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

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B.2 Symmetric group and Young tableaux 223 where ∑ p indicates the sum over all permutations and ǫ p = + or − in correspondence to an even or odd permutation (a permutation is even or odd if it contains an even or odd number of transpositions). The state with the symmetry of the τ standard Young tableau is obtained by means of |Φ τ > = Y τ |a 1 b 2 . . .z n > . (B.12) Particular importance have the states which are totally symmetrical or totally antisymmetrical, because they are those occurring in nature. They correspond to the Young tableaux consisting of only one row or one column, so that they are simply given by |Φ s > = S n |a 1 b 2 . . . z n > , (B.13) |Φ a > = A n |a 1 b 2 . . . z n > . (B.14) An assembly of particles which occurs only in symmetrical states (B.13) is described by the Bose-Einstein statistics. Such particles occur in nature and are called bosons. A comparison between the Bose statistics and the usual classical (Boltzmann) statistics (which considers particles distinguishable) shows e.g. that the probability of two particles being in the same state is greater in Bose statistics than in the classical one. Let us examine the antisymmetrical state (B.14). It can be written in the form of a determinant ⎛ ⎞ a 1 a 2 . . . a n ⎜ b 1 b 2 . . . b n ⎟ ⎝. . . . . . . . . . . . ⎠ . (B.15) z 1 z 2 . . . z n It then appears that if two of the states |a>, |b>, . . ., |z n > are the same, the state (B.14) vanishes. This means that two particles cannot occupy the same state: this rule corresponds to the Pauli exclusion principle. In general, the occupied states must be all independent, otherwise (B.15) vanishes. An assembly of particles which occur only in the antisymmetric states (B.14) is described by the Fermi-Dirac statistics. Such particles occur in nature: they are called fermions. Other quantum statistics exist which allow states with more complicated symmetries than the complete symmetry or antisymmetry. Until now, however, all experimental evidence indicates that only the Bose and the Fermi statistics occur in nature; moreover, systems with integer spin obey the Bose statistics, and systems with half-integer spin the Fermi statistics. This connection of spin with statistics is shown to hold in quantum field theory (spinstatistics theorem), in which bosons and fermions are described by fields which commute and anticommute, respectively, for space-like separations. Then, unless new particles obeying new statistics are experimentally detected, all known particles states belong to the one-dimensional representation of the symmetric group. We note, however, that the states of n-particles have
224 B Symmetric group and identical particles to be singlet (totally symmetric or totally antisymmetric) under permutations of all variables (space-time coordinates, spin, internal quantum numbers); if one takes into account only a subset of variables, e.g. the internal ones, a state can have mixed symmetry. In this case, one has to make use of the other – in general higher dimensional – IR’s of the symmetric group S n .
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Lecture Notes in Physics Volume 823
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Giovanni Costa • Gianluigi Fogli
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To our children Alessandra, Fabrizi
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VIII Preface The book is divided in
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X Preface Notation The natural syst
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XII Contents Problems . . . . . . .
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1 Introduction to Lie groups and th
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1.1 Basic definitions 3 common. The
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1.1 Basic definitions 5 In other wo
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1.2 Lie groups and Lie algebras 7 T
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1.2 Lie groups and Lie algebras 9 W
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1.2 Lie groups and Lie algebras 11
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1.3 Semi-simple Lie algebras and th
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2 The rotation group In this Chapte
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2.1 Basic properties 29 ⎛ ⎞ n 2
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2.1 Basic properties 31 ( ) ( ) ( )
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2.3 Irreducible representations of
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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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Page 208 and 209: 9.7 The group SU(2) ⊗ U(1) and th
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Page 228 and 229: A Rotation matrices and Clebsch-Gor
Page 230 and 231: A.2 Clebsch-Gordan coefficients 215
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Page 235 and 236: 220 B Symmetric group and identical
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Page 241 and 242: 226 C Young tableaux and irreducibl
Page 253 and 254: 238 Solutions ∣ ∣1, 1 2 ; 3 2 ,
Page 255 and 256: 240 Solutions ⎛ |α| 2 + |β| 2
Page 257 and 258: 242 Solutions Λ ρ σ = g ρ σ
Page 259 and 260: 244 Solutions If A ′ µν is the
Page 261 and 262: 246 Solutions 4.4 Starting from U(
Page 263 and 264: 248 Solutions a) Product: R a R b =
Page 265 and 266: 250 Solutions After some algebra, m
Page 267 and 268: 252 Solutions U(a, Λ)|p, λ> = e
Page 269 and 270: 254 Solutions E p = p , ẼgE = g ,
Page 271 and 272: 256 Solutions and the rotation angl
Page 273 and 274: 258 Solutions 6.2 Knowing that all
Page 275 and 276: 260 Solutions term changes its sign
Page 277 and 278: 262 Solutions For the pseudoscalar
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Page 283 and 284: 268 Solutions so that, accordingly
Page 285 and 286: 270 Solutions I + I − ψ max = 2
Page 287 and 288: 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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