1. 6. One-Dimensional Continuous Groups 6.1 The Rotation Group SO(2) 6.2 The Generator of SO(2) 6.3 Irreducible Representations of SO(2) 6.4 Invariant Integration Measure, Orthonormality and Completeness Relations 6.5 Multi-Valued Representations 6.6 Continuous Translational Group in One Dimension 6.7 Conjugate Basis Vectors

2. Introduction Lie Group, rough definition: Infinite group that can be parametrized smoothly & analytically. Exact definition: A differentiable manifold that is also a group. Linear Lie groups = Classical Lie groups = Matrix groups E.g. O(n), SO(n), U(n), SU(n), E(n), SL(n), L, P, … Generators, Lie algebra Invariant measure Global structure / Topology

3. 6.1. The Rotation Group SO(2) 2-D Euclidean space Rotations about origin O by angle  :

4. by  Rotation is length preserving:  i.e., R(  ) is special orthogonal.  If O is orthogonal,

5. Theorem 6.1: There is a 1–1 correspondence between rotations in E n & SO(n) matrices. Proof: see Problem 6.1 Geometrically: and Theorem 6.2:2-D Rotational Group R 2 = SO(2) is an Abelian group under matrix multiplication with and inverse identity element Proof: Straightforward. SO(2) group manifold SO(2) is a Lie group of 1 (continuous) parameter 

6. 6.2. The Generator of SO(2) Lie group: elements connected to E can be acquired by a few generators. For SO(2), there is only 1 generator J defined by R(  ) is continuous function of    with J is a 2  2 matrix Theorem 6.3:Generator J of SO(2)

7. Comment: Structure of a Lie group ( the part that's connected to E ) is determined by a set of generators. These generators are determined by the local structure near E. Properties of the portions of the group not connected to E are determined by global topological properties.  Pauli matrix J is traceless, Hermitian, & idempotent ( J 2 = E )

8. 6.3. IRs of SO(2) Let U(  ) be the realization of R(  ) on V.  U(  ) unitary  J Hermitian SO(2) Abelian  All of its IRs are 1-D The basis |   of a minimal invariant subspace under SO(2) can be chosen as so that  IR U m : m = 0: Identity representation

9. m = 1: SO(2) mapped clockwise onto unit circle in C  plane m =  1: … counterclockwise … m =  n:SO(2) mapped n times around unit circle in C  plane Theorem 6.4:IRs of SO(2) Single-valued IRs of SO(2) are given by Only m =  1 are faithful Representationis reducible has eigenvalues  1 with eigenvectors  Problem 6.2

10. 6.4.Invariant Integration Measure, Orthonormality & Completeness Relations Finite group  g  Continuous group  d  g Issue 1: Different parametrizations Remedy: Introduce weight  : so that  Changing parametrization to  =  (  ), we have, where  = (  1, …  n ) & f is any complex-valued function of g. Let G = { g(  ) } & define

11. Issue 2: Rearrangement Theorem Let   Since R.T. is satisfied by setting M = G if d  g is (left) invariant, i.e., ( Notation changed ! )

12. From one can determine the (vector) function  :  where  e (0) is arbitrary Theorem 6.5: SO(2) Proof:  Setting  e (0) = 1 completes proof.

13. Theorem 6.6:Orthonormality & Completeness Relations for SO(2) Orthonormality Completeness Proof: These are just the Fourier theorem since Comments: These relations are generalizations of the finite group results with  g   d  g Cf. results for T d ( roles of continuous & discrete labels reversed )

14. 6.5. Multi-Valued Representations Consider representation 2-valued representation m-valued representations : ( if n,m has no common factor ) Comments: Multi-connected manifold  multi-valued IRs: For SO(2): group manifold = circle  Multi-connected because paths of different winding numbers cannot be continuously deformed into each other. Only single & double valued reps have physical correspondence in 3-D systems ( anyons can exist in 2-D systems ).

15. 6.6. Continuous Translational Group in 1-D R(  ) ~ translation on unit circle by arc length   Similarity between reps of R(2) & T d Let the translation by distance x be denoted by T(x) Given a state | x 0  localized at x 0, is localized at x 0 +x     is a 1-parameter Abelian Lie group = Continuous Translational Group in 1-D

16. Generator P:   For a unitary representation T(x)  U p (x), P is Hermitian with real eigenvalue p. Basis of U p (x) is the eigenvector | p  of P: Comments: 1. IRs of SO(2), T d & T 1 are all exponentials: e –i m , e –i  k n b & e –i p x, resp. Cause: same group multiplication rules. 2. Group parameters are continuous & bounded for SO(2) = { R(  ) } discrete & unbounded for T d = { T(n) } continuous & unbounded for T 1 = { T(x) }

17. Invariant measure for T 1 :  Orthonormality Completeness C = (2  ) –1 is determined by comparison with the Fourier theorem. SO(2)TdTd T1T1 Orthonormality mnmn  (k–k)  (p–p) Completeness  (  –  )  nn  (x–x)

18. 6.7. Conjugate Basis Vectors Reminder:2 kind of basis vectors for T d. | x  localized state | E k  extended normal mode For SO(2): |   = localized state at ( r=const,  ) | m  = eigenstate of J & R(  ) Settinggives  m  transfer matrix elements  m |   = representation function e –i m 

19. 2 ways to expand an arbitrary state |   :

20.    in the x-representation J is Hermitian:   J = angular momentum  component  plane of rotation

21. For T 1 : | x  = localized state at x | p  = eigenstate of P & T(x)  p | 0  set to 1 T is unitary

22. 2 ways to expand an arbitrary state |   : P + = P : on V = span{ | x  }  P = linear momentum