Part III | Quantum Field Theory Theorems with proof Based on lectures by B. Allanach Notes taken by Dexter Chua Michaelmas 2016 These notes are not endorsed by the lecturers, and I have modified them (often significantly) after lectures. They are nowhere near accurate representations of what was actually lectured, and in particular, all errors are almost surely mine. Quantum Field Theory is the language in which modern particle physics is formulated. It represents the marriage of quantum mechanics with special relativity and provides the mathematical framework in which to describe the interactions of elementary particles. This first Quantum Field Theory course introduces the basic types of fields which play an important role in high energy physics: scalar, spinor (Dirac), and vector (gauge) fields. The relativistic invariance and symmetry properties of these fields are discussed using Lagrangian language and Noether's theorem. The quantisation of the basic non-interacting free fields is firstly developed using the Hamiltonian and canonical methods in terms of operators which create and annihilate particles and anti-particles. The associated Fock space of quantum physical states is explained together with ideas about how particles propagate in spacetime and their statistics. How these fields interact with a classical electromagnetic field is described. Interactions are described using perturbative theory and Feynman diagrams. This is first illustrated for theories with a purely scalar field interaction, and then for a couplings between scalar fields and fermions. Finally Quantum Electrodynamics, the theory of interacting photons, electrons and positrons, is introduced and elementary scattering processes are computed. Pre-requisites You will need to be comfortable with the Lagrangian and Hamiltonian formulations of classical mechanics and with special relativity. You will also need to have taken an advanced course on quantum mechanics. 1 Contents III Quantum Field Theory (Theorems with proof) Contents 0 Introduction 3 1 Classical field theory 4 1.1 Classical fields . .4 1.2 Lorentz invariance . .4 1.3 Symmetries and Noether's theorem for field theories . .4 1.4 Hamiltonian mechanics . .5 2 Free field theory 6 2.1 Review of simple harmonic oscillator . .6 2.2 The quantum field . .6 2.3 Real scalar fields . .6 2.4 Complex scalar fields . .7 2.5 The Heisenberg picture . .7 2.6 Propagators . .8 3 Interacting fields 10 3.1 Interaction Lagrangians . 10 3.2 Interaction picture . 10 3.3 Wick's theorem . 11 3.4 Feynman diagrams . 12 3.5 Amplitudes . 12 3.6 Correlation functions and vacuum bubbles . 12 4 Spinors 14 4.1 The Lorentz group and the Lorentz algebra . 14 4.2 The Clifford algebra and the spin representation . 14 4.3 Properties of the spin representation . 15 4.4 The Dirac equation . 16 4.5 Chiral/Weyl spinors and γ5 ..................... 16 4.6 Parity operator . 16 4.7 Solutions to Dirac's equation . 17 4.8 Symmetries and currents . 17 5 Quantizing the Dirac field 18 5.1 Fermion quantization . 18 5.2 Yukawa theory . 18 5.3 Feynman rules . 18 6 Quantum electrodynamics 19 6.1 Classical electrodynamics . 19 6.2 Quantization of the electromagnetic field . 19 6.3 Coupling to matter in classical field theory . 19 6.4 Quantization of interactions . 19 6.5 Computations and diagrams . 19 2 0 Introduction III Quantum Field Theory (Theorems with proof) 0 Introduction 3 1 Classical field theory III Quantum Field Theory (Theorems with proof) 1 Classical field theory 1.1 Classical fields Proposition (Euler-Lagrange equations). The equations of motion for a field are given by the Euler-Lagrange equations: @L @L @µ − = 0: @(@µφa) @φa 1.2 Lorentz invariance 1.3 Symmetries and Noether's theorem for field theories Theorem (Noether's theorem). Every continuous symmetry of L gives rise to a conserved current jµ(x) such that the equation of motion implies that µ @µj = 0: More explicitly, this gives 0 @0j + r · j = 0: A conserved current gives rise to a conserved charge Z Q = j0d3x; 3 R since dQ Z dj0 = d3x dt 3 dt RZ = − r · j d3x 3 R = 0; assuming that ji ! 0 as jxj ! 