Particles in the Early Universe: High-energy Limit of the Standard Model from the Contraction of Its Gauge Group

Contents
Preface
1. The Cayley–Klein groups and algebras
	1.1 Dual numbers and the Pimenov algebra
		1.1.1 Dual numbers
		1.1.2 The Pimenov algebra
	1.2 The Cayley–Klein orthogonal groups and algebras
		1.2.1 Three fundamental geometries on a line
		1.2.2 Nine Cayley–Klein groups
		1.2.3 Extension to higher dimensions
	1.3 The Cayley–Klein unitary groups and algebras
		1.3.1 Definitions, generators, commutators
		1.3.2 The unitary group SU(2; j1)
		1.3.3 Representations of the group SU(2; j1)
		1.3.4 The unitary group SU(3; j)
		1.3.5 Invariant operators
	1.4 Classification of transitions between the Cayley–Klein spaces and groups
2. Space–time models
	2.1 Kinematics groups
	2.2 Carroll kinematics
	2.3 Non-relativistic kinematics
3. The Jordan–Schwinger representations of Cayley–Klein groups
	3.1 The second quantization method and matrix elements
	3.2 The rotation groups in Cayley–Klein spaces
	3.3 The Jordan–Schwinger representations of the orthogonal Cayley–Klein groups
		3.3.1 Representations of SO2(j1) groups
		3.3.2 Representations of SO3(j) groups
		3.3.3 Representations of SO4(j) groups
	3.4 The Jordan–Schwinger representations of the special unitary Cayley–Klein groups
		3.4.1 Representations of SU2(j1) groups
		3.4.2 Representations of SU3(j1, j2) groups
	3.5 The Jordan–Schwinger representations of the symplectic Cayley–Klein groups
		3.5.1 The symplectic group Spn
		3.5.2 The symplectic Cayley–Klein groups Spn(j)
		3.5.3 Representations of Sp1 group
		3.5.4 Representations of Sp2(j2) groups
	3.6 Concluding remarks
4. The Gel’fand–Tsetlin representations of Cayley–Klein algebras
	4.1 Representations of unitary algebras u(2; j1) and su(2; j1)
		4.1.1 Finite-dimensional irreducible representationsof algebras u(2) and su(2)
		4.1.2 Transition to the representations of algebras u(2; j1) and su(2; j1)
		4.1.3 Contractions of irreducible representations
			4.1.3.1 Analytical continuation of irreducible representations
	4.2 Representations of unitary algebras u(3; j1, j2)
		4.2.1 Description of representations
		4.2.2 Contraction over the first parameter
		4.2.3 Contraction over the second parameter
		4.2.4 Two-dimensional contraction
	4.3 Representations of unitary algebras u(n; j)
		4.3.1 Operators of representation
		4.3.2 Spectrum of Casimir operators
		4.3.3 Possible variants of contractions of irreducible representations
	4.4 Representations of orthogonal algebras
		4.4.1 Algebra so(3; j)
		4.4.2 Algebra so(4; j)
		4.4.3 Contractions of representations of algebra so(4; j)
		4.4.4 so(n; j)
5. High-temperature limit of the Standard Model
	5.1 Introduction
	5.2 Electroweak Model
	5.3 High-temperature Lagrangian of EWM
	5.4 Lagrangian of Quantum Chromodynamics
	5.5 QCD with contracted gauge group
	5.6 Estimation of boundary values
	5.7 Concluding remarks
Bibliography
Index