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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

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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

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

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

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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

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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)

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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 )

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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

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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

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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

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

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

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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 )

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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 ).

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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

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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) }

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Invariant measure for T 1 : Orthonormality Completeness C = (2 ) –1 is determined by comparison with the Fourier theorem. SO(2)TdTd T1T1 Orthonormality mnmn (k–k) (p–p) Completeness ( – ) nn (x–x)

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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

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2 ways to expand an arbitrary state | :

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in the x-representation J is Hermitian: J = angular momentum component plane of rotation

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

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

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