FORCE AND NATURE
CONTENTS 1 WHAT MAKES THE WORLD TICK? MOTION GRAVITATION THE FORCE ?ELD EQUIVALENCE PRINCIPLE ENERGY MOMENTUM LEAST ACTION NEWTON CANONIZED THE MECHANICAL UNIVERSE 2 ELECTROMAGNETISM ELECTRIC ?ELD LINES OF FORCE MULTIPOLES SCALAR POTENTIAL ELECTRIC CURRENT MAGNETIC ?ELD VECTOR POTENTIAL ELECTROMAGNETIC INDUCTION MAXWELL’S EQUATIONS 2 10 RADIATION 3 THE VACUUM IS THE MEDIUM THE ETHER REFERENCE FRAMES THE LIGHT CONE LORENTZ TRANSFORMATION RELATIVITY OF SPACE AND TIME FOUR VECTORS E = MC2 FASTER THAN LIGHT? MAXWELL’S TRUE FORM 3 10 THE GAUGE ?ELD 3 11 WHO WROTE THESE SIGNS 3 12 LORENTZ AND EINSTEIN 4 LET THERE BE LIGHT LOCAL GAUGE INVARIANCE A CREATION THE GAUGE PRINCIPLE HERMANN WEYL AND THERE WAS LIGHT 5 HEROIC AGE: THE STRUGGLE FOR QUANTUM THEORY ALIEN SIGNALS BOHR’S ATOM PURELY IMAGINARY QUANTUM MECHANICS THE WAVE FUNCTION QUANTUM THEORY AND RELATIVITY SILLY QUESTION 6 QUANTUM REALITY THE UNCERTAINTY RELATION WAVE NATURE OF MATTER ENTANGLEMENT ALL VIRTUAL REALITIES THE QUANTUM CENTURY THE WASTE LECTURE 7 WHAT IS CHARGE? THE QUANTUM GAUGE COVARIANT DERIVATIVE AHARONOV–BOHM EXPERIMENT U(1) QUANTUM GAUGE PRINCIPLE GLOBAL VS LOCAL GAUGE INVARIANCE 8 THE ZEN OF ROTATION ROTATIONS DO NOT COMMUTE HAMILTON’S ?ASH OF INSIGHT GENERATORS OF ROTATION GROUPS SU(2): FUNDAMENTAL REPRESENTATION THE ADJOINT REPRESENTATION 9 YANG–MILLS FIELD: NON-COMMUTING CHARGES GAUGING SU(2) PICTURING LOCAL GAUGE INVARIANCE MAXWELL GENERALIZED GAUGE PHOTONS MAGNETIC CHARGE MONOPOLE: THE GAUGE HEDGEHOG INTO THE DEEP FREEZE 10 PHOTONS REAL AND VIRTUAL 10 1 REAL PHOTONS 10 2 QUANTUM JUMPS 10 3 VIRTUAL PHOTONS 11 CREATION AND ANNIHILATION 11 1 THE QUANTUM ?ELD 11 2 PARTICLE AND ANTIPARTICLE 11 3 THE DIRAC EQUATION 11 4 THE DIRAC SEA 11 5 REVERSING TIME 11 6 FEYNMAN DIAGRAM 11 7 THE ?NE-STRUCTURE CONSTANT 12 THE DYNAMICAL VACUUM 12 1 QED 12 2 INTERACTION VERTEX 12 3 SELF-ENERGY 12 4 VACUUM POLARIZATION 12 5 THE DRESSED ELECTRON 12 6 THE ULTRAVIOLET CATASTROPHE 12 7 REALITY OF VACUUM ?UCTUATIONS 12 8 WHEN PHYSICISTS WERE HEROES 12 9 THE ENDURING QED 13 ELEMENTARY PARTICLES 13 1 BEGINNINGS 13 2 BOSONS AND FERMIONS 13 3 SPIN AND STATISTICS 13 4 INTERACTIONS 14 THE FALL OF PARITY 14 1 DAWN OF THE POST-MODERN ERA 14 2 NEUTRINO: A LEFT-HANDED SCREW 14 3 CP 14 4 IS NOTHING SACRED? 