A MODERN INTRODUCTION TO QUANTUM FIELD THEORY
CONTENTS INTRODUCTION OVERVIEW TYPICAL SCALES IN HIGH-ENERGY PHYSICS FURTHER READING EXERCISES LORENTZ AND POINCAR´E SYMMETRIES IN QFT LIE GROUPS THE LORENTZ GROUP THE LORENTZ ALGEBRA TENSOR REPRESENTATIONS DECOMPOSITION OF LORENTZ TENSORS UNDER SO(3) SPINORIAL REPRESENTATIONS SPINORS IN NON-RELATIVISTIC QUANTUM MECHANICS SPINORS IN THE RELATIVISTIC THEORY FIELD REPRESENTATIONS SCALAR ?ELDS WEYL ?ELDS DIRAC ?ELDS MAJORANA ?ELDS VECTOR ?ELDS THE POINCAR´E GROUP REPRESENTATION ON ?ELDS REPRESENTATION ON ONE-PARTICLE STATES SUMMARY OF CHAPTER CLASSICAL ?ELD THEORY THE ACTION PRINCIPLE NOETHER’S THEOREM THE ENERGY–MOMENTUM TENSOR REAL SCALAR ?ELDS; KLEIN–GORDON EQUATION COMPLEX SCALAR ?ELD; U(1) CHARGE SPINOR ?ELDS THE WEYL EQUATION; HELICITY THE DIRAC EQUATION CHIRAL SYMMETRY MAJORANA MASS THE ELECTROMAGNETIC ?ELD COVARIANT FORM OF THE FREE MAXWELL EQUATIONS GAUGE INVARIANCE; RADIATION AND LORENTZ GAUGES MINIMAL AND NON-MINIMAL COUPLING TO MATTER FIRST QUANTIZATION OF RELATIVISTIC WAVE EQUATIONS SOLVED PROBLEMS THE ?NE STRUCTURE OF THE HYDROGEN ATOM RELATIVISTIC ENERGY LEVELS IN A MAGNETIC ?ELD QUANTIZATION OF FREE ?ELDS REAL SCALAR ?ELDS FOCK SPACE COMPLEX SCALAR ?ELD; ANTIPARTICLES SPIN 1/2 ?ELDS DIRAC ?ELD MASSLESS WEYL ?ELD C, P, T ELECTROMAGNETIC ?ELD QUANTIZATION IN THE RADIATION GAUGE COVARIANT QUANTIZATION PERTURBATION THEORY AND FEYNMAN DIAGRAMS THE S-MATRIX THE LSZ REDUCTION FORMULA SETTING UP THE PERTURBATIVE EXPANSION THE FEYNMAN PROPAGATOR WICK’S THEOREM AND FEYNMAN DIAGRAMS A FEW VERY EXPLICIT COMPUTATIONS LOOPS AND DIVERGENCES SUMMARY OF FEYNMAN RULES FOR A SCALAR ?ELD FEYNMAN RULES FOR FERMIONS AND GAUGE BOSONS RENORMALIZATION VACUUM ENERGY AND THE COSMOLOGICAL CONSTANT PROBLEM THE MODERN POINT OF VIEW ON RENORMALIZABILITY THE RUNNING OF COUPLING CONSTANTS CROSS-SECTIONS AND DECAY RATES RELATIVISTIC AND NON-RELATIVISTIC NORMALIZATIONS DECAY RATES CROSS-SECTIONS TWO-BODY ?NAL STATES RESONANCES AND THE BREIT–WIGNER DISTRIBUTION BORN APPROXIMATION AND NON-RELATIVISTIC SCATTERING THREE-BODY KINEMATICS AND PHASE SPACE INELASTIC SCATTERING OF NON-RELATIVISTIC ELECTRONS ON ATOMS QUANTUM ELECTRODYNAMICS THE QED LAGRANGIAN ONE-LOOP DIVERGENCES E+E??? ?µ+µ? ELECTROMAGNETIC FORM FACTORS THE LOW-ENERGY LIMIT OF THE ELECTROWEAK THEORY A FOUR-FERMION MODEL CHARGED AND NEUTRAL CURRENTS IN THE STANDARD MODEL SOLVED PROBLEMS: WEAK DECAYS µ??E?¯?E?µ ?