Problem Book Quantum Field Theory Voja Radovanovic Problem Book Quantum Field Theory ABC Voja Radovanovic Faculty of Physics University of Belgrade Studentski trg 12-16 11000 Belgrade Yugoslavia Library of Congress Control Number: 2005934040 ISBN-10 3-540-29062-1 Springer Berlin Heidelberg New York ISBN-13 978-3-540-29062-9 Springer Berlin Heidelberg New York This work is subject to copyright All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer Violations are liable for prosecution under the German Copyright Law Springer is a part of Springer Science+Business Media springeronline.com c Springer-Verlag Berlin Heidelberg 2006 Printed in The Netherlands The use of general descriptive names, registered names, trademarks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use Typesetting: by the author and TechBooks using a Springer LATEX macro package Cover design: design & production GmbH, Heidelberg Printed on acid-free paper SPIN: 11544920 56/TechBooks 543210 To my daughter Natalija Preface This Problem Book is based on the exercises and lectures which I have given to undergraduate and graduate students of the Faculty of Physics, University of Belgrade over many years Nowadays, there are a lot of excellent Quantum Field Theory textbooks Unfortunately, there is a shortage of Problem Books in this field, one of the exceptions being the Problem Book of Cheng and Li [7] The overlap between this Problem Book and [7] is very small, since the latter mostly deals with gauge field theory and particle physics Textbooks usually contain problems without solutions As in other areas of physics doing more problems in full details improves both understanding and efficiency So, I feel that the absence of such a book in Quantum Field Theory is a gap in the literature This was my main motivation for writing this Problem Book To students: You cannot start to problems without previous studying your lecture notes and textbooks Try to solve problems without using solutions; they should help you to check your results The level of this Problem Book corresponds to the textbooks of Mandl and Show [15]; Greiner and Reinhardt [11] and Peskin and Schroeder [16] Each Chapter begins with a short introduction aimed to define notation The first Chapter is devoted to the Lorentz and Poincar´e symmetries Chapters 2, and deal with the relativistic quantum mechanics with a special emphasis on the Dirac equation In Chapter we present problems related to the Euler-Lagrange equations and the Noether theorem The following Chapters concern the canonical quantization of scalar, Dirac and electromagnetic fields In Chapter 10 we consider tree level processes, while the last Chapter deals with renormalization and regularization There are many colleagues whom I would like to thank for their support and help Professors Milutin Blagojevi´c and Maja Buri´c gave many useful ideas concerning problems and solutions I am grateful to the Assistants at the Faculty of Physics, University of Belgrade: Marija Dimitrijevi´c, Duˇsko Latas and Antun Balaˇz who checked many of the solutions Duˇsko Latas also drew all the figures in the Problem Book I would like to mention the contribution of the students: Branislav Cvetkovi´c, Bojan Nikoli´c, Mihailo Vanevi´c, Marko VIII Preface Vojinovi´c, Aleksandra Stojakovi´c, Boris Grbi´c, Igor Salom, Irena Kneˇzevi´c, Zoran Ristivojevi´c and Vladimir Juriˇci´c Branislav Cvetkovi´c, Maja Buri´c, Milutin Blagojevi´c and Dejan Stojkovi´c have corrected my English translation of the Problem Book I thank them all, but it goes without saying that all the errors that have crept in are my own I would be grateful