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E-grāmata: Particles In The Early Universe: High-energy Limit Of The Standard Model From The Contraction Of Its Gauge Group

(Russian Academy Of Sci, Russia)
  • Formāts: 172 pages
  • Izdošanas datums: 16-Dec-2019
  • Izdevniecība: World Scientific Publishing Co Pte Ltd
  • Valoda: eng
  • ISBN-13: 9789811209741
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Particles In The Early Universe: High-energy Limit Of The Standard Model From The Contraction Of Its Gauge GroupPalielināt
  • Formāts: 172 pages
  • Izdošanas datums: 16-Dec-2019
  • Izdevniecība: World Scientific Publishing Co Pte Ltd
  • Valoda: eng
  • ISBN-13: 9789811209741
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The aim of this book is to develop a contraction method for classical orthogonal and unitary groups (algebras), and apply it to the investigation of physical structures, offering a new and unique interpretation to the high-energy limit of the Standard Model. Readers will find a comprehensive and rigorous study, summarized as follows: The space-time models (or kinematics) are described on the motion group level. The Jordan-Schwinger representations of the groups are shown to be closely connected to the properties of stationary quantum systems, whose Hamiltonians are quadratic in creation and annihilation operators. The high-temperature limit of the Standard Model is associated with the contraction of its gauge group, and the tending-to-zero contraction parameter is connected to the inverse average energy (temperature) of the Universe. This makes it possible to re-establish the evolution of particles and their interactions in the early Universe up to Planck energy, where readers shall discover that the properties of elementary particles change drastically in the infinite temperature limit: all particles lose mass, all quarks are monochromatic, electroweak interactions become long range and are mediated by neutral currents.Absent in all known literature, this book would be the first in describing the dynamics of particle properties and their interactions at different stages in the evolution of the Universe. Presenting both physical and mathematical approaches to various problems, and their applications to the physics of the early Universe, this book will be a valuable addition to the sparse literature on the subject matter.
Preface v
1 The Cayley-Klein groups and algebras
1(34)
1.1 Dual numbers and the Pimenov algebra
1(3)
1.1.1 Dual numbers
1(2)
1.1.2 The Pimenov algebra
3(1)
1.2 The Cayley-Klein orthogonal groups and algebras
4(13)
1.2.1 Three fundamental geometries on a line
4(4)
1.2.2 Nine Cayley-Klein groups
8(6)
1.2.3 Extension to higher dimensions
14(3)
1.3 The Cayley-Klein unitary groups and algebras
17(15)
1.3.1 Definitions, generators, commutators
17(3)
1.3.2 The unitary group SU(2;j1)
20(2)
1.3.3 Representations of the group SU(2;j1)
22(6)
1.3.4 The unitary group SU(3;j)
28(3)
1.3.5 Invariant operators
31(1)
1.4 Classification of transitions between the Cayley-Klein spaces and groups
32(3)
2 Space---time models
35(12)
2.1 Kinematics groups
35(3)
2.2 Carroll kinematics
38(5)
2.3 Non-relativistic kinematics
43(4)
3 The Jordan-Schwinger representations of Cayley-Klein groups
47(38)
3.1 The second quantization method and matrix elements
48(2)
3.2 The rotation groups in Cayley-Klein spaces
50(3)
3.3 The Jordan-Schwinger representations of the orthogonal Cayley-Klein groups
53(10)
3.3.1 Representations of SO2(j1) groups
56(2)
3.3.2 Representations of SO3(j) groups
58(2)
3.3.3 Representations of SO4(j) groups
60(3)
3.4 The Jordan-Schwinger representations of the special unitary Cayley-Klein groups
63(10)
3.4.1 Representations of SU2(j1) groups
66(3)
3.4.2 Representations of SU3(j1, j2) groups
69(4)
3.5 The Jordan-Schwinger representations of the symplectic Cayley-Klein groups
73(10)
3.5.1 The symplectic group Spn
74(1)
3.5.2 The symplectic Cayley-Klein groups Spn(j)
75(3)
3.5.3 Representations of Sp1 group
78(1)
3.5.4 Representations of Sp2(j2) groups
79(4)
3.6 Concluding remarks
83(2)
4 The Gel'fand--Tsetlin representations of Cayley-Klein algebras
85(42)
4.1 Representations of unitary algebras u(2; j1) and su(2;j1)
85(8)
4.1.1 Finite-dimensional irreducible representations of algebras u(2) and su(2)
85(2)
4.1.2 Transition to the representations of algebras u(2; j1) and su(2; j1)
87(2)
4.1.3 Contractions of irreducible representations
89(4)
4.2 Representations of unitary algebras u(3; j1, j2)
93(11)
4.2.1 Description of representations
93(4)
4.2.2 Contraction over the first parameter
97(2)
4.2.3 Contraction over the second parameter
99(3)
4.2.4 Two-dimensional contraction
102(2)
4.3 Representations of unitary algebras u(n; j)
104(6)
4.3.1 Operators of representation
104(3)
4.3.2 Spectrum of Casimir operators
107(1)
4.3.3 Possible variants of contractions of irreducible representations
108(2)
4.4 Representations of orthogonal algebras
110(17)
4.4.1 Algebra so(3; j)
110(2)
4.4.2 Algebra so(4; j)
112(4)
4.4.3 Contractions of representations of algebra so(4; j)
116(4)
4.4.4 so(n; j)
120(7)
5 High-temperature limit of the Standard Model
127(24)
5.1 Introduction
127(2)
5.2 Electroweak Model
129(5)
5.3 High-temperature Lagrangian of EWM
134(6)
5.4 Lagrangian of Quantum Chromodynamics
140(2)
5.5 QCD with contracted gauge group
142(5)
5.6 Estimation of boundary values
147(1)
5.7 Concluding remarks
148(3)
Bibliography 151(6)
Index 157
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