+ ?L+?L ISOSPIN AND ?AVOR SU(3) K0 ???L+?L PATH INTEGRAL QUANTIZATION PATH INTEGRAL FORMULATION OF QUANTUM MECHANICS PATH INTEGRAL QUANTIZATION OF SCALAR ?ELDS PERTURBATIVE EVALUATION OF THE PATH INTEGRAL EUCLIDEAN FORMULATION QFT AND CRITICAL PHENOMENA QFT AT ?NITE TEMPERATURE INSTANTONS AND TUNNELING 10 NON-ABELIAN GAUGE THEORIES NON-ABELIAN GAUGE TRANSFORMATIONS YANG–MILLS THEORY QCD FIELDS IN THE ADJOINT REPRESENTATION 11 SPONTANEOUS SYMMETRY BREAKING DEGENERATE VACUA IN QM AND QFT SSB OF GLOBAL SYMMETRIES AND GOLDSTONE BOSONS ABELIAN GAUGE THEORIES: SSB AND SUPERCONDUCTIVITY 12 SOLUTIONS TO EXERCISES CHAPTER 1 1 OVERVIEW 1 2 TYPICAL SCALES IN HIGH-ENERGY PHYSICS OVERVIEW QUANTUM ?ELD THEORY IS A SYNTHESIS OF QUANTUM MECHANICS AND SPECIAL RELATIVITY, AND IT IS ONE OF THE GREAT ACHIEVEMENTS OF MODERN PHYSICS QUANTUM MECHANICS, AS FORMULATED BY BOHR, HEISENBERG, SCHR¨ODINGER, PAULI, DIRAC, AND MANY OTHERS, IS AN INTRINSICALLY NON-RELATIVISTIC THEORY TO MAKE IT CONSISTENT WITH SPECIAL RELATIVITY, THE REAL PROBLEM IS NOT TO ?ND A RELATIVISTIC GENERALIZATION OF THE SCHR¨ODINGER EQUATION 1 WAVE 1ACTUALLY, SCHR¨ODINGER ?RST FOUND A RELATIVISTIC EQUATION, THAT TODAY WE CALL THE KLEIN–GORDON EQUATION HE THEN DISCARDED IT BECAUSE IT GAVE THE WRONG ?NE STRUCTURE FOR THE HYDROGEN ATOM, AND HE RETAINED ONLY THE NON- RELATIVISTIC LIMIT SEE WEINBERG (1995), PAGE 4 EQUATIONS, RELATIVISTIC OR NOT, CANNOT ACCOUNT FOR PROCESSES IN WHICH THE NUMBER AND THE TYPE OF PARTICLES CHANGES, AS IN ALMOST ALL REACTIONS OF NUCLEAR AND PARTICLE PHYSICS EVEN THE PROCESS OF AN ATOMIC TRANSITION FROM AN EXCITED ATOMIC STATE A?TO A STATE A WITH EMISSION OF A PHOTON, A??A + ?, IS IN PRINCIPLE UNACCESSIBLE TO THIS TREATMENT (ALTHOUGH IN THIS CASE, DESCRIBING THE ELECTROMAGNETIC ?ELD CLASSICALLY AND THE ATOM QUANTUM MECHANICALLY, ONE CAN GET SOME CORRECT RESULTS, EVEN IF IN A NOT VERY CONVINCING MANNER) FURTHERMORE, RELATIVISTIC WAVE EQUATIONS SU?ER FROM A NUMBER OF PATHOLOGIES, LIKE NEGATIVE-ENERGY SOLUTIONS A PROPER RESOLUTION OF THESE DI?CULTIES IMPLIES A CHANGE OF VIEWPOINT, FROM WAVE EQUATIONS, WHERE ONE QUANTIZES A SINGLE PARTICLE IN AN EXTER- NAL CLASSICAL POTENTIAL, TO QUANTUM ?ELD THEORY, WHERE ONE IDENTI?ES THE PARTICLES WITH THE MODES OF A ?ELD, AND QUANTIZES THE ?ELD ITSELF THE PROCEDURE ALSO GOES UNDER THE NAME OF SECOND QUANTIZATION THE METHODS OF QUANTUM ?ELD THEORY (QFT) HAVE GREAT GENERALITY AND ?EXIBILITY AND ARE NOT RESTRICTED TO THE DOMAIN