ADVANCED QUANTUM FIELD THEORY

Transcription

1 Imperial College London MSc EXAMINATION May 2016 This paper is also taken for the relevant Examination for the Associateship ADVANCED QUANTUM FIELD THEORY For Students in Quantum Fields and Fundamental Forces Friday, 6th May 2016: 14:00 to 17:00 Answer THREE out of the following FOUR questions. Marks shown on this paper are indicative of those the Examiners anticipate assigning. General Instructions Complete the front cover of each of the THREE answer books provided. If an electronic calculator is used, write its serial number at the top of the front cover of each answer book. USE ONE ANSWER BOOK FOR EACH QUESTION. Enter the number of each question attempted in the box on the front cover of its corresponding answer book. Hand in THREE answer books even if they have not all been used. You are reminded that Examiners attach great importance to legibility, accuracy and clarity of expression. c Imperial College London Go to the next page for questions

2 You may use the following results without proof: Chain rule for functional differentiation: δf (x) δb(y) = d 4 z δf (x) δa(z) δa(z) δb(y). Minkowski-space loop integral in d = 4 ɛ dimensions: I n (m 2 ) := µ ɛ d d p 1 (2π) d (p 2 m 2 + iɛ ) n = ( 1) n iµɛ m d 2n Γ(n 1 2 d), (4π) d/2 Γ(n) where Γ(z + 1) = zγ(z), such that Γ(m) = (m 1)! if m is a positive integer and Γ(δ) = δ 1 γ + O(δ), Γ( 1 + δ) = (δ γ) + O(δ), for small δ where γ = is the Euler Mascheroni constant. Gaussian path integrals: real: Grassmannian: Dφ e 1 2 d d x d d y φ(x)m(x,y)φ(y) = const. det M, Dθ Dθ e d d x d d y θ (x)m(x,y)θ(y) = det M 2 Go to the next page for questions

3 1. Consider a particle with action tb S[q; t a, t b ] = dt [ 1 2 m q2 V (q) ] t a Let q, t be the eigenstate of the position operator ˆq(t) q; t = q q; t at time t. The amplitude for the particle to evolve from position q a at time t a to position q b at time t b is given by the path integral U(q a, q b ; t b t a ) := q b ; t b q a ; t a = q(tb )=q b q(t a)=q a Dq e is[q:ta,t b]/. (i) Discuss briefly the physical interpretation of this path integral. In the limit 0, which paths q(t) in the integral give the largest contribution to the amplitude? Why? By inserting factors of 1 = dq q, t q, t, show that q b ; t b ˆq(t) q a ; t a = dq U(q, q b ; t b t) qu(q a, q; t t a ) = q(tb )=q b q(t a)=q a Dq q(t)e is[q;ta,t b]/. Hence argue that in general one gets the time-ordered expression q(tb )=q b q(t a)=q a Dq q(t 1 ) q(t n ) e is[q;ta,t b]/ = q b ; t b T ˆq(t 1 ) ˆq(t n ) q a ; t a. lim T (1 iɛ) [7 marks] (ii) Setting = 1 and given q; t = e iĥt q where Ĥ is the Hamiltonian, show that lim T (1 iɛ) q; T Ω where Ω is the ground state. Hence show that Dq q(t1 ) q(t n ) e is[q; T,T ] Ω T ˆq(t 1 ) ˆq(t n ) Ω =. Dq e is[q; T,T ] (iii) The generating functional is given by Z[J] = lim T (1 iɛ) Dq e is[q]+i dt q(t)j(t). [6 marks] By considering a change of integration variables in Z[J] from q(t) to q (t) = q(t) + α(t) for arbitrary, small α(t), show that ( ) δs[q] dt α(t) lim Dq T (1 iɛ) δq(t) + J(t) e is[q]+i dt q(t)j(t) = 0. (You may assume Dq = Dq.) (iv) Hence show that m d2 dt Ω T dv (t) 2 ˆq(t)ˆq(t ) Ω + Ω T dq ˆq(t ) Ω = iδ(t t ). [3 marks] [Total 20 marks] 3 Please go to the next page

