PAPER 44 ADVANCED QUANTUM FIELD THEORY

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1 MATHEMATICAL TRIPOS Part III Friday, 3 May, 203 9:00 am to 2:00 pm PAPER 44 ADVANCED QUANTUM FIELD THEORY Attempt no more than THREE questions. There are FOUR questions in total. The questions carry equal weight. STATIONERY REQUIREMENTS Cover sheet Treasury Tag Script paper SPECIAL REQUIREMENTS None You may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator.

2 2 The action for a complex z(t), defined over time interval 0 t T, is S[z] = T 0 dt(ż ż ω 2 z z), ż(t) = d dt z(t). Show how to evaluate the functional integral over z(t) K(z f,z i ;T) = d[z] e is[z], z(0) = z i, z(t) = z f, by expanding z(t) about the classical solution z c (t), for which S[z] is stationary at z = z c under independent variations of z and z, requiring z c (0) = z i, z(t) = z f, to obtain K(z f,z i ;T) = e is[zc], ω = d det ω dt 2 ω2, S[z c ] = ω ) (( z f 2 + z i 2 )cosωt z f z i z i z f. sinωt Assume d[z] is defined so that det 0 = πit and show that then det ω = πi sinωt/ω. Show that d 2 z K(±z,z; it) = e ωt ( e ωt ) 2 = n= (±) n ne nωt. What is the interpretation of this result? [You may assume for integration over complex z d 2 z e iλz z = π λ real, λ > 0, d 2 z = dxdy for z = x+iy. ] iλ

3 3 2 For the Lagrangian, in d-dimensions, L F = 2 µ φ µ φ 2 m2 φ 2 ψ(γ µ µ +M)ψ, where φ is a real scalar field and ψ a Dirac spinor field, define the two point correlation functions φ(x)φ(0) and ψ(x) ψ(0) intermsof afunctionalintegral. Derive theequations ( 2 +m 2 ) φ(x)φ(0) = iδ d (x), (γ µ µ +M) ψ(x) ψ(0) = iδ d (x), and hence obtain the momentum space propagators for φ, ψ, i p 2 +m 2 iǫ, i iγµ p µ +M p 2 +M 2 iǫ. Let ˆτ n (p,...,p n ), i p i = 0, be the amplitude corresponding to connected one particle irreducible graphs with n external φ lines after factoring off i(2π) d δ d ( i p i). For L F, ˆτ 2 (p, p) = p 2 m 2. What is ˆτ n for n > 2? For an interaction L I = y ψψφ, show that there is a one loop contribution to ˆτ 2 of the form ˆτ 2 (p, p) () = 4y2 (2π) d d d M 2 k (k p) k i (k 2 +M 2 iǫ)((k p) 2 +M 2 iǫ). Using k (k p) = 2 (k2 +M 2 )+ 2 ((k p)2 +M 2 ) 2 p2 M 2 show that ˆτ 2 (p, p) () has a pole as ε = 4 d 0 of the form ˆτ 2 (p, p) () ε y 2 6π 2(ap2 +bm 2 ), and determine a, b. Show how this divergence may be cancelled by adding counterterms L c.t. = 2 A µ φ µ φ 2 Bφ2 with A a,b b. Show that in the bare Lagrangian φ φ 0 = Z φ 2φ and m 2 m 2 0 where Z φ = 4 ε y 2 6π 2. Sketch the one loop graph which gives rise to a contribution to ˆτ 4 () and show, without any detailed calculation, that the resulting integral is divergent in 4 dimensions. Assume ˆτ 4 () 8 ε y 4 6π 2. How must L = L F +L I be modified if the one loop divergences are to be all cancelled by letting φ φ 0 and also by suitable redefinitions of the couplings in L? [The gamma matrices are assumed to satisfy {γ µ,γ ν } = 2η µν and also tr(γ µ γ ν ) = 4η µν, tr(γ µ ) = 0. The metric is η µν = diag(,,,). You may also use (2π) d d d k i k 2 +m 2 iǫ 2 m 2 ε 6π 2. ] [TURN OVER

4 4 3 Consider a renormalisable quantum field theory which has a single dimensionless coupling g and no mass parameters. Let φ(x )...φ(x n ) be the finite n-point correlation function for scalar fields φ determined by perturbation expansion of the quantum field theory as a series in g. Explain how the perturbative result for φ(x )...φ(x n ) depends on a mass scale µ and obtain, with suitable assumptions, the RG equation ( µ µ +β(g) ) g +nγ(g) φ(x )...φ(x n ) = 0. ( ) Assume the function C is defined by and obtain the solution of ( ) in the form d 4 xe ip x φ(x)φ(0) = i C(p2 /µ 2,g) p 2, C(e 2t p 2 /µ 2,g) = f(t)c(p 2 /µ 2,g(t)), ( ) for suitable g(t) and f(t). How does this equation imply that µ is essentially arbitrary. What is the behaviour of C(p 2 /µ 2,g) for large p 2 if (i) β(g) > 0, 0 < g < g, β(g ) = 0, (ii) for small g, β(g) = bg 3, γ(g) = cg 2, b > 0? In this case show that f(t) t c/b for large t. Suppose that the theory depends on a mass m and ( ) is modified to ( µ µ +β(g) g +δ(g)m ) m +nγ(g) φ(x )...φ(x n ) = 0. Assuming now C(p 2 /µ 2,m/µ,g) show that ( ) becomes C(e 2t p 2 /µ 2,m/µ,g) = f(t)c(p 2 /µ 2,e t m(t)/µ,g(t)), for suitable m(t). If C(p 2 /µ 2,m/µ,g) has a well defined limit as m 0 why does this imply that masses can be neglected at large energies if δ(g) is not too large?

5 5 4 For a theory with fields φ which has a continuous symmetry such that the action is invariant, δ ǫ S[φ] = 0, subject to a transformation δ ǫ φ = ǫ a t a φ for ǫ a infinitesimal show how, by letting ǫ a ǫ a (x), to define an associated conserved current j µ a. How can j µ a be used to construct a conserved charge Q a? If for any X(φ) and X(φ) = d[φ] X(φ)e is[φ], =, show how to obtain, with appropriate assumptions, the Ward identity i d d x ǫ a (x) x µ jµ a(x)x = δ ǫ X. What is i x µ j µ a(x)φ(x )...φ(x n )? For a non abelian gauge theory with gauge field A µa (x) (a is a group index), anti-commuting ghost fields c a, c a and an auxiliary scalar field b a the quantum action is determined by the Lagrangian L q = 4 Fµν F µν + µ b A µ + 2 ξb b µ c D µ c, where F µνa = µ A νa ν A µa +f abc A µb A νc, (D µ c) a = µ c a +f abc A µb c c, X Y = X a Y a and f abc is antisymmetric. Show that L q has a symmetry, with suitable assumptions, under δ θ (c a, c a ) = θ(c a, c a ), δ ǫ (A µa,c a, c a,b a ) = ǫ((d µ c) a, 2 f abcc b c c,b a,0), for infinitesimal θ and anti-commuting ǫ. Verify that δ ǫ δ ǫ (A µa,c a, c a,b a ) = 0. [You need to use δ ǫ F µνa = ǫf abc F µνb c c, δ ǫ D µ c = 0 but you also need to show why these results are true.] What are the corresponding conserved currents j µ G and jµ B? In the quantum field theory there are associated conserved charges Q G and Q B where Q B 2 = 0. Outline how Q B can be used to construct the space of physical states of the theory. Why do we expect the physical states to be annihilated by Q G? END OF PAPER