for any readers’ comments Belgrade, August 2005 Voja Radovanovi´c Contents Part I Problems Lorentz and Poincar´ e symmetries The Klein–Gordon equation The γ–matrices 13 The Dirac equation 17 Classical field theory and symmetries 25 Green functions 31 Canonical quantization of the scalar field 35 Canonical quantization of the Dirac field 43 Canonical quantization of the electromagnetic field 49 10 Processes in the lowest order of perturbation theory 55 11 Renormalization and regularization 61 Part II Solutions Lorentz and Poincar´ e symmetries 67 The Klein–Gordon equation 77 The γ–matrices 85 X Contents The Dirac equation 93 Classical fields and symmetries 121 Green functions 131 Canonical quantization of the scalar field 141 Canonical quantization of the Dirac field 161 Canonical quantization of the electromagnetic field 179 10 Processes in the lowest order of the perturbation theory 191 11 Renormalization and regularization 211 References 239 Index 241 Part I Problems Lorentz and Poincar´ e symmetries • Minkowski space, M4 is a real 4-dimensional vector space with metric tensor defined by   0 0   −1 gµν =  (1.A)  0 −1 0 0 −1 Vectors can be written in the form x = xµ eµ , where xµ are the contravariant components of the vector x in the basis         0 0 1 0 0 e0 =   , e1 =   , e2 =   , e3 =   0 0 0 The square of the length of a vector in M4 is x2 = gµν xµ xν The square of the line element between two neighboring points xµ and xµ + dxµ takes the form (1.B) ds2 = gµν dxµ dxν = c2 dt2 − dx2 The space M4 is also a manifold; xµ are global (inertial) coordinates The covariant components of a vector are defined by xµ = gµν xν • Lorentz transformations, (1.C) x µ = Λµν xν , leave the square of the length of a vector invariant, i.e x = x2 The matrix Λ is a constant matrix1 ; xµ and x µ are the coordinates of the same event in two different inertial frames In Problem 1.1 we shall show that from the previous definition it follows that the matrix Λ must satisfy the condition ΛT gΛ = g The transformation law of the covariant components is given by xµ = (Λ−1 )νµ xν = Λµν xν (1.D) The first index in Λµν is the row index, the second index the column index Chapter 11 Renormalization and regularization 229 Fig 11.5 Feynman rules in renormalized Yukawa theory Fig 11.6 The one–loop correction to fermionic propagator Since γ5 / aγ5 = −/ a and (γ5 )2 = we have −iΣ2 (p) = − g2 µ (2π)D dD k =− g2 µ (2π)D dD k −/p + k /+M (k − m2 + i0)((p − k)2 − M + i0) dx g2 µ =− iπ D/2 Γ (2π)D −/p + k /+M (k − px)2 − ∆ + i0)2 dx /p(x − 1) + M , ∆ /2 (11.40) where ∆ = M x + m2 (1 − x) − p2 x + p2 x2 Since µ 2D π D/2 = (4πµ2 ) 16π /2 = 1 + ln(4πµ2 ) + 16π 2 , we have ∆ ig 2 − γ + o( ) dx [M + (x − 1)/ p] − ln 16π 4πµ2 ig (11.41) = − (M − /p) + fin part 8π −iΣ2 (p) = − The full one–loop correction to the fermionic propagator is −iΣ(p) = − ig (M − /p) − iδM + iδZψ /p + fin part 8π 2 From the renormalization conditions: Σ(/ p = M) = , dΣ =0, d/p /p=M (11.42) 230 Solutions follows that g2 + fin part , 16π g2 M δM = − + fin part 8π δZψ = − (11.43) (c) The one–loop correction to the scalar propagator is represented in Fig 11.7 Fig 11.7 The one-loop correction to the scalar propagator The first diagram is −iΠ1 (p2 ) = − = i2 g µ (2π)D dD k k + M )γ5 (/ p+k / + M )] tr[γ5 (/ (k − M + i0)((p + k)2 − M + i0) g2 µ (2π)D dD k dx g2 µ = (2π)D dD k dx tr[(−/k + M )(/ p+k / + M )] (k + 2k · px − M + p2 x)2 4(−k · p − k + M ) , (k + 2k · px − M + p2 x)2 where we use the Feynman parametrization formula (11.G) Introducing a new variable l = k + px we further have −iΠ1 (p2 ) = 4g µ ig = 4π × dx dD l 2M − ∆ − l2 (2π)D (l2 − ∆ + i0)2 ∆ 4πµ2 (M − p2 (x2 − x))( − γ + o( ))+ dx − ln D (− − + γ + o( ))(M + p2 (x2 − x)) p2 ig − M + fin part , = 2π 2 + where ∆ = M + p2 (x2 − x) The second diagram is −iΠ2 = Summing, we obtain iλm2 + fin part 16π (11.44) Chapter 11 Renormalization and regularization −iΠ(p2 ) = ig 2π 231 p2 iλm2 − M2 + +iδZφ p2 −iδm2 +fin.part (11.45) 16π Using the renormalization conditions: Π(p2 = m2 ) = dΠ =0, dp2 p2 =m2 (11.46) we get g2 + fin part 4π 2 g2 M λm δm2 = − + fin part 16π 2π δZφ = − (11.47) (d) The amplitude of the diagram is iM3 = (ig)3 µ3 =− /2 γ5 (/ k + q/ + M )γ5 (/ k + M )γ5 dD k D 2 2 (2π) ((k + q) − M )(k − M )((k − p)2 − m2 ) 2ig µ3 /2 γ5 (2π)D 2ig µ3 /2 =− γ5 (2π)D 1−x dx dD k dz dD l 1−x dx dz M − /q /k + M /q − k ((k + qx − pz)2 − ∆)3 N , (l2 − ∆)3 where ∆ = x2 q + z p2 + (1 − z)M − xq + zm2 − p2 z − 2xzq · p and N = M − (l − xq + zp)2 + M /q − /q (/ l − x/q + z/p) In the previous formulae we introduced a variable l = k + xq − zp As we are interested to find only the divergent part of iM3 , it is useful to note that only l2 –term in the numerator of the integrand is divergent So, by using (11.C) we get: 232 Solutions iM3 = 2ig µ3 =− g3 µ /2 γ5 dD l (2π)D dz − ∆ 4πµ2 ln 1−x dx (4 − ) γ5 32π /2 1−x × dz l2 + (l2 − ∆)3 − γ + dx Finally iM3 = − g µ /2 γ5 + fin part 8π (11.48) The vertex correction is so, from iV3 = gγ5 µ /2 + δgγ5 µ /2 − g µ /2 + fin.part 8π q =0 = gγ5 follows g3 + fin part 8π (e) Let us first calculate the following diagram δg = Since we have to find the divergent part of this diagram we can put that the external momenta are equal to zero Then, iM4 (k1 = k2 = k3 = k4 = 0) = −g µ2 p + M )]4 dD p tr[γ5 (/ (2π)D (p2 − M )4 (11.49) Since p + M )γ5 (/ p + M ) = (−/p + M )(/ p + M ) = M − p2 γ5 (/ we have Chapter 11 Renormalization and regularization iM4 (k1 = k2 = k3 = k4 = 0) = −4g µ2 dD p D (2π) (p − M )2 M2 −γ − ln 4πµ2 ig µ 4π ig µ = − + fin part 2π =− 233 (11.50) The previous result should be multiplied by a factor as there are six diagrams of this type The complete four vertex is −iλµ − iδλµ − iV4 = 3iλ2 µ 6ig µ + + fin part 2π 16π s=4m2 ,t=u=0 = −iλ , (11.51) and finally 3λ2 3g + + fin part π 16π 11.13 In this problem dimension of spacetime is D = − δλ = − (11.52) (a) The polarization of vacuum is given by: −iΠµν (p) = (ie)2 (−i2 ) q − /p)γν /q γµ ] dD q tr[(/ D (2π) q (q − p)2 (11.53) In D-dimensional space trace identities necessary to calculate the previous expression read: tr(γµ γν ) = f (D)gµν , tr(γµ γν γρ γσ ) = f (D)(gµν gρσ − gµρ gνσ + gµσ gρν ) , where f (D) is any analytical function which satisfies the condition f (2) = Instead of f (D) we will write as we did in the previous problems (of course, there f (D) = 4) The Feynman parametrization gives 2e2 dx dD q (2π)D 2qµ qν − q gµν − pµ qν − pν qµ + (p · q)gµν (11.54) × (q − 2p · qx + p2 x)2 −iΠµν (p) = By using (11.A–C) in (11.54) we obtain x2 pµ pν 2ie2 π D/2 dx D (2π) (−p x + p2 x2 )1+ gµν − Γ( ) (−p2 x + p2 x2 ) /2 −iΠµν = − /2 Γ (1 + ) 234 Solutions x2 p2 Γ (1 + ) (−p2 x + p2 x2 )1+ /2 2− − Γ( ) (−p2 x + p2 x2 ) /2 xpµ pν −2 Γ (1 + ) 2 (−p x + p2 x2 )1+ /2 p2 x + gµν Γ (1 + ) (−p2 x + p2 x2 )1+ /2 − gµν From the previous expression (for D → i.e → 0) we obtain −iΠµν (p) = −i(pµ pν − p2 gµν )Π(p2 ) =− ie2 (pµ pν − p2 gµν ) , πp2 (11.55) from which we see that the polarization of vacuum is a finite quantity (b) The full photon propagator is obtained by summing the diagrams in the Figure −igµρ ρσ −igµν −igσν + [p g − pρ pσ ]iΠ(p2 ) + p2 + i0 p2 + i0 p + i0 i pµ pν ipµ pν =− (gµν − )(1 + Π(p2 ) + Π (p2 ) + ) − p + i0 p p4 pµ pν i(gµν − p2 ) , (11.56) =− p (1 − Π(p2 ) + i0) iDµν (p) = were we discarded the ipµ pν /p4 -term