Groups, Systems and Many-Body Physics

I Fundamentals on Semigroups, Groups and Representations.- 1 Groups and group action.- 1.1 Preliminaries.- 1.2 Algebraic composition, semigroups and groups.- 1.3 Transformation groups.- 2 Examples of groups.- 2.1 The symmetric group S(n).- 2.2 Matrix semi-groups and inner products.- 2.3 Inhomogeneous matrix groups.- 3 Subgroup structures of groups and semigroups.- 3.1 Preliminaries: complexes and lattices.- 3.2 Subgroup diagrams.- 3.3 Morphisms and subgroups.- 3.4 Direct and semidirect products.- 3.5 Subnormal chains.- 3.6 Subgroups of semigroups.- 4 Groups and topology.- 4.1 Topological groups.- 4.2 Lie groups.- 4.3 Invariant measures.- 5 Representations of groups.- 5.1 General properties of representations.- 5.2 Irreducible representations of the symmetric group.- 5.3 Finite irreducible representations of the general linear group.- 5.4 Group algebras and representations.- 5.5 Induced and subduced representations.- 6 Induced representations of the Poincaré group.- 6.1 Poincaré transformations.- 6.2 Representations.- 6.3 Induced representations.- 6.4 UIR’s of the Poincaré group.- 6.5 The Dirac equation.- References.- II Fundamentals of Algebraic Quantum Theory.- 1 Algebra of observables.- 2 States.- 3 Symmetry transformations.- 4 Represented systems and symmetries.- 5 Dynamical symmetries and equilibrium states.- Appendix: Jordan homomorphisms.- References.- III Pauli Principle and Indirect Exchange Phenomena in Molecules and Solids.- 1 Permutation symmetry and chemical bonding in molecules and solids.- 1.1 Permutation symmetry and exchange interactions.- 1.2 Exchange interactions and chemical bonding.- 1.3 The Heisenberg (effective spin-) Hamiltonian approach to exchange interactions.- References.- 2 Group-theoretical aspects pertaining to the quantum-mechanical N-particle system.- 2.1 Introduction.- 2.1.1 Hamilton operator and Hilbert space.- 2.1.2 Symmetry and Wigner’s theorem. The irreducibility postulate, the identity postulate and the Pauli postulate.- 2.1.3 The spin free Pauli principle.- Appendix A The Schur-Weyl theorem.- Electrbn spin and permutation symmetry.- 2.2 Symmetry adaptation.- 2.2.1 Matric bases.- 2.2.2 Some fundamental expressions pertaining to matric bases.- 2.2.3 Some special choices of matric bases, G = SN.- 2.2.4 Symmetry adaptation of Vn ?N to SN.- 2.2.5 Antisymmetrization.- Appendix B Sequence adaption. Double coset decomposition of unitary matric bases.- Appendix C Tableau operators.- Invariance groups for tableaux operators.- 2.3 Matrix elements and their evaluation.- 2.3.1 Introduction.- 2.3.2 Matrix elements over Young-Yamanouchi N-electron spin (S, M) eigenfunctions.- 2.3.3 Matrix elements of a spin-free observable over sequence-symmetryadapted bases.- 2.3.4 Matrix elements of the spin-free N-electron Hamilton operator over Young unil-type Y?-projected N-electron bases. Pauling numbers.- 2.3.5 Summary and Discussion.- Appendix D The triple double coset symbol, double cosets, DC.- References.- Appendix E The canonical double coset symbol.- References.- Appendix F General references.- Appendix G Special references.- 3 The effective electron model: Applications.- 3.1 Introduction.- 3.2 Effective-electron model for weak chemical bonding.- 3.3 Applications of the model.- 3.3.1 Indirect exchange interactions in magnetic solids.- 3.3.2 Stability of rare-gas halides: A case of selective valence.- 3.3.3 Rotational barriers in simple molecules.- 3.3.4 Magnetic structures of the manganese pyrites.- References.- IV Groups and Semigroups for Composite Nucleon Systems.- 1 Introduction.- 2 Exchange and double cosets of the symmetric group.- 3 Orbital symmetry and the representation of the symmetric and general linear groups.- 4 Weyl operators, linear canonical transformations and Bargmann Hilbert space.- 5 Canonical transformations for interacting n-body systems.- 6 Interaction of composite particles.- 7 Configurations of simple composite particles.- 8 Composite particles with a closed-shell configuration.- 9 Conclusion.- References.- V An Algebraic Approach for Spontaneous Symmetry Breaking in Quantum Statistical Mechanics.- 1 Introduction.- 2 General theory.- 2.1 Finite statistical mechanics.- 2.2 Thermodynamical limit.- 2.3 KMS and thermodynamical stability.- 2.4 Properties of KMS states.- 2.5 What is a pure thermodynamical phase?.- 2.6 Cluster properties and ergodic theory.- 2.7 Uniform clustering.- 2.8 Symmetry-breaking and decomposition theory.- 3 Exactly solvable models.- 3.1 The BCS model.- 3.2 The Weiss-Ising (anti-) ferromagnet.- 3.3 Ising and Heisenberg models.- 3.4 Conclusion.- 4 Existence of crystals.- 4.1 A general theory.- 4.2 Physical interpretation.- 4.3 Related theories.- References.- VI Dynamical Groups for the Motion of Relativistic Composite Systems.- 1 Introduction.- 2 The general framework.- 3 Composite systems and reducible representations.- 4 The method of induced representations from dynamical groups.- 5 Electron-positron complex.- 6 Composite systems. Inductive approach.- 7 Dynamical algebras, their contraction and generalizations.- 8 Passage to relativistic wave equations.- 9 Principles on the choice of infinite component wave equations.- 10 The Majorana “particle”.- 11 An inverse problem.- 12 The class of composite systems based on conformal group.- 13 Electromagnetric interactions of composite systems.- 14 Other interactions of composite systems.- 15 A characteristic property of relativistic composite systems: Space-like states.- Appendix A: List of unitary irreducible representations of
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$$\overline {SO\left( {2,1} \right)} \sim \overline {SU\left( {1,1} \right)} $$.- Appendix B: List of most degenerate unitary irreducible representations of SO (4, 2).- References.- VII New Representation Spaces of the Poincaré Group and Functional Quantum Theory.- 1 Quantum theoretical fundamentals.- 2 Poincaré group representations in field theory.- 3 Functional map.- 4 Functional calculation methods.- 5 Noncanonical quantization and unitarization.- References.- VIII The Algebraic Theory of Automata.- 1 Systems, automata and switching networks.- 1.1 Input-output systems.- 1.2 Automata.- 1.3 Sequential switching networks.- 2 Transformation monoids and cascades.- 2.1 Simulation of automata and covering of transformation monoids.- 2.2 Cascades and wreath products.- 3 Structures of finite automata.- 3.1 Simulation of group automata.- 3.2 Examples.- 3.3 The Krohn-Rhodes-Eilenberg decomposition of finite automata.- 3.4 Realizations of group automata.- References.- IX On the (Internal) Symmetry Groups of Linear Dynamical Systems.- 1 Introduction and statement of the main definitions and results.- 2 Complete Teachability and complete observability.- 3 Nice selections and the local structure of Lm,n,pcr/GLn(IR).- 4 Realization theory.- 5 Feedback splits the external description degeneracy.- 6 Description of Lm,n,pco,cr(IR)/GLn(IR). Invariants.- 7 On the (non-)existence of canonical forms.- 8 On the geometry of Mm,n,pco,cr(IR). Holes and (partial) compactifications.- References.