15 THE PARTICLE EXPLOSION 15 1 THE ACCELERATOR BOOM 15 2 DARKNESS AT NOON 15 3 THE ONTOLOGICAL BOOTSTRAP 15 4 THE ULTIMATE TEMPERATURE 15 5 ECHOS OF AN ERA 16 QUARKS 16 1 STRANGENESS 16 2 OCTET AND DECAPLET 16 3 THE EIGHTFOLD WAY 16 4 “THREE QUARKS FOR MUSTER MARK!” 16 5 CHARM AND BEYOND 16 6 PARTONS 16 7 CHARMONIUM 16 8 COLOR 17 ALL INTERACTIONS ARE LOCAL 17 1 YANG–MILLS AWAKENS 17 2 UNIFYING ELECTROMAGNETIC AND WEAK INTERACTIONS 17 3 GENERATING MASS 17 4 MAKING THE PHOTON 17 5 HISTORICAL NOTE 17 6 THE LEPTON-QUARK FAMILY 17 7 QCD 17 8 TWO MORE FAMILIES: WHO ORDERED THEM? 17 9 THE STANDARD MODEL 18 BROKEN SYMMETRY 18 1 WHAT IS MASS? 18 2 HOW A MAGNET GETS MAGNETIZED 18 3 THE ORDER PARAMETER 18 4 THE GOLDSTONE MODE 18 5 SUPERCONDUCTIVITY: THE PHOTON GETS MASS 18 6 HISTORICAL NOTE 19 QUARK CON?NEMENT 19 1 MONOPOLE CON?NEMENT 19 2 ELECTRIC ?UX TUBE 19 3 THE QCD STRING 19 4 ASYMPTOTIC FREEDOM 20 HANGING THREADS OF SILK 20 1 MASS 20 2 CHIRALITY 20 3 THE PION AS GOLDSTONE BOSON 20 4 PCAC 20 5 THE TRIANGLE ANOMALY 20 6 LEPTON-QUARK FAMILY STRUCTURE 20 7 WAITING FOR CLOSURE 21 THE WORLD IN A GRAIN OF SAND 21 1 A MATTER OF SCALE 21 2 RENORMALIZATION 21 3 THE RUNNING COUPLING 21 4 FIXED POINT: THEORETICAL MODEL 21 5 UV ?XED POINT: QCD 21 6 IR ?XED POINT: QED 21 7 CROSSOVER: SCIENTI?C REVOLUTION 22 IN THE SPACE OF ALL POSSIBLE THEORIES 22 1 THE PHYSICS IS IN THE CUTO? 22 2 THE RG TRAJECTORY 22 3 THE SPACE OF LAGRANGIANS 22 4 OF TIME AND TEMPERATURE 22 5 TIAN WEN 22 6 TIAN WEN UPDATED EPILOGUE: BEAUTY IS TRUTH WHAT MAKES THE WORLD TICK? 1 1 MOTION WE SEE MOTION ALL AROUND US LEAVES FALL; WAVES BREAK; HEAVENLY BODIES MOVE WHAT CAUSES MOTION? THE ANSWER IS INTERACTION INTERACTION MAKES THE WORLD TICK IF THERE WERE NO INTERACTIONS, BODIES WOULD STAND STILL, OR MOVE WITH UNCHANGING VELOCITY ANY CHANGE REQUIRES FORCE, AND THAT MEANS INTERACTION NEWTON’S LAW, THE FOUNDATION OF CLASSICAL MECHANICS, STATES F = MA HERE, F IS THE FORCE ACTING ON A BODY, M IS THE INERTIAL MASS OF THE BODY, AND A IS ITS ACCELERATION — THE RATE OF CHANGE OF THE VELOCITY WE CAN USE THIS EQUATION IN TWO WAYS: AS DE?NITION OF FORCE; AS EQUATION OF MOTION IN THE ?RST INSTANCE, WE OBTAIN THE FORCE F(X) BY MEASURING THE ACCELERATION OF THE BODY AT