in the last line since the propagator is coupled to a conserved current Then the photon propagator is iDµν (p) = − i(gµν − p2 − √ Photon mass is e/ π pµ pν p2 ) e2 π (11.57) 11.14 The dimension of spacetime is D = − (a) The renormalized Lagrangian density is Lren = L + Lct , where L= m2 gµ /2 (∂φ)2 − φ − φ − hµ− 2 3! (11.58) /2 φ, (11.59) Chapter 11 Renormalization and regularization δm2 µ /2 δg δZ(∂φ)2 − φ − φ − µ− 2 3! By introducing new quantities Lct = /2 δhφ Z = + δZ , m20 Z = m2 + δm2 , 235 (11.60) (11.61) (11.62) g0 Z 3/2 = (g + δg)µ /2 , (11.63) 1/2 − /2 h0 Z = (h + δh)µ , (11.64) √ and rescaling the field, φ0 = Zφ, the renormalized Lagrangian density becomes m2 g0 Lren = (∂φ0 )2 − φ20 − φ30 − h0 φ0 2 3! The quantities with index are called bare The Feynman rules are given in Figure 11.8 Fig 11.8 Feynman rules in φ3 theory Superficially divergent amplitudes are: Fig 11.9 Divergent amplitudes in φ3 theory (b) The tadpole diagram in one–loop order is shown in the following fihure 236 Solutions The second term is − igµ = −i i dD k D (2π) k − m2 + i0 /2 π D/2 gµ /2 D (2π) (m )−2+ − /2 =− igm µ 128π =− igm4 µ− 64π /2 Γ −2 + 2 + ln 4πµ m2 + −γ /2 + fin part , and it does not depend on momentum Summing all diagrams we get iH = −ihµ− /2 − igm4 µ− 64π /2 − iδhµ− /2 + fin part (11.65) Hence, gm4 + fin part (11.66) 64π Finite part in the previous expression can be chosen so that H = and we can ignore all diagrams which contain tadpoles (c) The full one–loop propagator is shown in Fig 11.10 δh = − Fig 11.10 The one–loop propagator in φ3 theory The second diagram is −iΠ2 = (ig)2 µ i2 dD k D 2 (2π) (k − m + i0)((k − p)2 − m2 + i0) 1 dD k g2 µ dx D (k − 2k · px + p2 x − m2 + i0)2 (2π) ig + − γ + o( ) =− 128π = × dx(m2 + p2 x(x − 1)) − ig =− 64π p2 m2 − ln + fin part m2 + p2 x(x − 1) 4πµ2 (11.67) Chapter 11 Renormalization and regularization 237 Propagator correction is −iΠ(p2 ) = − ig 64π m2 − p2 + ip2 δZ − iδm2 + fin part (11.68) From the condition −iΠ(p2 ) = finite we get δZ = − g2 + fin part , 384π (11.69) m2 g + fin part 64π In MS scheme the finite parts in (11.69) and (11.70) are zero (d) The vertex correction is given in the Fig 11.11 δm2 = − (11.70) Fig 11.11 Vertex correction in φ3 theory The second diagram is i3 dD k (2π)D (k − m2 )((k + p2 )2 − m2 )((k − p1 )2 − m2 ) (11.71) By applying (11.H) and integrating over the momentum k we get iΓ = (−ig)3 µ3 /2 π D/2 Γ dx (2π)D × 2 (m − p2 x − p1 z + p22 x2 + p21 z ) iΓ = − (−ig)3 µ3 1−x /2 dz /2 1−x ig µ /2 + dx dz 26− π 3− /2 0 m2 − p22 x − p21 z + p22 x2 + p21 z − ln µ2 = − × (11.72) From the last formula we find that the divergent part of iΓ is given by − ig µ /2 64π The full one–loop vertex in the renormalized theory is (11.73) 238 Solutions iV3 = −igµ /2 − iδgµ /2 + iΓ In minimal subtraction scheme δg is δg = − g3 64π (11.74) (e) From (11.61), (11.69) and (11.70) follows Z =1− g2 , 384π g2 384π 5m20 g , = m20 + 384π m2 = m20 − + (11.75) m2 g 64π (11.76) in the one–loop order Similarly, from (11.69) and (11.74) we have g0 = (g + δg)µ Z 3/2 = gµ /2 = gµ /2 /2 g2 g2 + 64π 256π 3g 1− 256π 1− The last expression is important for calculation of the β function (11.77) (11.78) (11.79) References D Bailin and A Love, Introduction to Gauge Field Theory, Adam Hilger, Bristol, 1986 J Bjorken and S Drell, Relativistic Quantum Mechanics, McGraw-Hill, New York, 1964 J Bjorken and S Drell, Relativistic Quantum Fields, McGraw-Hill, New York, 1965 N N Bogoljubov and D.V Shirkov, Introduction to the Theory of Quantized Fields, Wiley-Interscience, New York, 1980 M Blagojevi´c, Gravitation and Gauge Symmetries, IOP Publishing, Bristol, 2002 T.P Cheng and