. Lecture notes: Group theory and its applications in physics Boris Gutkin Faculty of Physics, University Duisburg-Essen School of Physics, Georgia Tech |||||||||||||||||||||- version 2.3 printed April 29, 2016 comments to: [email protected] Contents Part 1. Finite and discrete groups 1 Lecture 1. Symmetries in Physics3 1. Classical physics3 2. Continues symmetries & Noether theorem4 3. Quantum mechanics6 4. Continues symmetries6 5. Discrete symmetries6 Lecture 2. Basics of Group Theory9 1. Groups definitions9 2. Subgroups 10 3. Conjugate classes. Normal subgroups 11 4. Point groups 12 5. Non-special transformations 13 Lecture 3. Representation Theory I 15 1. Basic notions 15 2. Schur's lemmas 16 3. Orthogonality theorem 17 Lecture 4. Representation Theory II 19 1. Characters 19 2. Classification of irreducible representations 19 3. How to find characters of irreducible representations? 20 4. Dual orthogonality relationship 20 5. Three types of representations 21 6. Representations of cross products 22 7. Representations of subgroups 22 8. Appendix. Proof of the theorem (4.1) 23 Lecture 5. Applications I. Vibration modes 25 1. Vibration spectrum of molecules 25 2. 3-dimensional symmetries 30 Lecture 6. Applications II. Quantum Mechanics 31 1. Spectral decomposition 31 2. Perturbation theory 32 3. Selection rules 33 4. Appendix A: Density of states or why singlets are rare in the energy spectrum of systems with high symmetry 34 5. Appendix B: Spectral statistics or why compex representation matter 35 References 35 References 35 ii LECTURE 0. CONTENTS Lecture 7. Applications III. Energy Band Structure 37 1. Lattice symmetries 37 2. Band structure 38 3. Band structure of graphene 40 References 41 References 41 Part 2. Continuous groups 43 Lecture 8. Lie groups & Lie algebras 45 1. Basic definitions and properties 45 2. Representations 46 Lecture 9. SU(2), SO(3) and their representations 49 1. Connection between SU(2) and SO(3) 49 2. Representations of SO(3) and SU(2) 50 3. Applications 51 4. Spinors 53 5. Product representations of SO(3) 55 6. Clebsch-Gordan series 55 7. Clebsch-Gordan coefficients. Adding angular momenta 55 8. Wigner-Eckart theorem 56 9. Applications 58 Lecture 10. Representations of simple algebras, general construction. Application to SU(3) 61 1. Adjoint representation and Killing form 61 2. Cartan sub-algebra and roots 61 3. Main properties of root systems 62 4. Building up representations of g 62 5. Representations of su(3) 63 Lecture 11. Strong interactions: flavor SU(3) 67 1. Internal symmetries: isospin 67 2. Internal symmetries: strangeness 69 3. Quarks 70 References 73 References 73 Lecture 12. Many particle systems. Young tableaux 75 1. Irreducible representations of Sn 75 2. Irreducible representations of SU(N) 78 References 78 References 78 Lecture 13. Classification of semi-simple Lie algebras 79 1. Semisimple algebras 79 2. Roots 79 3. A B C D E F G classification 81 4. Representation 83 Lecture 14. Infinite dimensional symmetries. String theory. 85 2015-01-24 iii Lecture notes: Group theory Part 1 Finite and discrete groups LECTURE 1 Symmetries in Physics 1. Classical physics What does it mean that a dynamical system has symmetries? Consider standard dynamical equa- tions: d @L @L Z t2 (1.1) = ;S = L(q; q_)dt dt @q_ @q t1 for a time evolution of a number of classical particles. To find solution q(t) we have to solve a complicated (system) of differential equations, which in exact form only in few exceptional cases (of integrable systems) is possible. On the other hand, it often happens that given an arbitrary solution q(t) one can construct a family of solutions by application of certain transformation to it: q(t; ) := g · q(t); such that q(t) = q(t; 0): In such a case the transformations g, form a symmetry group G of the system. By the definition −1 g1 ·g2 2 G if g1 ; g2 2 G and there exists an inverse transformation g for each g 2 G. In math- ematical language the last two properties in combination with the associativity of transformations imply that G is a group. x g ’ γ γ t1t 2 t’1t’ 2 t Figure 1. Symmetry transformation. Example 1.1. Continuous symmetries: Translational symmetries q(t; ) = q(t)+. Rotation symmetries q(t; ) = R() · q(t), R()RT () = I. The existence of such symmetries is directly follows from the