Solutions Manual for Teachers Introduction to Quantum Field Theory: Classical Mechanics to Gauge Field Theories ETHAN N. CARRAGHER AND ANTHONY G. WILLIAMS Contents Preface to Solutions Manual page vi 1 Lorentz and Poincar´e Invariance 1 1.1 Problem 1 1 1.2 Problem 2 2 1.3 Problem 3 3 1.4 Problem 4 5 1.5 Problem 5 5 1.6 Problem 6 7 1.7 Problem 7 8 1.8 Problem 8 8 1.9 Problem 9 9 1.10 Problem 10 10 1.11 Problem 11 11 1.12 Problem 12 12 1.13 Problem 13 13 1.14 Problem 14 13 1.15 Problem 15 14 1.16 Problem 16 15 2 Classical Mechanics 18 2.1 Problem 1 18 2.2 Problem 2 19 2.3 Problem 3 20 2.4 Problem 4 22 2.5 Problem 5 24 2.6 Problem 6 29 2.7 Problem 7 30 2.8 Problem 8 34 2.9 Problem 9 35 2.10 Problem 10 36 2.11 Problem 11 37 2.12 Problem 12 40 2.13 Problem 13 42 2.14 Problem 14 43 2.15 Problem 15 45 iii iv Contents 3 Relativistic classical fields 48 3.1 Problem 1 48 3.2 Problem 2 52 3.3 Problem 3 53 3.4 Problem 4 57 3.5 Problem 5 60 3.6 Problem 6 61 3.7 Problem 7 61 3.8 Problem 8 64 4 Relativistic Quantum Mechanics 68 4.1 Problem 1 68 4.2 Problem 2 69 4.3 Problem 3 72 4.4 Problem 4 75 4.5 Problem 5 76 4.6 Problem 6 77 4.7 Problem 7 79 4.8 Problem 8 81 4.9 Problem 9 82 4.10 Problem 10 86 4.11 Problem 11 88 4.12 Problem 12 89 5 Introduction to Particle Physics 91 5.1 Problem 1 91 5.2 Problem 2 92 5.3 Problem 3 94 5.4 Problem 4 95 5.5 Problem 5 97 5.6 Problem 6 98 5.7 Problem 7 100 5.8 Problem 8 102 5.9 Problem 9 103 5.10 Problem 10 104 5.11 Problem 11 108 5.12 Problem 12 110 6 Formulation of Quantum Field Theory 117 6.1 Problem 1 117 6.2 Problem 2 118 6.3 Problem 3 120 6.4 Problem 4 121 6.5 Problem 5 123 v Contents 6.6 Problem 6 127 6.7 Problem 7 130 6.8 Problem 8 133 6.9 Problem 9 135 6.10 Problem 10 147 6.11 Problem 11 150 6.12 Problem 12 151 6.13 Problem 13 154 7 Interacting Quantum Field Theories 163 7.1 Problem 1 163 7.2 Problem 2 164 7.3 Problem 3 164 7.4 Problem 4 165 7.5 Problem 5 166 7.6 Problem 6 167 7.7 Problem 7 168 7.8 Problem 8 170 7.9 Problem 9 171 7.10 Problem 10 173 8 Symmetries and Renormalization 176 8.1 Problem 1 176 8.2 Problem 2 177 8.3 Problem 3 179 8.4 Problem 4 180 8.5 Problem 5 181 8.6 Problem 6 182 8.7 Problem 7 189 8.8 Problem 8 191 9 Gauge Field Theories 192 9.1 Problem 1 192 9.2 Problem 2 194 9.3 Problem 3 198 9.4 Problem 4 200 9.5 Problem 5 202 9.6 Problem 6 205 9.7 Problem 7 208 9.8 Problem 8 220 9.9 Problem 9 227 References 231 Preface to Solutions Manual Thissolutionsmanualhasbeenprovidedtoassistteacherswhoadoptthetextbook as a teaching resource for their classes. It is not intended for broad distribution as thatwoulddefeatthepurposeoftheproblemsetsattheendofeachchapter.Some of the problems are challenging for students and such problems could be assigned to a small group or as small projects for individual students. As can be seen from the length of this manual, it took some time and effort for us to prepare and typeset these solutions and it is almost certain that we have not caught every error. So we ask the reader’s forgiveness for any mistakes found and we would be very grateful if these could be reported at the website below. In that way corrections can be made and an appropriate acknowledgement given. Ethan N. Carragher and Anthony G. Williams Adelaide, May 28, 2022 Numbering of equations Two-part equation numbers, such as Eq. (1.42) and Eq. (9.51), refer to equations in this Solutions Manual. Three-part equation numbers, such as Eq. (3.1.16) and Eq. (9.4.23), refer to equations in the book. Corrections to this book As is the case for the book itself, the current list of corrections for this Solutions Manual along with the names of those who suggested them can be found at: www.cambridge.org/WilliamsQFT. Itwouldbegreatlyappreciatedifanyonefindingadditionalerrorsinthebookitself or in this Solutions Manual could please report them using the relevant corrections link provided on this website. vi 1 Lorentz and Poincar´e Invariance 1.1 Problem 1 Problem: A muon is a more massive version of an electron and has a mass of 105.7 MeV/c2. The dominant decay mode of the muon is to an electron, an electron antineutrino and a muon neutrino, µ e + ν¯ + ν . If we have N(t) un- − − e µ → stable particles at time t then the fraction of particles decaying per unit time is a constant, i.e., we have dN/N = (1/τ)dt for some constant τ. This gives − dN/dt= (1/τ)N, which has the solution N(t)=N e t/τ where we have N un- 0 − 0 − stableparticlesatt=0.Thefractionofparticlesdecayingintheintervalttot+dt is dN/N = ( dN/dt)dt/N = (1/τ)e t/τdt. So the mean lifetime (or lifetime) 0 0 − −(cid:82) − (cid:82) is 0∞t(−dN/dt)dt/N0 = τ 0∞xe−xdx = τ, where x = t/τ. The half-life, t1/2, is