POSITION X THE FORCE CAN BE REPRESENTED BY A TABLE OF DATA, OR BY A FORCE LAW WE DEDUCE FROM THE DATA WHEN THE FORCE IS GIVEN, NEWTON’S EQUATION TAKES THE FORM OF A DI?ERENTIAL EQUATION THAT CAN BE SOLVED, EITHER ANALYTICALLY USING CALCULUS, OR THROUGH NUMERICAL INTEGRATION ON A COMPUTER: ¨X = F(X) M GALILEO DROPPED TWO BALLS FROM THE TOP OF THE LEANING TOWER OF PISA, ONE LIGHT, THE OTHER HEAVY THEY HIT THE GROUND SIMULTANEOUSLY, SHOWING THAT THE ACCELERATION DUE TO GRAVITY IS INDEPENDENT OF MASS AN OVERHEAD DOT DENOTES TIME DERIVATIVE THUS, ?X DENOTES VELOCITY, AND ¨X IS ACCELERATION TIME HAS ENTERED THE PICTURE, AND THE EQUATION DESCRIBES DYNAMICAL EVOLUTION 1 2 GRAVITATION THE EARLIEST KNOWN INTERACTION IS GRAVITY AS LEGEND HAS IT, GALILEO DROPPED TWO BALLS FROM THE TOP OF THE LEANING TOWER OF PISA, ONE HEAVY, THE OTHER LIGHT THEY HIT THE GROUND SIMULTANEOUSLY, SHOWING THAT THE ACCELERATION DUE TO GRAVITY IS INDEPENDENT OF THE MASS THAT FORCE AND NATURE WHAT MAKES THE WORLD TICK? 1THE CONSTANT G IS CALLED “ACCELERATION DUE TO GRAVITY”, OR SIMPLY BUT MISLEADINGLY “G-FORCE” ITS VALUE IS 9 8 M S?2, OR 32 FT S?2 2THE VALUE OF THE GRAVITATIONAL CONSTANT IS ? = 6 670 × 10?11 M3 KG?1 S?2 FORCE AND NATURE FUNDAMENTAL FORCES OF NATURE: THE STORY OF GAUGE FIELDS THE EARTH’S PULL ON A PERSON CAN BE OBTAINED BY PUTTING M = EARTH’S MASS, M? = PERSON’S MASS, R = DISTANCE BETWEEN PERSON AND CENTER OF EARTH THUS, R IS VERY NEARLY THE RADIUS OF EARTH, EVEN FOR A HIGH JUMPER; WHENCE THE APPROXIMATE CONSTANCY OF THE ACCELERATION OF GRAVITY: G = ?M R2 , WHERE R IS EARTH’S RADIUS THE SAME INVERSE-SQUARE LAW GIVES THE FORCE BETWEEN JUPITER AND MARS, THE FORCE ACTING ON A COMET BY THE SUN, AND INDEED ON ANY TWO MASSES IN THE UNIVERSE THIS IS WHY IT IS CALLED UNIVERSAL GRAVITATION 1 3 THE FORCE ?ELD A MASS M EXERTS A GRAVITATIONAL FORCE ON ANY OTHER MASS, PROPORTIONAL TO THE LATTER’S MASS THE FORCE PER UNIT MASS IS CALLED THE GRAVITATIONAL ?ELD: GRAVITATIONAL ?ELD = ?M R2 ANY OTHER MASS AT A DISTANCE R FROM IT WILL FEEL A FORCE EQUAL TO THIS ?ELD TIMES ITS MASS IN A SENSE THE MASS ALTERS THE PROPERTY OF SPACE, FOR IT CREATES A FORCE ?ELD PERMEATING ALL SPACE THE ?ELD IS TO BECOME A CENTRAL CONCEPT IN MODERN PHYSICS 1 4 EQUIVALENCE PRINCIPLE THE MASS