L.F Li, Gauge Theory of Elementary Particle Physics, Oxford University Press, New York, 1984 T.P Cheng and L.F Li, Gauge Theory of Elementary Particle Physics, Problems and Solutions, Oxford University Press, New York, 2000 M Damnjanovi´c, Hilbert spaces and group theory, Faculty of Physics, Beograd, 2000 (in Serbian) I.S Gradshteyn and I.M Ryzhnik, Table of Integrals, Series and Products, (trans and ed by Alan Jeffrey), Academic Press, Orlando, Florida, 1980 10 W Greiner and J, Reinhardt, Quantum Electrodinamics, Springer, Berlin, Heidelberg, New York, 1996 11 W Greiner and J, Reinhardt, Field Quantization, Springer, Berlin, Heidelberg, New York, 1996 12 F Gross, Relativistic Quantum Mechanics and Field Theory, Wiley, New York, 1993 13 C Itzykson and J.B Zuber, Quantum Field Theory, McGraw-Hill, New York, 1980 14 M Kaku, Quantum Field Theory: A Modern Introduction, Oxford University Press, New York, 1993 15 F Mandl and G Show, Quantum Field Theory, New York, 1999 16 M E Peskin and D V Schroeder, An Introduction to Quantum Field Theory, Addison Wesley, 1995 17 P Ramond, Field Theory: A Modern Primer (second edition), Addison-Wesley, RedwoodCity, California, 1989 240 References 18 L Rayder, Quantum Field Theory, Cambridge University Press, Cambridge, 1985 19 J J Sakurai, Advanced Quantum Mechanics, Addison-Wesley, Reading, 1967 20 S S Schweber, An Introduction to Relativistic Quantum Field Theory, Harpen and Row, New York, 1962 21 A G Sveshnikov and A N Tikhonov, The Theory of Functions of a Complex Variable, Mir Publisher, Moscow, 1978 22 G Sterman, Introduction to Quantum Field Theory, Cambridge University Press, Cambridge, 1993 23 S Weinberg, The Quantum Theory of Fields I and II, Cambridge University Press, New York, 1996 Index Action 25 Einstein–Hilbert 27 Advanced Green function Dirac equation 138 Klein–Gordon equation 132 Angular momentum tensor Dirac field 44, 45, 164 electromagnetic field 52, 183–185 Klein–Gordon field 36, 37, 144 Anticommutation relations Dirac field 43 Baker–Hausdorff formula Bianchi identity 49 91, 144 Casimir effect 53, 187–190 Casimir operator Charge Dirac field 45, 162 Klein–Gordon field 37, 142 Charge conjugation Dirac equation 18 bilinears 23–24, 115–118 Dirac field 45 bilinears 47, 175–177 scalar field 41, 159 Chiral transformations 28 Coherent states 40, 156–158 Commutation relations electromagnetic field 50 scalar field 35 Conformal group 75 Conformal transformations Continuity equation 10 Cross section 55 Cutkosky rule 62, 225 Decay rate 218 Differential cross section 192 Dilatations Dirac field 30, 46, 129, 168 scalar field 29, 38, 129, 148–150 Dimensional regularization 63 Dirac equation 17 helicity 99, 118 helicity basic 20, 95 plane wave solutions 17, 18, 93–95 spinor basic 20 Dirac field quantization 43 Dirac particle in a hole 22, 110–111 in a magnetic field 23, 113 Dyson Green function Klein–Gordon equation 133 Electromagnetic field quantization 49 Energy–momentum tensor 26, 126 symmetric or Belinfante tensor 29, 127 Euler–Lagrange equations 25, 121 Feynman parametrization 62, 211 Feynman propagator Dirac equation 138, 139 Dirac field 44 242 Index Klein–Gordon equation 31, 33, 132, 136 Klein–Gordon field 36, 153 Foldy–Wouthuysen transformation 24, 118–119 Functional derivative 25, 121 Furry theorem 225 Galilean algebra 39, 156 Gamma matrices 13 contraction identities 14, 86–87 Dirac representation 13 Majorana representation 13 trace identities 15, 87–89 Weyl representation 13 Gamma–function 62, 213 γ5 –matrix 13, 86, 102 s–operator 98 γ5 / gauge transformations 49 Gordon identity 21, 104 Grassmann variable 173 Green function Dirac equation 31, 