derived the Euler–Lagrange equationsfor the massive vector field. In this problem, we develop the Hamiltonian formalism forthe A µ (x). Our first step is to identify the canonically conjugate “momentum” fields.(a) Show that ∂L/∂Ȧ = −E but ∂L/∂ ˙ A 0 ≡ 0.In other words, the canonically conjugate field to A(x) is −E(x) but the A 0 (x) does nothave a canonical conjugate! Consequently,∫ ()H = d 3 x −Ȧ(x) · E(x) − L . (11)(b) Show that in terms of the A, E, and A 0 fields, and their space derivatives,∫ { }H = d 3 x 12E 2 + A 0 (J 0 − ∇ · E) − 1 2 m2 A 2 0 + 1 2 (∇ × A)2 + 1 2 m2 A 2 − J · A .Because the A 0 field does not have a canonical conjugate, the Hamiltonian formalism doesnot produce an equation for the time-dependence of this field. Instead, it gives us a timeindependentequation relating the A 0 (x, t) to the values of other fields at the same time t.Specifically, we haveδHδA 0 (x) ≡∂H∂A 0∣ ∣∣∣x− ∇ ·(12)∂H∂(∇A 0 ) ∣ = 0. (13)xAt the same time, the vector fields A and E satisfy the Hamiltonian equations of motion,∂A(x, t) = −δH[ ]∂H∂t δE(x) ∣ ≡ −t∂E − ∇ ∂Hi,∂(∇ i E)(x,t)∂E(x, t) = +δH[ ](14)∂H∂t δA(x) ∣ ≡ +t∂A − ∇ ∂Hi.∂(∇ i A)(c) Write down the explicit form of all these equations.(x,t)3

(d) Verify that the equations you have just written down are equivalent to the relativisticEuler–Lagrange equations for the A µ (x), namely(∂ µ ∂ µ + m 2 )A ν = ∂ ν (∂ µ A µ ) + J ν (15)and hence ∂ µ A µ (x) = 0 and (∂ ν ∂ ν + m 2 )A µ = 0 when ∂ µ J µ ≡ 0, cf. homework #1.3. Next, let’s quantize the massive vector fields. Since classically the −E(x) fields are canonicallyconjugate momenta to the A(x) fields, the corresponding quantum fields Ê(x) andÂ(x) satisfy the canonical equal-time commutation relations[Âi(x, t), Âj(y, t)] = 0,[Êi(x, t), Êj(y, t)] = 0,(16)[Âi(x, t), Êj(y, t)] = −iδ ij δ (3) (x − y)(in the ¯h = c = 1 units). The currents also become quantum fields Ĵµ (x, t), but theyare composed of some kind of charged degrees of freedom rather than the vector fieldsin question. Consequently, at equal times the currents Ĵµ (x, t) commute with both theÊ(y, t) and the Â(y, t) fields.The classical A 0 (x, t) field does not have a canonical conjugate and its equation of motiondoes not involve time derivatives. In the quantum theory, Â 0 (x, t) satisfies a similar timeindependentconstraintm 2 Â 0 (x, t) = Ĵ0 (x, t) − ∇ · Ê(x, t), (17)but from the Hilbert space point of view this is an operatorial identity rather than anequation of motion. Consequently, the commutation relations of the scalar potential fieldfollow from eqs. (16); in particular, at equal times the Â0 (x, t) commutes with the Ê(y, t)but does not commute with the Â(y, t). 4

Finally, the Hamiltonian operator follows from the classical eq. (12), namely∫ {)(Ĥ = d 3 1x 2 Ê 2 +(Ĵ0 Â) }2 Â0 − ∇ · Ê −2 1m2  2 0 + 1 2∇ × +12m 2  2 − Ĵ · Â∫ {= d 3 1x 2 Ê 2 + 1) 2 ((Ĵ0 Â) }22m 2 − ∇ · Ê +12∇ × +12m 2  2 − Ĵ · Âwhere the second line follows from the first and eq. (17).Your task is to calculate the commutators [Âi(x, t), Ĥ] and [Êi(x, t), Ĥ] and write downthe Heisenberg equations for the quantum vector fields. Make sure those equations aresimilar to the Hamilton equations for the classical fields.(18)5