1. Proof. Consider making an arbitrary transformation of the field φa 7! φa + δφa. We then have @L @L δL = δφa + δ(@µφa) @φa @(@µφa) @L @L @L = − @µ δφa + @µ δφa : @φa @(@µφa) @(@µφa) When the equations of motion are satisfied, we know the first term always vanishes. So we are left with @L δL = @µ δφa : @(@µφa) If the specific transformation δφa = Xa we are considering is a symmetry, then δL = 0 (this is the definition of a symmetry). In this case, we can define a conserved current by µ @L j = Xa; @(@µφa) and by the equations above, this is actually conserved. 4 1 Classical field theory III Quantum Field Theory (Theorems with proof) 1.4 Hamiltonian mechanics 5 2 Free field theory III Quantum Field Theory (Theorems with proof) 2 Free field theory 2.1 Review of simple harmonic oscillator 2.2 The quantum field 2.3 Real scalar fields Proposition. We have Z d3p e−ip·x = δ3(x): (2π)3 Proposition. The canonical commutation relations of φ, π, namely [φ(x); φ(y)] = 0 [π(x); π(y)] = 0 [φ(x); π(y)] = iδ3(x − y) are equivalent to [ap; aq] = 0 y y [ap; aq] = 0 y 3 3 [ap; aq] = (2π) δ (p − q): Proof. We will only prove one small part of the equivalence, as the others are similar tedious and boring computations, and you are not going to read it anyway. We will use the commutation relations for the ap to obtain the commutation relations for φ and π. We can compute [φ(x); π(y)] Z 3 3 r d p d q (−i) !q y ip·x−iq·y y −ip·x+iq·y = 6 −[ap; aq]e + [ap; aq]e (2π) 2 !p Z 3 3 r d p d q (−i) !q 3 3 ip·x−iq·y 3 −ip·x+iq·y = 6 (2π) −δ (p − q)e − δ (q − p)e (2π) 2 !p (−i) Z d3p = −e−ip·(x−y) − eip·(y−x) 2 (2π)3 = iδ3(x − y): Note that to prove the inverse direction, we have to invert the relation between y y φ(x); π(x) and ap; ap and express ap and ap in terms of φ and π by using Z 3 ip·x 1 y d x φ(x) e = p a−p + ap 2!p Z r! d3x π(x) eip·x = (−i) p a − ay : 2 −p p Proposition. The expression Z d3p 2Ep 6 2 Free field theory III Quantum Field Theory (Theorems with proof) is Lorentz-invariant, where 2 2 2 Ep = p + m for some fixed m. Proof. We know R d4p certainly is Lorentz invariant, and 2 µ 2 2 2 m = pµp = p = p0 − p is also a Lorentz-invariant quantity. So for any m, the expression Z 4 2 2 2 d p δ(p0 − p − m ) is also Lorentz invariant. Writing 2 2 2 2 Ep = p0 = p + m ; integrating over p0 in the integral gives Z d3p Z d3p = ; 2p0 2Ep and this is Lorentz invariant. Proposition. The expression 3 2Epδ (p − q) is Lorentz invariant. Proof. We have Z 3 d p 3 · (2Epδ (p − q)) = 1: 2Ep Since the RHS is Lorentz invariant, and the measure is Lorentz invariant, we 3 know 2Epδ (p − q) must be Lorentz invariant. 2.4 Complex scalar fields 2.5 The Heisenberg picture Proposition. Let A and B be operators. Then 1 1 eABe−A = B + [A; B] + [A; [A; B]] + [A; [A; [A; B]]] + ··· : 2! 3! In particular, if [A; B] = cB for some constant c, then we have eABe−A = ecB: 7 2 Free field theory III Quantum Field Theory (Theorems with proof) Proof. For λ a real variable, note that d e(λ+")ABe−(λ+")A − eλABe−λA (eλABe−λA) = lim dλ "!0 " e"ABe−"A − B = lim eλA e−λA "!0 " (1 + "A)B(1 − "A) − B + o(") = lim eλA e−λA "!0 " ("(AB − BA) + o(")) = lim eλA e−λA "!0 " = eλA[A; B]e−λA: So by induction, we have dn (eλABe−λA) = eλA[A; [A; ··· [A; B] ··· ]]e−λA: dλn Evaluating these at λ = 0, we obtain a power series representation λ2 eλABe−λA = B + λ[A; B] + [A; [A; B]] + ··· : 2 Putting λ = 1 then gives the desired result. 2.6 Propagators Proposition. Z 3 d p 1 −ip·(x−y) D(x − y) = 3 e : (2π) 2Ep Proposition. We have ∆(x − y) = D(x − y) − D(y − x): Proof. ∆(x − y) = [φ(x); φ(y)] = h0j [φ(x); φ(y)] j0i = D(x − y) − D(y − x); where the second equality follows as [φ(x); φ(y)] is just an ordinary function. Proposition. We have Z d4p i ∆ (x − y) = e−ip·(x−y): F (2π)4 p2 − m2 This expression is a priori ill-defined since for each p, the integrand over p0 has a pole whenever (p0)2 = p2 + m2. So we need a prescription for avoiding this. We replace this with a complex contour integral with contour given by −Ep Ep 8 2 Free field theory III Quantum Field Theory (Theorems with proof) Proof. To compare with our previous computations of D(x − y), we evaluate the p0 integral for each p. Writing 1 1 1 2 2 = 0 2 2 = 0 0 ; p − m (p ) − Ep (p − Ep)(p + Ep) 0 1 we see that the residue of the pole at p = ±Ep is ± . 2Ep When x0 > y0, we close the contour in the lower plane p0 ! −i1, so −p0(x0−t0) −∞ R 0 0 e ! e = 0.