4 2. The generating functionals Z[J], E[J] and Γ[φ cl ] for a scalar theory are related by i ln Z[J] = E[J] = Γ[φ cl ] d 4 xφ cl (x)j(x), where φ cl (x) = δe[j]/δj(x), and the corresponding correlation functions are given by G n (x 1,..., x n ) = ( i)n δ n Z[J] Z[0] δj(x 1 ) δj(x n ), J=0 G n (x 1,..., x n ) = ( i) n+1 δ n E[J] δj(x 1 ) δj(x n ), J=0 δ n Γ[φ cl ] Γ n (x 1,..., x n ) = i. δφ cl (x 1 ) δφ cl (x n ) φcl =0 (i) In a perturbation expansion, what kind of Feynman diagrams contribute to G n? What about G n and Γ n? Consider λφ 4 theory. Argue that G 2 (x 1, x 2 ) = G 2 (x 1, x 2 ) and draw the tree-level and one-loop Feynman diagrams that contribute to the functions G 4, G 4 and Γ 4. [6 marks] (ii) If A is a constant, the effective potential V eff (A) is defined by Γ[φ cl ] φcl =A = d 4 x V eff (A). Ignoring terms independent of A, show that 1 V eff (A) = i n! An Γn (0,..., 0), n=1 where the Fourier transform of Γ n (x 1,..., x n ) is (2π) 4 δ (4) ( i p i) Γ n (p 1,..., p n ). (iii) Draw the Feynman diagrams that give the one-loop contribution to V eff (A) in λφ 4 theory. Using the standard Feynman rules show that ( ) V 1-loop i d 4 n p λ 0 eff (A) = 2 n+1 n (2π) 4 p 2 m 2 n=1 0 + A 2n, iɛ ( = 1 2 i d 4 p p 2 (2π) log m λ 0A 2 + iɛ ) 4 p 2 m0 2 + iɛ (You do not have justify the symmetry factors.) Which terms in the power series expression are divergent? (iv) Show that, using dimensional regularisation, V 1-loop eff = λ [ ( ) 0 ( m 2 A 2 64π λ 0A 2) 2 ɛ + log 4πµ 2 m λ 0A 2 ] γ + 1. [Total 20 marks] 4 Please go to the next page

5 3. The action for electromagnetism is S[A µ ] = 1 4 d 4 x F µν F µν where F µν = µ A ν ν A µ. It is invariant under gauge transformations A µ A (α) µ = A µ + 1 e µα. (i) Show that in momentum space S[à µ ] = 1 2 i d 4 p d 4 q (2π) 4 (2π) 4 à µ(p) M µν (p, q)ã ν (q), where M µν (p, q) = i(2π) 4 δ (4) (p q) (p 2 η µν p µ p ν ) and à µ (p) = à µ( p) is the Fourier transform of A µ (x). Show that M µν (p, q) is not invertible and hence argue that the standard procedure for defining the propagator 0 T A µ (x)a ν (y) 0 fails. (ii) Let G[A µ ](x) = w(x) be some (local) gauge-fixing condition. By inserting ( ) δg[a µ (α) ] 1 = Dα det δ(g[a µ (α) ] w) δα show that the partition function of A µ (x) can be written as ( ) Z[J µ δg[a µ (α) ] ] = DA µ det e is ξ[a µ]+i d 4 xj µ (x)a µ(x), δα where S ξ [A µ ] = S[A µ ] 1 2ξ d 4 x G[A µ ] 2 for some constant ξ. Do you expect correlation functions of A µ to depend on G[A µ ] and ξ? What about correlation functions of F µν? (iii) Locality implies that δg[a µ (α) ](x)/δα(y) = δ (4) (x y) (x) for some operator (x). Show that Z[J µ ] can be rewritten as Z[J µ ] = DA µ Dc Dc e is[aµ,c,c ]+i d 4 xj µ (x)a µ(x) where S[A µ, c, c ] = S ξ [A µ ] d 4 x c c. What are the properties of the new fields c(x) and c (x)? What type of particle do they describe? [3 marks] (iv) Show that Z[J µ ] DA µ Dc Dc DB e is[aµ,c,c,b]+i d 4 xj µ (x)a µ(x) where S[A µ, c, c, B] = d 4 x ( 1 4 F µνf µν c c ξb2 + BG[A µ ] ). What are the properties of the new field B(x)? [3 marks] 5 [This question continues on the next page... ]