invariance of the action S under the transformation g: (1.2) S(q(t)) = S(g · q(t)): This is clear, since g transforms extremum of S into another extremum, see figure1. In its own turn the invariance of the action would follow from the invariance of the Lagrangian: (1.3) L(q(t); q_(t))dt = L (g · q(t); g · q_(t)) d(g · t): 3 4 Lecture 1. SYMMETRIES IN PHYSICS Here we can also add an additional dF which changes transformed action by a constant (and therefore does not affect the above arguments). This leads to the following (sufficient) symmetry condition: d(g · t) dF (1.4) L(q(t); q_(t)) = L(g · q(t); g · q_(t)) + : dt dt To summarize, the transformations g leaving Ldt invariant up to a closed form dF form a group of symmetries of the system. They transform one solution of the dynamical equation to another: q(t) ! q(t; ); q(t; 0) = q(t): Note that typically we do not know explicitly any solution of this family. Nevertheless, knowl- edge of symmetries of the system might be very useful: With any continues symmetry of a Lagrangian system we can associate a conserved quantity. 2. Continues symmetries & Noether theorem Let g be a continues symmetry, i.e., 2 R. Consider an infinitesimal transformation q ! q + η(q) q(t; ) = q(t) + η(q(t)) which leaves L invariant up to a full time derivative, see (1.4): @L @L (2.1) L(q(t) + η(t); q_(t) + η_(t)) = L(q(t); q_(t)) + η(t) + η_(t) : @q @q_ It follows immediately from eq. (1.1) that: @L dJ (2.2) J = η(t) + F (t); = 0: @q_ dt From this also follows that dJ (2.3) fH; J g = η = 0; η dt @f @g @f @g where ff; gg = @q @p − @p @q is the Poisson bracket. Currents Jη form Lie algebra with the Poisson brackets playing the role of \commutator operation". Symmetry transformations including time. Considering more general transformation: 0 0 t ! t + φ(q; t) = t ; qi ! qi + ηi(q; t) = qi leaving S invariant (one can also consider φ and η depending on time derivatives of q = (q1; : : : qn)) we can get a generalized version of Noether theorem: X @L @L dJ (2.4) J = η (q; t) + L − q_ φ(q; t) + F (t); = 0: @q_ i @q_ i dt i i i Proof : If the system is invariant under the above transformation dq (2.5) L q ; i dt + dF = i dt dq0 = L q0; i dt0 = L (q + η (q; t); q_ (1 − φ(q; t)) + η_ (q; t)) (1 + φ(q; t))dt: i dt0 i i i i symmPhys - 2016-01-04 2. CONTINUES SYMMETRIES & NOETHER THEOREM 5 After expansion of the right hand side we have: X _ _ (2.6) [@qi Lηi + @q_i L(_ηi − q_iφ)] + Lφ = F i Now using the equation of motion (1.1) the left part of this equation can be written down as full time derivative: 2 3 d X 6 @L @L 7 6 η + L − q_ φ7 : dt 6@q_ i @q_ i 7 i 4 i i 5 | {z } Hamiltonian P 2 P Applications L = i mq_ i =2 + i6=j V (jqi − qjj) A) Shift in space: (x) (x) X (x) qi ! qi + , Jx = pi (Momentum). i B) Rotations: qi ! Rx;y;zqi: 0 cos() sin() 01 00 −1 01 2 (2.7) Rz = @− sin() cos() 0A = I + @1 0 0A + O( ) 0 0 1 0 0 0 X (x) (y) (y) (x) (2.8) Jz = −pi qi + pi qi (Angular Momentum). i C) Shift in time: X @L t ! t + , J = L − q_ = H (Energy). t @q_ i i i D) Some less standard example for the potential V (q) ∼ 1=jqj2. The transformation: 2 t ! λ t; qi ! λqi is an obvious symmetry of the action (consider Ldt). In the infinitesimal form λ = 1 + this symmetry is given by: 2 t ! λ t(1 + 2); qi ! qi(1 + ); implying that X 2 J = (1=2)dt qi − 2Ht; J = const: This leads to the conclusion that X 2 2 qi = 2Et + t · const: Depending on the sign of energy the particle either flies away with a constant radial velocity or collapses to the center in a finite time! symmPhys - 2016-01-04 6 Lecture 1. SYMMETRIES IN PHYSICS 3. Quantum mechanics Now consider action of symmetry group in the framework of quantum mechanics. By the same principle, if Ψ(x; t) is solution of the Schrodinger equation: (3.1) i~@tΨ(x; t) = HbΨ(x; t) the function Ψ0(x; t) = g · Ψ(x; t) ≡ Ψ(g · x; t) is also solution of the same equation. This is automatically satisfied if (3.2) g · Hb = Hb · g: 4. Continues symmetries As in the classical case we can consider an infinitesimal group action: (4.1) g · Ψ(x) ≈ Ψ(x) + JbΨ(x): By (3.2) we then immediately obtain: dJb (4.2) [H;b Jb] = = 0: dt Example 1.2. Translation and rotation symmetries: (a) Translation symmetry Ψ(x + ) = Ψ(x) + ∂xΨ; −i~Jb = −i~@x =p; ^ dtp^ = 0: (b) Rotation symmetry Ψ(x + y; y − x) = Ψ(x) + (y@x − x@y)Ψ ; −i~Jb = L;b dtLb = 0 5.