M APPEARS BOTH AS A MEASURE OF INERTIA, AND A MEASURE OF ?ELD STRENGTH THESE TWO ROLES ARE CONCEPTUALLY DISTINCT, AND WE SHOULD REALLY DENOTE THEM WITH DI?ERENT SYMBOLS: THE INERTIAL MASS MINERTIA IS THE QUANTITY APPEARING IN F = MINERTIA A FORCE AND NATURE SOME THREE HUNDRED YEARS AFTER GALLILEO’S PISA EXPERIMENT, EINSTEIN EXPLAINED IT IN TERMS OF THE GEOMETRY OF SPACE-TIME, IN HIS THEORY OF GENERAL RELATIVITY IT MEASURES THE BODY’S RESPONSE TO AN EXTERNAL FORCE THE GRAVITATIONAL MASS MGRAV APPEARS IN ?MGRAV/R2, AND MEASURES THE ?ELD STRENGTH IT PRODUCES EXPERIMENTALLY, THEY HAVE THE SAME NUMERICAL VALUE: MINERTIA = MGRAV THIS IS KNOWN AS THE EQUIVALENCE PRINCIPLE, AND APPEARS TO BE ACCI- DENTAL EINSTEIN COULD NOT ACCEPT THE ACCIDENTAL EXPLANATION HE HELD THAT THE TWO MASSES CAN BE CONSIDERED EQUIVALENT ONLY WHEN THEIR DE?NING CONCEPTS ARE SHOWN TO BE EQUIVALENT IN 1917, NEARLY THREE HUNDRED YEARS AFTER GALILEO’S EXPERIMENT, HE TURNED THE ACCIDENT INTO AN IMPERATIVE THROUGH THE THEORY OF GENERAL RELATIVITY FORCE AND NATURE IN A ROLLER COASTER, KINETIC ENERGY 2MV2 AND POTENTIAL EN- ERGY MGH CONVERT INTO EACH OTHER DURING THE RIDE, BUT THEIR SUM REMAINS CONSTANT IN EINSTEIN’S GENERAL RELATIVITY, MASS GENERATES CURVATURE IN SPACE- TIME A BODY IN ITS NEIGHBORHOOD SIMPLY ROLLS ALONG A GROOVE IN CURVED SPACE, FOLLOWING THE SHORTEST POSSIBLE PATH (A GEODESIC) THUS, THE MASS HAS NO BEARING ON MOTION IN A GRAVITATIONAL ?ELD THE ACTUAL CURVATURE OF SPACE-TIME IS VERY SLIGHT, AND DISCERNIBLE ONLY OVER COSMIC DISTANCES AT RELATIVELY SMALL SCALES, SUCH AS IN THE SOLAR SYSTEM, OR EVEN IN GALAXIES, ORDINARY NETWONIAN MECHANICS IS QUITE ADEQUATE 1 5 ENERGY A BODY HAS MORE “MOTION” WHEN IT GOES FASTER, AND A MEASURE OF THE VIGOR IS THE KINETIC ENERGY KINETIC ENERGY = 2MV2, WHERE V IS THE VELOCITY WHEN THE BODY MOVES IN A FORCE ?ELD, THE VELOCITY CHANGES FROM POINT TO POINT FOR EXAMPLE, A ROLLER COASTER MOVES UNDER GRAVITY, AT VARYING HEIGHTS CONSTRAINED BY THE TRACK THE VELOCITY IS SMALL NEAR THE TOP, AND LARGE NEAR THE BOTTOM, AS ILLUSTRATED IN FIG 1 4 WE CAN DE?NE A POTENTIAL ENERGY MGH, WHERE H IS THE HEIGHT ABOVE GROUND WHEN ADDED TO THE KINETIC ENERGY, WE OBTAIN A CONSTANT TOTAL FORCE AND NATURE ENERGY, WHEN FRICTION IS NEGLECTED: TOTAL