6 (v) Take G[A µ ] = µ A µ and show that S[A µ, c, c, B] is invariant under the infinitesimal BRST transformations, parameterised by a constant Grassmannian variable ɛ, δ ɛ A µ = 1 e ɛ µc, δ ɛ c = 0, δ ɛ c = ɛb, δ ɛ B = 0. Defining an operator Q such that acting on fields ɛq A µ = δ ɛ A µ, and ɛq c = δ ɛ c etc., show that Q 2 = 0. Outline how the physical states of the theory are defined using Q. [Total 20 marks] 6 Please go to the next page

7 4. The Lagrangian density for λφ 4 theory is given by L = 1 2 µφ µ φ 1 2 m2 0φ 2 1 4! λ 0φ 4. (i) In renormalised perturbation theory one defines φ = Z 1/2 φ R, Zm 2 0 = m 2 R + δm 2, Z 2 λ 0 = λ R + δλ, where Z = 1+δZ. Rewrite the Lagrangian density in terms of the renormalised field, renormalised couplings, and counter terms. [2 marks] (ii) Let G2 R (x, y) = Ω T φ R (x)φ R (y) Ω be the renormalised two-point function. Using the standard Feynman rules, show that the Fourier transform can be written as ( ) k G 2 R (p, q) = (2π) 4 δ (4) i i Γ 2 (p) (p + q) p 2 mr 2 + iɛ p 2 mr 2 + iɛ k=0 = (2π) 4 δ (4) i (p + q) p 2 mr 2 i Γ 2 (p) + iɛ where Γ 2 (p) is the one-particle irreducible (1PI) two-point function. (iii) By considering one-loop and counter-term diagrams, show that Γ 2 (p) = 1 2 λ RI 1 (m 2 R) + i ( p 2 δz δm 2) + O(λ 2 R). (where I 1 (m 2 ) is defined on page 2). Hence show that in the MS scheme δz = 0 + O(λ 2 R) δm 2 = λ ( ) RmR 2 2 4πµ2 + log γ + O(λ 2 32π 2 ɛ M R). 2 (iv) Now consider the renormalised four-point vertex function Γ R 4 (p 1, p 2, p 3, p 4 ). By considering one-loop and counter-term diagrams, show that Γ R 4 (0, 0, 0, 0) = iλ R λ2 RI 2 (m 2 R) iδλ + O(λ 3 R). (where I 2 (m 2 ) is defined on page 2). Hence show that in the MS scheme ( ) δλ = 3λ2 R 2 4πµ2 + log γ + O(λ 3 32π 2 ɛ M R). 2 (v) Using the relation between bare and renormalised quantities given in part (i), show that γ 2 = (M/m 2 R )( m2 R / M) and β = M( λ R/ M) are given by γ 2 = M m 2 R δm 2 R M = λ R 16π 2 + O(λ2 R), β = M δλ M = 3λ2 R 16π 2 + O(λ3 R), 7 [This question continues on the next page... ]

8 where the derivatives are defined holding the bare quantities fixed. Hence argue that λ R is a marginally irrelevant coupling. These equations also imply that the mass mr 2 is a marginal coupling. Is this correct? Would we have found the same result if we had used a simple cut-off to regulate the theory? [Total 20 marks] 8 End of examination paper