33 Klein-Gordon equation 31 massive vector field 33, 140 massless vector field 34, 140 Schr¨ odinger equation 154 Gupta–Bleuler quantization 50 Hamiltonian Dirac field 44, 45, 162 Klein–Gordon field 36, 37, 142 Helicity 94, 165, 181 Klein paradox Dirac particle 109 scalar particle 82 Klein–Gordon equation plane wave solutions 77 Klein–Gordon particle in a hole 10, 79 in a magnetic field 10, 81 in the Coulomb potential 10, 83 Lagrangian density Dirac field 43 massive vector field 27 massless vector field 49 Schr¨ odinger field 39 sigma model 28 Left/right spinors 102–103 Levi-Civita tensor 4, 5, 68 Little group 74 Lorentz group 5, 67 generators in defining repr Lorentz transformations Dirac equation 17 bilinears 23–24, 115–118 Dirac field 44, 170 bilinears 47, 174–177 scalar field 158–159 69 Majorana spinor 47, 173 Maxwell equations 49 Metric tensor Minkowski space Momentum Dirac field 44, 45 Klein–Gordon field 36, 37, 142 MS scheme 237 Noether theorem 26 Normal ordering Dirac field 44, 47, 172 Klein–Gordon field 36 Optic theorem 220 Parity Dirac equation 18 bilinears 23–24, 115–118 Dirac field 44 bilinears 47, 174–177 scalar field 41, 159 Pauli matrices Pauli–Lubanski vector 7, 19, 72–74, 98 Pauli–Villars regularization 62, 215 Phase transformations 28, 125 φ3 theory in 4D 58 φ3 theory in 6D 64, 234–238 Poincar´e algebra 6, 71, 72 Poincar´e group 4, Poincar´e transformations scalar field 40 Projection operators energy 19, 95–96 spin 100 QED processes Index scattering in an external electromagnetic field 202 QED processes µ− µ+ → e− e+ 58, 196–198 e− µ+ → e− µ+ 58 e− µ+ → e− µ+ 198 Compton scattering 58, 199 scattering in an external electromagnetic field 58, 200 Reflection and transmission coefficients Dirac equation 22 Klein–Gordon equation 10 Reiman ζ–function 53 Retarded Green function Klein–Gordon equation 132, 137 S–matrix 55 Scalar electrodynamics 64, 226 Scalar field quantization 35 Scalar product Scattering of polarized particles 203–205 Schr¨ odinger equation 153 Schwinger model 64, 233 Σ–vector 96 σµν –matrices 14, 85, 87 59, 243 SL(2, C) group Superficial degree of divergence 64, 227 Symmetry factor in φ4 theory 57, 194–195 Tensor of rank (m, n) Time reversal Dirac equation 18 bilinears 23–24, 115–118 Dirac field 44 bilinears 47, 175–178 scalar field 41, 159 Vacuum polarization 63, 225 Vector contravariant components covariant components dual vector or one–form Vertex correction 231–232, 237 Virasora algebra 38 Weyl fields 20 Wick rotation 212 Wick theorem 55, 57, 152, 172, 193–196 Yukawa theory 64, 206, 227–233 [...]... In addition to the spinor basis, one often uses the helicity basis The helicity basis is obtained by taking n = p/|p| in the rest frame Find the equations for the spin in this case 4.13 Find the form of the equations for the spin, defined in Problem 4.12 in the ultrarelativistic limit 4.14 Show that the operator γ5 /s commutes with the operator /p, and that the eigenvalues of this operator are ±1 Find... the previous problem Find the nonrelativistic limit of this Hamiltonian 2.14 Determine the velocity operator v = i[H, x], where H is the Hamiltonian obtained in Problem 2.12 Solve the eigenvalue problem for v 2.15 In the space of two–component wave functions the scalar product is defined by 1 ψ1 |ψ2 = d3 xψ1† σ3 ψ2 2 (a) Show that the Hamiltonian H obtained in Problem 2.12 is Hermitian (b) Find expectation... moves in the Coulomb field of a nucleus Find the energy spectrum of the bounded states for this system if the charge of the nucleus is Ze θ , where θ = 12 (φ + mi ∂φ ∂t ) χ and χ = 12 (φ − mi ∂φ ∂t ), instead of φ rewrite the Klein–Gordon equation in the