ENERGY = KINETIC ENERGY + POTENTIAL ENERGY THIS RELATION IS KNOWN AS THE CONSERVATION OF ENERGY THE SPEEDUP AND SLOWDOWN OF A ROLLER COASTER SIGNI?ES THE CONVERSION OF POTENTIAL ENERGY TO KINETIC ENERGY AND VICE VERSA 1 6 MOMENTUM MOMENTUM IS DE?NED AS MASS TIMES VELOCITY: MOMENTUM = MV NEWTON’S LAW SAYS FORCE IS THE RATE OF CHANGE OF MOMENTUM THUS, THE MOMENTUM REMAINS CONSTANT IN THE ABSENCE OF FORCE THIS UNDERLIES THE INTUITIVE NOTION THAT MOMENTUM IS WHAT KEEP THINGS ON THE MOVE IF A SYSTEM IS COMPOSED OF MORE THAN ONE BODY, THEN EACH BODY HAS AN INDIVIDUAL MOMENTUM, AND THEIR SUM IS CALLED THE TOTAL MOMENTUM: TOTAL MOMENTUM = M1V1 + M2V2 + WHEN THERE IS NO OVERALL EXTERNAL FORCE ACTING ON THE SYSTEM, THE SUM OF THE INTERNAL FORCES MUST BE ZERO, AND THE TOTAL MOMENTUM IS CONSERVED IF TWO PARTICLES COLLIDE IN FREE SPACE, THEIR INDIVIDUAL MOMENTA WILL SU?ER CHANGES, BUT THE SUM OF THE MOMENTA MUST BE THE SAME BEFORE AND AFTER THE COLLISION 1 7 LEAST ACTION THE MAGIC FORMULA F = MA EXPLAINS THE CLASSICAL WORLD WHY IS IT TRUE? TO PROPERLY POSE THE QUESTION, CONSIDER THE MOTION DEPICTED SCHEMATICALLY IN FIG 1 5 THE SOLID LINE REPRESENTS A PARTICLE’S AC- TUAL PATH, WHICH IS GOVERNED BY NEWTON’S EQUATION THE DOTTED LINES REPRESENT OTHER “VIRTUAL” PATHS WITH THE SAME ENDPOINTS HOW DOES FORCE AND NATURE 3THE PRINCIPLE OF LEAST ACTION HAD BEEN PROPOSED IN VARIOUS FORMS BY PIERRE FER- MAT (1601–1665), PIERRE-LOUIS MOREAU DE MAUPERTUIS (1698–1759), AND LEONHARD EULER (1707–1783) 4ACTUALLY, THE SIGN OF THE ACTION IS IMMATERIAL, AND THE ACTION COULD BE MAXIMAL INSTEAD OF MINIMAL FOR THIS REASON PURISTS PREFER THE NAME “PRINCIPLE OF STATIONARY ACTION” FORCE AND NATURE UNLOCKING THE POWER AND BEAUTY OF NEWTONIAN MECHANICS AN EARLY PHILOSOPHICAL UNDERPINNING OF THE PRINCIPLE CAME FROM LAOZI5: LEAST ACTION ACHIEVES ALL ACTIONS 1 8 NEWTON CANONIZED LAGRANGE PAVED THE WAY FOR WILLIAM HAMILTONIAN, WHO BASED HIS APPROACH ON WHAT WE NOW CALL THE “HAMILTONIAN”: HAMILTONIAN = KINETIC ENERGY + POTENTIAL ENERGY ITS VALUE IS NONE OTHER THAN THE TOTAL ENERGY, BUT THE FORMALISM REQUIRES THAT THE HAMILTONIAN BE EXPRESSED IN TERMS OF “CANONICAL VARIABLES” — THE COORDINATE Q AND ITS “CANONICALLY CONJUGATE” MOMENTUM P ACCORDINGLY WE WRITE IT AS H(P, Q) NEWTON’S