Symmetries and Group Theory in Particle Physics: An Introduction to ...

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Lecture Notes in Physics Volume 823 Founding Editors W. Beiglböck J. Ehlers K. Hepp H. Weidenmüller Editorial Board B.-G. Englert, Singapore U. Frisch, Nice, France F. Guinea, Madrid, Spain P. Hänggi, Augsburg, Germany W. Hillebrandt, Garching, Germany M. Hjorth-Jensen, Oslo, Norway R. A. L. Jones, Sheffield, UK H. v. Löhneysen, Karlsruhe, Germany M. S. Longair, Cambridge, UK M. L. Mangano, Geneva, Switzerland J.-F. Pinton, Lyon, France J.-M. Raimond, Paris, France A. Rubio, Donostia, San Sebastian, Spain M. Salmhofer, Heidelberg, Germany D. Sornette, Zurich, Switzerland S. Theisen, Potsdam, Germany D. Vollhardt, Augsburg, Germany W. Weise, Garching, Germany For further volumes: http://www.springer.com/series/5304
The Lecture Notes in Physics The series Lecture Notes in Physics (LNP), founded in 1969, reports new developments in physics research and teaching—quickly and informally, but with a high quality and the explicit aim to summarize and communicate current knowledge in an accessible way. Books published in this series are conceived as bridging material between advanced graduate textbooks and the forefront of research and to serve three purposes: • to be a compact and modern up-to-date source of reference on a well-defined topic • to serve as an accessible introduction to the field to postgraduate students and nonspecialist researchers from related areas • to be a source of advanced teaching material for specialized seminars, courses and schools Both monographs and multi-author volumes will be considered for publication. Edited volumes should, however, consist of a very limited number of contributions only. Proceedings will not be considered for LNP. Volumes published in LNP are disseminated both in print and in electronic formats, the electronic archive being available at springerlink.com. The series content is indexed, abstracted and referenced by many abstracting and information services, bibliographic networks, subscription agencies, library networks, and consortia. Proposals should be sent to a member of the Editorial Board, or directly to the managing editor at Springer: Christian Caron Springer Heidelberg Physics Editorial Department I Tiergartenstrasse 17 69121 Heidelberg/Germany christian.caron@springer.com
Page 4 and 5: Giovanni Costa • Gianluigi Fogli
Page 6: To our children Alessandra, Fabrizi
Page 9 and 10: VIII Preface The book is divided in
Page 11 and 12: X Preface Notation The natural syst
Page 13 and 14: XII Contents Problems . . . . . . .
Page 16 and 17: 1 Introduction to Lie groups and th
Page 18 and 19: 1.1 Basic definitions 3 common. The
Page 20 and 21: 1.1 Basic definitions 5 In other wo
Page 22 and 23: 1.2 Lie groups and Lie algebras 7 T
Page 24 and 25: 1.2 Lie groups and Lie algebras 9 W
Page 26 and 27: 1.2 Lie groups and Lie algebras 11
Page 32 and 33: 1.3 Semi-simple Lie algebras and th
Page 42 and 43: 2 The rotation group In this Chapte
Page 44 and 45: 2.1 Basic properties 29 ⎛ ⎞ n 2
Page 46 and 47: 2.1 Basic properties 31 ( ) ( ) ( )
Page 48 and 49: 2.3 Irreducible representations of
Page 50 and 51: 2.3 Irreducible representations of
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2.4 Matrix representations of the r
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2.5 Addition of angular momenta and
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2.5 Addition of angular momenta and
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44 3 The homogeneous Lorentz group
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62 4 The Poincaré transformations
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72 5 One particle and two particle
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84 6 Discrete operations [ I s , J
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86 6 Discrete operations generality
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88 6 Discrete operations For a stat
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90 6 Discrete operations parity of
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92 6 Discrete operations and U(R)T
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94 6 Discrete operations Tests of t
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7 Relativistic equations In this Ch
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7.1 The Klein-Gordon equation 99 We
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7.3 The Maxwell equations 101 which
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7.3 The Maxwell equations 103 ǫ(k,
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7.4 The Dirac equation 105 ( ) ( )(
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7.4 The Dirac equation 107 and the
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7.5 The Dirac equation for massless
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7.6 Extension to higher half-intege
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8 Unitary symmetries This Chapter i
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8.2 Generalities on symmetries of e
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8.3 U(1) invariance and Additive Qu
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8.3 U(1) invariance and Additive Qu
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8.4 Isospin invariance 121 Table 8.