Schr¨ odinger form 2.12 Using the two-component wave function 1 Actually this is current density Chapter 2 The Klein–Gordon equation 11 2.13 Find the... for an observer moving with momentum p = pez , 24 Problems (c) the wave functions which are obtained after space and time inversion, (d) the wave function in a frame which is obtained from S by a rotation about the x–axis through θ 4.43 Find the matrices C and P in the Weyl representation of the γ–matrices 4.44 Prove that the helicity of the Dirac particle changes sign under space inversion, but not... relativistic particle with zero spin The transformation law of a scalar field φ(x) under Lorentz transformations is given by φ (Λx) = φ(x) • The equation for the spinless particle in an electromagnetic field, Aµ is obtained by changing ∂µ → ∂µ + iqAµ in equation (2.A), where q is the charge of the particle 2.1 Solve the Klein–Gordon equation 2.2 If φ is a solution of the Klein–Gordon equation calculate the... solution of Klein–Gordon equation in external electromagnetic potential Aµ 2.7 A scalar particle in the s–state is moving in the potential qA0 = −V, r < a , 0, r>a where V is a positive constant Find the dispersion relation, i.e the relation between energy and momentum, for discrete particle states Which condition has to be satisfied so that there is only one bound state in the case V < 2m? 2.8 Find the energy... action in flat spacetime (anyone familiar with general relativity can easily prove this): S= d4 x 1 1 ∂σ hµν ∂ σ hµν − ∂σ hµν ∂ ν hµσ + ∂σ hµσ ∂µ h − ∂µ h∂ µ h 2 2 , where h = hµµ Derive the equations of motion for hµν These are the linearized Einstein equations Show that the linearized theory is invariant under the gauge symmetry: hµν → hµν + ∂µ Λν + ∂ν Λµ , where Λµ (x) is any four-vector field 5.10 Find... translation subgroup The Poincar´e algebra is given in Problem 1.11 • The Levi-Civita tensor, µνρσ is a totaly antisymmetric tensor We will use the convention that 0123 = +1 2 3 The tangent space is a vector space of tangent vectors associated to each point of spacetime Poincar´e transformations are very often called inhomogeneous Lorentz transformations Chapter 1 Lorentz and Poincare symmetries 5 1.1... and Nk = Mk0 Further, one can introduce the following linear combinations Ai = 12 (Mi + iNi ) and Bi = 12 (Mi − iNi ) Prove that [Ai , Aj ] = i ijl Al , [Bi , Bj ] = i ijl Bl , [Ai , Bj ] = 0 This is a well known result which gives a connection between the Lorentz algebra and ”two” SU(2) algebras Irreducible representations of the Lorentz group are classified by two quantum numbers (j1 , j2 ) which... j2 ) which come from above two SU(2) groups 1.10 The Poincar´e transformation (Λ, a) is defined by: x µ = Λµ ν xν + aµ Determine the multiplication rule i.e the product (Λ1 , a1 )(Λ2 , a2 ), as well as the unit and inverse element in the group 1.11 (a) Verify the multiplication rule U −1 (Λ, 0)U (1, )U (Λ, 0) = U (1, Λ−1 ) , in the Poincar´e group In addition, show that from the previous relation follows: ... Library of Congress Control Number: 2005934040 ISBN-10 3-5 4 0-2 906 2-1 Springer Berlin Heidelberg New York ISBN-13 97 8-3 -5 4 0-2 906 2-9 Springer Berlin Heidelberg New York This work is subject to copyright... excellent Quantum Field Theory textbooks Unfortunately, there is a shortage of Problem Books in this field, one of the exceptions being the Problem Book of Cheng and Li [7] The overlap between this Problem. .. doing more problems in full details improves both understanding and efficiency So, I feel that the absence of such a book in Quantum Field Theory is a gap in the literature This was my main motivation