LAW IS NOW RECAST IN HAMILTON’S CANONICAL EQUATIONS: ?Q = ? ?PH(P, Q), ?P = ?? ?Q H(P, Q) 5????? (DAO DE JING, CA 500 B C ): “?????? ” FORCE AND NATURE TRUIMPH OF NEWTONIAN MECHANICS: PREDICTION AND DISCOVERY OF THE PLANET NEPTUNE THE LAGRANGIAN AND HAMILTONIAN FORMULATIONS ARE EQUIVALENT THE MOST SUCCINCT WAY TO SPECIFY A SYSTEM IS TO GIVE ITS LAGRANGIAN OR HAMILTONIAN 1 9 THE MECHANICAL UNIVERSE THE CORRECTNESS OF NEWTONIAN MECHANICS HAD BEEN CON?RMED OVER AND OVER IN CELESTIAL MECHANICS, THROUGH THE E?ORT OF PIERRE SIMON LAPLACE AND OTHERS THE CROWNING MOMENT WAS SURELY THE PREDICTION AND DISCOVERY OF A HERETOFORE UNKNOWN PLANET — NEPTUNE ITS EX- ISTENCE WAS DEDUCED INDEPENDENTLY BY URBAIN LE VERRIER AND JOHN COUCH ADAMS, FROM PERTURBATIONS IN THE ORBIT OF URANUS A LETTER FROM LE VERRIER CONTAINING THE PREDICTED PLANET’S COORDINATES REACHED JOHANN GALLE ON SEPTEMBER 23, 1846 THE SAME EVENING, GALLE WANGLED OBSERVATION TIME ON THE BERLIN TELESCOPE POINTING IT TO THE PREDICTED POSITION, HE FOUND NEPTUNE THE UNDERSTANDING OF THE THE UNIVERSE SEEMED COMPLETE LAPLACE SAID THAT, GIVEN THE POSITIONS AND VELOCITIES OF ALL THE STARS AT ANY ONE INSTANT, HE WILL BE ABLE TO CALCULATE, IN PRINCIPLE, THE HISTORY OF THE UNIVERSE FOR ALL TIMES THE ABILITY TO QUANTITATIVELY UNDERSTAND NATURAL PHENOMENA LED TO PROFOUND PHILOSOPHICAL SHIFTS FORCE AND NATURE THE FOLLOWING EXCHANGE REPORTEDLY TOOK PLACE DURING A MEETING OF LAPLACE AND LAGRANGE WITH NAPOLEON BONAPARTE (1769–1821)6: NAPOLEON: HOW IS IT THAT, ALTHOUGH YOU SAY SO MUCH ABOUT THE UNIVERSE, YOU SAY NOTHING ABOUT ITS CREATOR? LAPLACE: NO, SIRE, I HAD NO NEED OF THAT HYPOTHESIS LAGRANGE: AH, BUT IT IS SUCH A GOOD HYPOTHESIS: IT EXPLAINS SO MANY THINGS! LAPLACE: INDEED, SIRE, MONSIEUR LAGRANGE HAS, WITH HIS USUAL SAGACITY, PUT HIS ?NGER ON THE PRECISE DI?CULTY WITH THE HYPOTH- ESIS: IT EXPLAINS EVERYTHING, BUT PREDICTS NOTHING LAPLACE MAY THINK THAT HE WAS ABLE TO PREDICT EVERYTHING; BUT HIS WAS A MECHANICAL UNIVERSE AN ESSENTIAL INGREDIENT OF THE REAL UNIVERSE WAS NOT YET CONSIDERED: ELECTROMAGNETISM 6A DE MORGAN, BUDGET OF PARADOXES (LONGMANS, GREEN, LONDON, 1872) FORCE AND NATURE THIS PAGE INTENTIONALLY LEFT BLANK THIS PAGE INTENTIONALLY LEFT BLANK PIONEERS IN ELECTRICITY AND MAGNETISM