The Physics of Ettore Majorana: Theoretical, Mathematical, and Phenomenological

394 Pages·2014·2.191 MB·English

by  Salvatore Esposito

THE PHYSICS OF ETTORE MAJORANA
Through just a handful of papers, Ettore Majorana left an indelible mark on the
fields of physics, mathematics, computer science, and even economics before his
mysteriousdisappearancein1938.ItisonlynowthattheimportanceofMajorana’s
work is being realized: Majorana fermions are intensely studied today, and his
work on neutrino physics has provided possible explanations for the existence of
darkmatter.
Inthisuniquevolume,SalvatoreEspositonotonlyexploresMajorana’sknown
papers but, even more interestingly, also unveils his unpublished works. These
include powerful methods and results, ranging from the atomic two-center prob-
lem, the Thomas–Fermi model, and ferromagnetism to quasi-stationary states,
n-componentrelativisticwaveequations,andquantumscalarelectrodynamics.
Featuring biographical notes and contributions from leading experts Evgeny
Akhmedov and Frank Wilczek, this fascinating book will captivate graduate stu-
dentsandresearchersinterestedinbothfrontierscienceandthehistoryofscience.
Salvatore Esposito is Professor of the History of Physics and Associate Pro-
fessor of Theoretical Physics, and Associate Researcher at the Naples Unit of the
Istituto Nazionale di Fisica Nucleare. His research interests range from neutrino
physicstofieldtheory,andheisconsideredtobetheworldexpertonMajorana’s
work.

THE PHYSICS OF ETTORE
MAJORANA
Phenomenological, Theoretical, and Mathematical
SALVATORE ESPOSITO
WithcontributionsbyE.AkhmedovandF.Wilczek
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©S.Esposito2015
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Firstpublished2015
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Contents
Acknowledgments pagexii
PartI Introducingthecharacter 1
1 Lifeandmyth 3
1.1 Fortunesandmisfortunesofagenius 3
1.2 Familyanduniversitytraining 6
1.3 LonephysicistintheFermigroup 9
1.4 Leipzig–Rome–Naples:thelateryears 12
2 Thevisibleside 17
2.1 Tenpapersdepictingthefuture 17
2.2 IntroducingtheDiracequationintoatomicspectroscopy 18
2.3 Spontaneousionization 19
2.3.1 Anomaloustermsinhelium 19
2.3.2 IncompleteP(cid:2) triplets 21
2.3.3 Majorana–Fano–Feshbachresonances 23
2.4 Chemicalbonding 24
2.4.1 Heliummolecularion 24
2.4.2 Majoranastructures 26
2.5 Non-adiabaticspin-flip 27
2.5.1 Majoranasphereandageneraltheorem 28
2.5.2 Landau–Zenerprobabilityformula 30
2.5.3 Majorana’sholes 31
2.5.4 Majorana–Brosseleffect 32
2.6 Nuclearforces 33
2.6.1 TheHeisenbergmodelofnuclearinteractions 33
2.6.2 Majorana’sexchangemechanism 35
vi Contents
2.6.3 Thomas–FermiformalismandYukawapotential 37
2.7 Infinite-componentequation 38
2.7.1 Asuccessfulrelativisticwaveequation 39
2.7.2 Majoranaequation 40
2.7.3 Infinite-dimensional representations of the
Lorentzgroup 41
2.7.4 AdifficultproblemforPauliandFierz 42
2.7.5 Furtherelaborations 44
2.8 Majorananeutrinotheory 47
2.8.1 “Symmetric”Diracequation 47
2.8.2 Neutrino–antineutrinoidentity 49
2.8.3 Racahandtheneutrinolessdoubleβ-decay 50
2.8.4 Pontecorvoandtheneutrinooscillations 51
