Continuous Groups for Physicists

Description

Continuous Groups for Physicists is written for graduate students as well as researchers working in the field of theoretical physics. The text has been designed uniquely and it balances coverage of advanced and non-standard topics with an equal focus on the basic concepts for a thorough understanding. The book describes the general theory of Lie groups and Lie algebras, the passage between them, and their unitary/ Hermitian representations in the quantum mechanical setting. The four infinite classical families of compact simple Lie groups and their representations are covered in detail. Readers will benefit from the discussions on topics like spinor representations of real orthogonal groups, the Schwinger representation of a group, induced representations, systems of coherent states, real symplectic groups important in quantum mechanics, Wigner's theorem on symmetry operations in quantum mechanics, ray representations of Lie groups, and groups associated with non-relativistic and relativistic space-time. Contents Preface Abbreviations 1. Basic Group Theory and Representation Theory 1.1 Definition of a Group 1.2 Some Examples 1.3 Operations within a Group 1.4 Operations with and Relations between Groups 1.5 Realisations and Representations of Groups 1.6 Group Representations 1.7 Equivalent Representations 1.8 Unitary/Orthogonal Cases – UR’s 1.9 Matrices of a Representation 1.10 Some Operations with Group Representations 1.11 Character of a Representation 1.12 Invariant Subspaces, Reducibility, Irreducibility – UIR’s 1.13 Schur’s Lemma: Proof and Applications 1.14 Group Algebra 1.15 Representations of G and Its Group Algebra 𝔽[G] Problems Bibliography 2. The Symmetric Group 2.1 Cycle Structure Notation 2.2 Signature of a Permutation: Alternating Subgroup 2.3 Conjugacy Classes 2.4 Young Frames and Young Tableaux 2.5 Young Subgroups of Sₙ 2.6 Young Symmetrisers 2.6.1 Primitive idempotence of yt λ 2.6.2 Orthogonality properties of Young symmetrisers 2.7 Irreducible Representations of Sₙ 2.8 Some Useful Explicit Constructions of Representations of Sₙ Bibliography 3. Rotations in 2 and 3 Dimensions, SU(2) 3.1 The Group SO(2) 3.2 The Group O(2) 3.3 The Group SO(3) 3.4 Inclusion of Parity – The Group O(3) 3.5 The Group SU(2) Bibliography 4. General Theory of Lie Groups and Lie Algebras 4.1 Local Coordinates, Group Composition, Inverses 4.2 Associativity as a System of (Nonlinear) PDE’s 4.3 One Parameter Subgroups, Canonical Coordinates of First Kind 4.4 Integrability Conditions, Passage to the Lie Algebra 4.5 Lie Algebras 4.6 Local Reconstruction of G from G 4.7 General Remarks on the G →G Relationship, Some Definitions Concerning Lie Algebras 4.8 Representations of Lie Algebras – A Brief Look 4.9 The Adjoint Representation 4.10 Summary Problems Bibliography 5. Compact Simple Lie Algebras – Classification and Irreducible Representations 5.1 From a Real Lie Algebra to Its Complexification 5.2 Properties of Roots and Root Space 5.3 The SO(2l) Family D_𝓵 5.4 The SO(2l +1) Family B_𝒍 5.5 The USp(2l) Family C_𝒍 5.6 The SU(l +1) Family A_𝒍 5.7 The Exceptional Groups 5.8 Representations of CSLA’s 5.9 Survey of UIR’s, Fundamental UIR’s, Elementary UIR’s 5.10 The General UIR {N𝓪}, Its Construction, Internal Structure, Reality 5.11 Orthogonality and Completeness of UIR Matrix Elements Problems Bibliography 6. Spinor Representations of the Orthogonal Groups 6.1 Spinor UIR’s for D_𝒍 = SO(2l) 6.2 Spinor UIR for B_𝒍 = SO(2l +1) 6.3 Conjugation Properties of Spinor UIR’s 6.4 Combined Results for D_𝒍 and B_𝒍 6.5 Some Properties of Antisymmetric Tensors Problems Bibliography 7. Properties of Some Reducible Group Representations, and Systems of Generalised Coherent States 7.1 The Schwinger Representation of a Group 7.2 Induced Representations on Coset Spaces, the Reciprocity Theorem 7.3 Generalised Coherent State Systems Problems Bibliography 8. Structure and Some Properties and Applications of the Groups 𝑆𝑝(2𝓃,ℝ) 8.1 The Group 𝑆𝑝(2,ℝ) 8.2 The Group 𝑆𝑝(2𝓃,ℝ) 8.3 Quantum Variance Matrices, 𝑆𝑝(2𝓃,ℝ) Invariant Uncertainty Principles 8.4 SO(2l) Spinor UIR’s and Metaplectic UR of 𝑆𝑝(2𝓃,ℝ) – A Comparison Problems Bibliography 9. Wigner’s Theorem, Ray Representations and Neutral Elements 9.1 Hilbert and Ray Space Descriptions of Pure Quantum States 9.2 Wigner Symmetry and Unitary–Antiunitary Theorem 9.3 Proofs of Wigner’s Theorem 9.4 Applications to Quantum Mechanics – Ray Representations and Neutral Elements 9.5 Neutral Elements in Classical Mechanics Problems Bibliography 10. Groups Related to Spacetime 10.1 SO(3) and SU(2) 10.2 The Euclidean Group E(3) 10.3 The Galilei Group 𝒢 10.4 Homogeneous Lorentz Group SO(3,1), and SL(2,ℂ) 10.5 The Poincaré Group P Problems Bibliography Index A-D E-L M-S T-Y