F. J. Dyson ADVANCED QUANTUM MECHANICS Second Edition arXiv:quant-ph/0608140v1 18 Aug 2006 ADVANCED QUANTUM MECHANICS Lecture notes by Professor F. J. Dyson for a course in Relativistic Quantum Mechanics given at Cornell University in the Fall of 1951. Second Edition The first edition of these notes was written by Professor Dyson. The second edition was prepared by Michael J. Moravcsik; he is responsible for the changes made in the process of re-editing. Generally used notation: A∗ : complex conjugate transposed (Hermitian conjugate) A+ : complex conjugate (not transposed) ∗ ∗ A : A β = A γ4 = adjoint A−1 = inverse AT = transposed I = identity matrix or operator i Table of Contents Introduction 1 SubjectMatter ...................................... ........ 1 DetailedProgram.................................... ......... 2 OneParticleTheories ................................ .......... 2 The Dirac Theory 4 TheFormoftheDiracEquation . .......... 4 LorentzInvarianceoftheDiracEquation . ............... 6 ToFindtheS......................................... ...... 7 TheCovariantNotation .. .. .. .. .. .. .. .. .. .. .. .. .. .......... 8 ConservationLaws–ExistenceofSpin . .............. 9 ElementarySolutions. .. .. .. .. .. .. .. .. .. .. .. .. .. ........... 10 TheHoleTheory...................................... ....... 11 PositronStates–chargeconjugation . ............... 11 ElectromagneticPropertiesof the Electron . ................. 12 TheHydrogenAtom .................................... ....... 13 Solutionoftheradialequation . ............. 14 Behavior of an Electron in non-relativistic approximation ...................... 17 Summary of Matrices in the Dirac Theory in our notation . ................. 20 Summary of Matrices in the Dirac Theory in the Feynman notation ................ 21 Scattering Problems and Born Approximation 22 GeneralDiscussion .................................. .......... 22 ProjectionOperators. .. .. .. .. .. .. .. .. .. .. .. .. .. ........... 23 CalculationofSpurs ................................. .......... 24 Scattering of two electrons in Born Approximation – The MøllerFormula ............. 27 Relation of Cross Sections to Transition Amplitudes . ................... 29 ResultsforMøllerScattering . ............. 30 NoteontheTreatmentofExchangeEffects . ............. 31 Relativistic Treatment of Several Particles . ................... 31 Field Theory 32 ClassicalRelativisticFieldTheory . ................ 32 QuantumRelativisticFieldTheory . .............. 34 TheFeynmanMethodofQuantization . ............ 35 TheSchwingerActionPrinciple. ............. 36 A.TheFieldEquations ................................ ...... 37 B. The Schr¨odinger Equation for the State-function . ................. 37 C.OperatorFormoftheSchwingerPrinciple . ............ 38 D.CanonicalCommutationLaws . ........ 39 E. The Heisenberg Equation of Motion for the Operators . .............. 39 F.GeneralCovariantCommutationLaws . .......... 39 G.AnticommutingFields .. .. .. .. .. .. .. .. .. .. .. .. .. ....... 40 ii Examples of Quantized Field Theories 41 I.TheMaxwellField.................................. ......... 41 MomentumRepresentations . ........ 42 FourierAnalysisofOperators . .......... 43 EmissionandAbsorptionOperators . .......... 44 GaugeInvarianceoftheTheory. ......... 45 TheVacuumState ..................................... .... 45 TheGupta-BleulerMethod . ....... 47 Example: Spontaneous Emission of Radiation . ............. 47 TheHamiltonianOperator . ....... 49 FluctuationsoftheFields . ......... 50 The Lamb Shift – Fluctuation of Position of Electron . ............... 51 Ia. TheoryofLineShiftandLineWidth. ............. 52 TheInteractionRepresentation . ........... 53 Application of Interaction Representation to Line Shift andLineWidth ........... 54 Calculation of Line Shift – Non-Relativistic Theory . ................. 57 TheIdeaofMassRenormalization . ......... 57 II. Field Theory of the Dirac Electron – Without Interaction .................... 59 CovariantCommutationRules . ........ 60 MomentumRepresentations . ........ 61 FourierAnalysisofOperators . .......... 62 EmissionandAbsorptionOperators . .......... 62 ChargeSymmetricalRepresentation . ........... 63 TheHamiltonian ..................................... ..... 64 Failure of Theory with Commuting Fields . ........... 64 TheExclusionPrinciple . .. .. .. .. .. .. .. .. .. .. .. .. ........ 65 TheVacuumState ..................................... .... 65 III. Field Theory of Dirac Electron in External Field . ................... 66 CovariantCommutationRules . ........ 66 TheHamiltonian ..................................... ..... 68 AntisymmetryoftheStates . ........ 69 PolarizationoftheVacuum . ........ 70 CalculationofMomentumIntegrals. ........... 