2.8.5 Majoranafermions 52
2.9 Complexsystemsinphysicsandeconomics 53
2.9.1 GenesisofpaperN.10 54
2.9.2 Statisticallawsinsocialsciences 55
2.9.3 Asensationalsuccessineconophysics 57
PartII Atomicphysics 61
3 Two-electronproblem 63
3.1 Along-lastingsuccessforquantummechanics 63
3.2 Knownsolutionstotheheliumatomproblem 64
3.2.1 Perturbativecalculations 64
3.2.2 VariationalmethodI 66
3.2.3 Self-consistentfieldmethod 68
3.2.4 Slater’srefinement 69
3.2.5 VariationalmethodII:Hylleraasvariables 70
3.2.6 Helium-likeions 71
3.3 Majoranaempiricalrelations 72
3.4 Heliumwavefunctionsandbroadrangeestimates 76
3.5 Accuratenumbersandageneraltheory 78
3.5.1 AsimpleralternativetoHylleraas’smethod 78
3.5.2 Majorana’svariantofthevariationalmethod 79
3.6 Conclusions 81
4 Thomas–Fermimodel 83
4.1 Fermiuniversalpotential 83
4.1.1 Thomas–Fermiequation 83
Contents vii
4.1.2 Numericalandapproximatesolutions 85
4.1.3 Mathematicalproperties 86
4.2 MajoranasolutionoftheThomas–Fermiequation 87
4.2.1 TransformationintoanAbelequation 87
4.2.2 Analyticseriessolution 89
4.2.3 Numericaltables 92
4.3 Mathematicalgeneralizations 93
4.3.1 Frobeniusmethod 93
4.3.2 Scale-invariantdifferentialequations 95
4.4 Physicalapplications 97
4.4.1 ModifiedFermipotentialforheavyatoms 98
4.4.2 Secondapproximationfortheatomicpotential 100
4.4.3 Atomicpolarizability 102
4.4.4 Applicationstomolecules 103
4.5 Conclusions 105
PartIII Nuclearandstatisticalphysics 107
5 Quasi-stationarynuclearstates 109
5.1 Probingtheatomicnucleuswithα particles 109
5.2 Scatteringofα particlesonaradioactivenucleus 111
5.2.1 Quantum-mechanicaltheory 111
5.2.2 Thermodynamicapproach 115
5.3 Transitionprobabilitiesofquasi-stationarystates 116
5.3.1 Transitionsfromadiscreteintoacontinuumstate 116
5.3.2 Transitionsintotwocontinuousspectra 118
5.3.3 Transitionsfromacontinuumstate 118
5.4 Nucleardisintegrationbyα particles 119
5.4.1 Statementoftheproblem 119
5.4.2 Theappropriatewavefunction 121
5.4.3 Crosssection 122
5.5 Conclusions 124
6 Theoryofferromagnetism 127
6.1 Towardsastatisticaltheoryofferromagnetism 128
6.1.1 Molecularfields 128
6.1.2 Heisenbergtheory 129
6.1.3 Laterrefinements 131
6.2 Majoranastatisticalmodel 131
6.2.1 Distributionfunction 134
viii Contents
6.3 Solutionofthemodelinthecontinuumlimit 136
6.3.1 Partitionfunctionatfinitetemperature 138
6.3.2 Meanmagnetization 139
6.4 Applicationsandfurtherresults 141
6.4.1 Particularferromagneticgeometries 141
6.4.2 Criticaltemperatureanddimensionality 143
6.5 Conclusions 144
PartIV Relativisticfieldsandgrouptheory 147
7 Groupsandtheirapplicationstoquantummechanics 149
7.1 The“Gruppenpest”inquantummechanics 150
7.2 Unitarytransformationsintwodimensions 153
7.2.1 D representationandgroupgenerators 154
j
7.3 Three-dimensionalrotations 156
7.3.1 Groupgenerators 157
7.4 Application:theanomalousZeemaneffect 160
7.5 Lorentzgroupanditsrepresentations 164
7.5.1 n-dimensionalDiracmatrices 164
7.5.2 Specialcase:maximumallowedpforfixedn 167
7.5.3 Non-Hermitianoperators 169
7.5.4 Infinite-dimensionalunitaryrepresentations 170
7.6 Conclusions 173
8 Diracequationsandsomealternatives 175
8.1 Searchingforanequation 175
8.1.1 MassivephotonsandtheDKPalgebra 178