OF PARTICLE PHYSICS IN A SENSE, ?ELD THEORY IS A UNIVERSAL LANGUAGE, AND IT PERMEATES MANY BRANCHES OF MODERN RESEARCH IN GENERAL, ?ELD THEORY IS THE CORRECT LAN- GUAGE WHENEVER WE FACE COLLECTIVE PHENOMENA, INVOLVING A LARGE NUMBER OF DEGREES OF FREEDOM, AND THIS IS THE UNDERLYING REASON FOR ITS UNIFYING POWER FOR EXAMPLE, IN CONDENSED MATTER THE EXCITATIONS IN A SOLID ARE QUANTA OF ?ELDS, AND CAN BE STUDIED WITH ?ELD THEORETICAL METHODS AN ESPECIALLY INTERESTING EXAMPLE OF THE UNIFYING POWER OF QFT IS GIVEN BY THE PHENOMENON OF SUPERCONDUCTIVITY WHICH, EXPRESSED IN THE ?ELD THEORY LANGUAGE, TURNS OUT TO BE CONCEPTUALLY THE SAME AS THE HIGGS MECHANISM IN PARTICLE PHYSICS AS ANOTHER EXAMPLE WE CAN MENTION THAT THE FEYNMAN PATH INTEGRAL, WHICH IS A BASIC TOOL OF MODERN QUAN- TUM ?ELD THEORY, PROVIDES A FORMAL ANALOGY BETWEEN ?ELD THEORY AND STATISTICAL MECHANICS, WHICH HAS STIMULATED VERY IMPORTANT EXCHANGES BETWEEN THESE TWO AREAS BESIDE PLAYING A CRUCIAL ROLE FOR PHYSICISTS, TH = ? 2? ?(0 328 478 965 ) ? ? + (1 176 11 ) ? ? ?(1 434 ) ? ? = 0 001 159 652 140(5)(4)(27) DI?ERENT SOURCES OF ERRORS ON THE LAST ?GURES ARE WRITTEN SEPARATELY IN PARENTHESES THE THEORETICAL ERROR IS DUE PARTLY TO THE NUMERICAL EVAL- UATION OF FEYNMAN DIAGRAMS (THERE ARE 891 OF THEM AT ORDER ?4!) AND PARTLY TO THE FACT THAT, AT THIS LEVEL OF PRECISION, HADRONIC CONTRIBUTIONS COME INTO PLAY WE ALSO NEED TO KNOW ? WITH SU?CIENT ACCURACY; THIS IS PROVIDED BY THE QUANTUM HALL E?ECT THE GYROMAGNETIC RATIO HAS BEEN MEASURED VERY PRECISELY ALSO FOR THE MUON, AND THE ACCURACY OF THIS MEASUREMENT HAS BEEN IMPROVED RECENTLY,2 WITH THE RESULT (G ?2)/2|EXP = 0 001 165 9208(6), AND A THEO- 2SEE HTTP://WWW G-2 BNL GOV/ THIS VALUES UPDATES THE VALUE REPORTED IN THE 2004 EDITION OF THE REVIEW OF PARTICLE PHYSICS RETICAL PREDICTION (G ?2)/2|TH = 0 001 165 9181(7) THE REMAINING DIS- CREPANCY HAS AROUSED MUCH INTEREST, IN THE HOPE THAT IT MIGHT BE A SIGNAL OF NEW PHYSICAL E?ECTS, BUT TO SEE WHETHER THIS IS ACTUALLY THE CASE RE- QUIRES ?RST A BETTER THEORETICAL UNDERSTANDING OF HADRONIC CONTRIBUTIONS, WHICH ARE MORE DI?CULT TO COMPUTE IN ANY CASE, AN AGREEMENT BETWEEN THEORY AND EXPERIMENT AT THE LEVEL OF 10 DECIMAL ?GURES FOR THE ELECTRON (OR EIGHT FOR THE MUON) IS SPECTACULAR, AND IT IS AMONG THE MOST PRECISE IN PHYSICS AS WE KNOW TODAY, QED IS ONLY A PART OF A LARGER THEORY AS WE APPROACH THE SCALES OF NUCLEAR PHYSICS, I E LENGTH SCALES R ?10?13 CM