the time taken for half the particles to decay, e−t1/2/τ = 21, which means that t = τln2. The decay rate, Γ, is defined as the probability per unit time that 1/2 a particle will decay, i.e., Γ = ( dN/dt)/N = 1/τ is the inverse mean lifetime. A − muonatresthasalifetimeofτ =2.197 10 6 s.Cosmic rays arehigh-energypar- − × ticlesthathavetraveledenormousdistancesfromoutsideoursolarsystem.Primary cosmic rays are particles that have been accelerated by some extreme astrophysi- cal event and secondary cosmic rays are those resulting from collisions of primary cosmic rays with interstellar gas or with our atmosphere. Most cosmic rays reach- ing our atmosphere will be stable particles such as photons, neutrinos, electrons, protons and stable atomic nuclei (mostly helium nuclei). Muon cosmic rays there- fore are secondary cosmic rays produced when primary or secondary cosmic rays collidewithoutatmosphere.Atypicalheightintheatmospherefortheproduction of cosmic ray muons is 15 km. What is the minimum velocity that this muon be ∼ produced with in order that it have a 50% chance of reaching the surface of the Earth before decaying? Solution: AstationaryobserveronEarthwillseethetimeexperiencedbyamuontraveling at speed v to be dilated by a factor of γ =(1 v2/c2) 1/2 compared to their own. − − So according to the observer, the half-life of the muon will be γt . If the muon 1/2 travels at a speed that allows it to traverse the L = 15 km of the atmosphere in this time, it will therefore have a 50% chance of reaching the Earth’s surface. That 1 2 Lorentz and Poincar´e Invariance is, the speed of the muon must satisfy (cid:113) v = L = 1− vc22 L (1.1) γt t 1/2 1/2 tohavea50%chanceofreachingthesurfacebeforedecaying.Thiscanberearranged to give L v = . (1.2) (cid:113) t2 + L2 1/2 c2 Now from the given information, the half-life of the muon is t =τln2=(2.197 10 6 s)ln2=1.523 10 6 s, (1.3) 1/2 − − × × so the necessary speed is 15 103 m v = (cid:113) × (1.523×10−6 s)2+ (3(151×0810m3sm−)12)2 × =299,653,700 ms 1 − =0.9995c. (1.4) An inertial observer traveling with the muon will see the height of the atmosphere contracted by a factor of γ, which at this speed gives a height of (cid:114) L v2 = 1 L γ − c2 (cid:112) = 1 0.99952(15 103 m) − × =456 m. (1.5) In other words, from the muon’s point of view, it only needs to travel 456 m to reach the Earth’s surface. 1.2 Problem 2 Problem: The diameter of our Milky Way spiral galaxy is approximately 100,000 - 180,000 lightyearsandoursolarsystemisapproximately25,000lightyearsfromthecenter of our galaxy. Recalling the effects of time dilation, approximately how fast would you have to travel to reach the center of the galaxy in your lifetime? Estimate how much energy would it take to accelerate your body to this speed. Solution: 3 Problem 3 From the solution to Problem 1.1, the speed needed for an observer to travel a proper distance L in a time T is L v = . (1.6) (cid:113) T2+ L2 c2 So for a human with a lifetime of, say, T =80 years, to travel to the center of the Milky Way L= 25,000 light years = 25,000c years away, a speed of 25,000c years v = (1.7) (cid:112) (80 years)2+(25,000 years)2 =0.999995c (1.8) would be required. The kinetic energy possessed by a human of mass m=70 kg at this speed is E =(γ 1)mc2 kin − (cid:18) (cid:19) 1 = 1 (70 kg)(3 108 ms 1) − √1 0.9999952 − × − =2 1021 J. (1.9) × For reference, it would take humanity slightly more than 3 years to consume this much energy at current rates. 1.3 Problem 3 Problem: Consider two events that occur at the same spatial point in the frame of some inertial observer . Explain why the two events occur in the same temporal order O in every inertial frame connected to it by a Lorentz transformation that does not invert time. Show that the time separation between the two events is a minimum in the frame of . (Hint: Consider Figs. 1.2 and 1.5.) O Solution: Since the two events E and E occur at the same spatial point in frame , the 1 2 O displacementvectorE E betweenthemwillpointentirelyalongthetimeaxisin 2 1 − this frame, in the positive direction, say. Under any given Lorentz transformation, displacement vectors will be moved along hyperbolae in a spacetime diagram, as illustratedinFig.1.1.Further,Lorentztransformationsthatdonotinverttimewill only move displacement vectors along branches of those hyperbolae. Our vector of interest E E lies on a positive-time branch, so under an orthochronus Lorentz 2 1 − transformation it will remain on that positive-time branch. That is, the temporal orderoftheeventsremainsthesame.Andthepointonsuchapositive-timebranch thathasthesmallesttimecomponentistheonethatliesonthetimeaxis,asE E 2 1 −