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8.4 Isospin invariance 123 method i
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8.4 Isospin invariance 125 A phenom
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8.4 Isospin invariance 127 other wo
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8.5 SU(3) invariance 129 For a few
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8.5 SU(3) invariance 131 Of course,
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8.5 SU(3) invariance 133 is irreduc
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8.5 SU(3) invariance 135 of IR’s
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8.5 SU(3) invariance 137 use the ei
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8.5 SU(3) invariance 139 Y n p 938.
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8.5 SU(3) invariance 141 d Y u Y s
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8.5 SU(3) invariance 143 ⎛ ⎞
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8.5 SU(3) invariance 145 Table 8.8.
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8.5 SU(3) invariance 147 Y U + V +
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8.5 SU(3) invariance 149 Y Y’ n p
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8.5 SU(3) invariance 151 book also
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8.5 SU(3) invariance 153 8.5.7 The
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8.5 SU(3) invariance 155 b) φ-ω m
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8.6 Beyond SU(3) 157 transforms as
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8.6 Beyond SU(3) 159 provided they
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8.6 Beyond SU(3) 161 Under each IR
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8.6 Beyond SU(3) 163 Table 8.11. Th
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8.6 Beyond SU(3) 165 X . + . . . .
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8.6 Beyond SU(3) 167 multiplet of v
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9 Gauge symmetries In this Chapter
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9.2 Invariance under group transfor
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9.2 Invariance under group transfor
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9.3 The gauge group U(1) and Quantu
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9.4 The gauge group SU(3) and Quant
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9.4 The gauge group SU(3) and Quant
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9.5 The mechanism of spontaneous sy
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9.6 Spontaneous breaking of the chi
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9.6 Spontaneous breaking of the chi
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9.7 The group SU(2) ⊗ U(1) and th
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9.8 Groups of Grand Unification 207
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A Rotation matrices and Clebsch-Gor
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A.2 Clebsch-Gordan coefficients 215
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A.2 Clebsch-Gordan coefficients 217
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220 B Symmetric group and identical
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226 C Young tableaux and irreducibl
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238 Solutions ∣ ∣1, 1 2 ; 3 2 ,
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240 Solutions ⎛ |α| 2 + |β| 2
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242 Solutions Λ ρ σ = g ρ σ
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244 Solutions If A ′ µν is the
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246 Solutions 4.4 Starting from U(
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248 Solutions a) Product: R a R b =
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250 Solutions After some algebra, m
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252 Solutions U(a, Λ)|p, λ> = e
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254 Solutions E p = p , ẼgE = g ,
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256 Solutions and the rotation angl
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258 Solutions 6.2 Knowing that all
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260 Solutions term changes its sign
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262 Solutions For the pseudoscalar
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264 Solutions x 1 = ξ 1 ξ 1 , x 2
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266 Solutions 27 = (3, S = 0) ⊕ (
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268 Solutions so that, accordingly
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270 Solutions I + I − ψ max = 2
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272 Solutions 21 ⊗ 6 = 56 ⊕ 70
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274 Solutions where D µ is the cov
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276 Solutions 9.3 The leading W ±
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278 Solutions component of φ l , o
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280 Solutions ⎛ ⎞ a a 0 = a
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282 Bibliography Sagle, A.A. and Wa
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284 Bibliography Bigi, I.I. and San
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Index Abelian group, 1, 118 ABJ ano
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Index 289 Neutrino helicity, 87 Neu
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Index 291 U(l,N − l), 11 U(1), 11
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