73 PhysicalMeaning of the Vacuum Polarization . ............. 75 Slowly Varying Weak Fields. The Uehling Effect. ............. 78 IV. Field Theory Of Dirac And Maxwell Fields In Interaction . ................... 78 The Complete Relativistic Quantum Electrodynamics . ............... 78 FreeInteractionRepresentation . ............ 80 Free Particle Scattering Problems 81 A.MøllerScatteringofTwoElectrons . .............. 81 Properties of the DF Function .................................. 82 TheMøllerFormula–Conclusion. ......... 83 Electron-PositronScattering. ............ 84 B. Scattering of a Photon by an Electron – The Compton Effect – The Klein-Nishina Formula. 84 CalculationofCrossSection. .......... 86 SumOverSpins....................................... .... 87 C.Two-QuantumPairAnnihilation . ............ 90 D. Bremsstrahlung and Pair Creation in the Coulomb Field of anAtom .............. 92 iii General Theory of Free Particle Scattering 94 ReductionofanOperatortoNormalForm . ............ 96 FeynmanGraphs...................................... ....... 98 FeynmanRulesofCalculation. ............ 100 TheSelf-EnergyoftheElectron. ............. 102 Second-Order Radiative Corrections to Scattering . .................... 104 Treatment of Low-Frequency Photons – The Infra-Red Catastrophe. ................ 116 Scattering By A Static Potential 118 A.TheMagneticMomentoftheElectron . ............ 121 B. Relativistic Calculation of the Lamb Shift . ................. 123 CovariantPartoftheCalculation. ........... 124 Discussion of the Nature of the Φ-Representation . ............... 126 Concluding Non-Covariant Part of the Calculation . ............... 127 AccuracyoftheLambShiftCalculation . ........... 130 Typist’s Afterword 132 Notes 134 References 138 Index 140 iv Lecture Course 491 — Advanced Quantum Theory F. J. Dyson — Fall Semester 1951 Introduction Books W. Pauli, “Die Allgemeinen Principien der Wellenmechanik”; Handbuch der Physik, 2 ed., Vol. 24, Part 1; Edwards reprint, Ann Arbor 1947. (In German) [1] W. Heitler, Quantum Theory of Radiation, 2nd Edition, Oxford. 3rd edition just published. [2] G. Wentzel, Introduction to the Quantum Theory of Wave-Fields, Interscience, N.Y. 1949 [3] I shall not expect you to have read any of these, but I shall refer to them as we go along. The later part of the course will be new stuff, taken from papers of Feynman and Schwinger mainly. [4], [5], [6], [7], [8] Subject Matter You have had a complete course in non-relativistic quantum theory. I assume this known. All the general principles of the NR theory are valid and true under all circumstances, in particular also when the system happens to be relativistic. What you have learned is therefore still good. You have had a course in classical mechanics and electrodynamics including special relativity. You know what is meant by a system being relativistic; the equations of motion are formally invariant under Lorentz transformations. General relativity we shall not touch. This course will be concerned with the development of a Lorentz–invariant Quantum theory. That is not a general dynamical method like the NR quantum theory, applicable to all systems. We cannot yet devise a general method of that kind, and it is probably impossible. Instead we have to find out what are the possible systems, the particular equations of motion, which can be handled by the NR quantum dynamics and which are at the same time Lorentz–invariant. In the NR theory it was found that almost any classical system could be handled, i.e.quantized. Now on the contrary we find there are very few possibilities for a relativistic quantized system. This is a most important fact. It means that starting only from the principles of relativity and quantization, it is mathe- matically possible only for very special types of objects to exist. So one can predict mathematically some important things about the real world. The most striking examples of this are: (i) Dirac from a study of the electron predicted the positron, which was later discovered [9]. (ii) Yukawa from a study of nuclear forces predicted the meson, which was later discovered [10]. These two examples are special cases of the general principle, which is the basic success of the relativistic quantum theory, that A Relativistic Quantum Theory of a Finite Number of Particles is Impossible. A RQ theory necessarily contains these features: an indefinite number of particles of one or more types, particles of each type being identical and indistinguishable from each other, possibility of creation and annihilation of particles. Thus the two principles of relativity and quantum theory when combined lead to a world built up out of various types of elementary particles, and so make us feel quite confident that we are on the right way to