8.1.2 Dirac–Fierz–Pauliformalism 180
8.1.3 Generalequationsforarbitraryspin 182
8.2 Majoranan-componentspinorequations 184
8.2.1 The16-componentequationforatwo-particlesystem 185
8.2.2 Equationforasix-componentspinor 187
8.2.3 Five-componentequation 189
8.3 Parallellives(andfindings) 189
8.4 Conclusions 191
PartV Quantumfieldtheory 193
9 Scalarelectrodynamics 195
9.1 Earlyquantumelectrodynamics 196
Contents ix
9.1.1 Quantumfieldformalism 196
9.1.2 Particlesandantiparticles 198
9.1.3 Pauli–Weisskopftheory 198
9.2 MajoranatheoryofscalarelectrodynamicsI 201
9.3 MajoranatheoryofscalarelectrodynamicsII 205
9.4 Applicationtothenuclearstructure 209
9.5 Conclusions 212
10 Photonsandelectrons 214
10.1 Photonwavemechanics 215
10.1.1 Majorana–Oppenheimer formulation of
electrodynamics 215
10.1.2 Lorentz-invariantwavetheory 217
10.1.3 Two-componenttheory 219
10.1.4 Fieldquantization 220
10.2 Dynamicaltheoryofelectronsandholes 221
10.3 Conclusions 224
PartVI Fundamentaltheoriesandothertopics 227
11 A“pathintegral”approachtoquantummechanics 229
11.1 DiracandFeynman’smathematicalapproach 230
11.2 Majorana’sphysicalapproach 232
11.3 Conclusions 234
12 Fundamentallengthsandtimes 236
12.1 Introducingelementaryspace-timelengths 236
12.2 Quasi-Coulombianscattering 239
12.3 Intrinsictimedelayandretardedelectromagneticfields 242
12.4 Conclusions 244
13 Majorana’smultifacetedlife 246
13.1 Majoranaasastudent 246
13.1.1 Meltingpointshiftduetoamagneticfield 246
13.1.2 Determinationofafunctionfromitsmoments 248
13.1.3 WKBmethodfordifferentialequations 251
13.2 Majorana as a phenomenologist: spontaneous and induced
ionizationofahydrogenatom 254
13.2.1 Hydrogenatomplacedinahighpotentialregion 255
13.2.2 Ionizationofahydrogen-likeatominanelectricfield 260
13.3 Majoranaasatheoretician:aunifyingmodelforthe
fundamentalconstants 263
x Contents
13.4 Majoranaasamathematician 264
13.4.1 Improperoperators 264
13.4.2 Cubicsymmetry 266
13.5 Majoranaasateacher 270
PartVII BeyondMajorana 277
14 Majoranaandcondensedmatterphysics 279
f.wilczek
14.1 Spinresponseanduniversalconnection 280
14.2 LevelcrossingandgeneralizedLaplacetransform 282
14.3 Majorana fermions and Majorana mass: from neutrinos to
electrons 285
14.3.1 Majorana’sequation 285
14.3.2 AnalysisofMajorananeutrinos 287
14.3.3 Majoranamass 289
14.3.4 Majoranaelectrons 292
14.4 Majorinos 293
14.4.1 Kitaevchain 294
14.4.2 Junctionsandthealgebraicgenesisofmajorinos 298
14.4.3 Continuummajorinos 301
15 MajorananeutrinosandotherMajoranaparticles:theory
andexperiment 303
e.akhmedov
15.1 Weyl,Dirac,andMajoranafermions 304
15.1.1 Particle−antiparticleconjugation 306
15.1.2 DiracdynamicsandtheMajoranacondition 307
15.1.3 FermionmasstermsandU(1)symmetries 312
15.1.4 FeynmanrulesforMajoranaparticles 313
15.2 C,P,CP,andCPTpropertiesofMajoranafermions 314
15.3 MixingandoscillationsofMajorananeutrinos 317
15.3.1 NeutrinoswithaMajoranamassterm 317
15.3.2 GeneralcaseofDirac+Majoranamassterm 322
15.3.3 Dirac and pseudo-Dirac neutrino limits
intheD+Mcase 325
15.4 Seesawmechanismofneutrinomassgeneration 327
15.5 ElectromagneticpropertiesofMajorananeutrinos 330
15.6 MajoranaparticlesinSUSYtheories 338