Representation theory and quantum mechanics tutorial Representation theory and quantum conservation laws Justin Campbell August 1, 2017 1 Generalities on representation theory 1.1 Let G ⊂ GLm(R) be a real algebraic group. Definition 1.1.1. A (finite-dimensional complex) representation of G is a finite-dimensional vector space V over C together with a smooth group homomorphism G ! GL(V ). Equivalently, a representation of G on V is a homomorphism G ! GL(V ) such that the action map G × V ! V is smooth. Yet another equivalent definition is an action of G on V such that G × V ! V is smooth and g : V ! V is C-linear for all g 2 G. A morphism of G-representations ' : V ! W is a C-linear map satisfying '(g · v) = g · '(v) for all v 2 V . Sometimes such a morphism is called an intertwining operator. The homomorphism × SO2(R) −! C = GL1(C) given by cos θ − sin θ 7! eiθ sin θ cos θ defines a one-dimensional complex representation of the circle group SO2(R). This homomorphism is injective with image the unitary group U1. 2 Another example of a representation is the tautological action of SU2 ⊂ GL2(C) on C , sometimes called thefundamental representation of SU2. The group SO3(R) also has a fundamental representation of sorts. Namely, the tautological action of 3 3 SO3(R) ⊂ GL3(R) on R , i.e. the action by rotations, extends to a linear action on C in the natural way. 1.2 Let V be a representation of G. A subspace W ⊂ V is called G-stable if for any w 2 W , we have g · w 2 W . In this case one says that W is a subrepresentation of V . Definition 1.2.1. A finite-dimensional representation V of a group G is called irreducible if the only subrepresentations of V are 0 and V itself. Obviously, any one-dimensional representation is irreducible. The tautological 2-dimensional representa- 2 tion of SU2 is irreducible: if it were not there would exist an SU2-stable one-dimensional subspace L ⊂ C . This is absurd, as it is not hard to see that SU2 acts transitively on the set of one-dimensional subspaces of C2, and in particular fixes none of them. Exercise 1.2.2. Show that the 3-dimensional representation of SO3(R) constructed above is irreducible. 1 2 On the other hand, the \fundamental representation" of SO2(R) ⊂ GL2(R) ⊂ GL2(C) on C is not irreducible. Namely, the one-dimensional subspaces spanned by (1; i) and (1; −i) are SO2(R)-stable, since cos θ − sin θ 1 1 = e−iθ sin θ cos θ i i and cos θ − sin θ 1 1 = eiθ : sin θ cos θ −i −i Exercise 1.2.3. Prove that a morphism between irreducible representations is either zero or an isomorphism. Hint: the kernel and image of a morphism of G-representations are G-stable. With a little additional work, we obtain the following. Lemma 1.2.4 (Schur). If V is a finite-dimensional irreducible representation of G, then the only G- endomorphisms of V are the scalars C. Proof. Suppose A : V ! V is a linear operator which commutes with the action of G. Since V is a finite- dimensional complex vector space, the operator A has an eigenvalue λ 2 C. This means that equation Av = λv has nonzero solutions, i.e. the operator A − λI has nonzero kernel. But then Exercise 1.2.3 implies that A − λI = 0, i.e. A is the scalar operator corresponding to λ. An immediate consequence of this is the following. Proposition 1.2.5. If G is abelian, then any finite-dimensional irreducible representation of G is one- dimensional. Proof. Since elements of G commute amongst each other, the image of the homomorphism G ! GL(V ) consists of G-morphisms. By Lemma 1.2.4, this implies that the image is contained in C×, i.e. G acts by scalars. Thus any linear subspace of V is G-stable, but since V is irreducible this is only possible if V is one-dimensional. A basic example of this is the following. ∼ iθ inθ Exercise 1.2.6. Prove that the irreducible representations of U1 = SO2(R) all have the form e 7! e for × some n 2 Z (as homomorphisms U1 ! C = GL1(C)). 2 For example, we saw in the previous section that the standard representation of SO2(R) on C decomposes as the sum of the one-dimensional representations corresponding to 1 and −1. 1.3 Now we introduce inner products. Definition 1.3.1. A unitary representation of G is a representation on a Hilbert space H such that the inner product is G-invariant, meaning hg · v; g · wi = hv; wi for all g 2 G and v; w 2 H . A slightly more flexible notion is the following. We say that a complex representation V of G is unitarizable if V admits a G-invariant Hermitian inner product. Choosing such an inner product and completing V if necessary, one obtains a unitary representation. To state the key consequence of unitarizability, we introduce the following terminology. A representation V of G is called completely decomposable if there exists a direct sum decomposition ∼ M V = Vi i 2 such that each Vi is an irreducible subrepresentation of V . For example, any irreducible representation is completely decomposable in a trivial fashion. The previously discussed standard representation of SO2(R) on C2 is a non-irreducible example. For a typical example of a representation which is not completely decomposable, let G = R under addition and consider the representation R ! GL2(C) given by 1 a a 7! : 0 1 Proposition 1.3.2. Let V be a unitarizable representation of G. Then V is completely decomposable. Proof. It suffices to show that any G-subrepresentation W ⊂ V admits a G-stable complement, meaning V = W ⊕ W 0 for some subrepresentation W 0 ⊂ V . Choose a G-invariant inner product h ; i : V × V −! C: The G-invariance of the inner product implies that W ? := fv 2 V j hv; wi = 0g is G-stable, and since V = W ⊕ W ? we are done. Finally, we have the following basic result on compact groups. Proposition 1.3.3. If G is compact, then any representation of G is unitarizable. Proof. Let V be a representation of G. Choose an inner product h ; i : V × V −! C arbitrarily. For example, one can choose a basis fvig of V , and then put X X X h aivi; bivii := aibi: i i i We define a new inner product by the formula Z hv; wiG := hg · v; g · wi: g2G The integral is well-defined because G is compact, and this inner product is G-invariant because Z Z hg · v; g · wiG = hhg · v; hg · wi = hhg · v; hg · wi = hv; wiG: h2G hg2G Corollary 1.3.3.1. If G is compact, then any representation of G is completely decomposable. 2 Lie algebra representations and quantum mechanics 2.1 Let g be a Lie algebra over R. Definition 2.1.1. A (complex) representation of g is a complex vector space V together with a homomor- phism g −! gl(V ) of Lie algebras over R. 3 Here the commutator bracket on gl(V ) := EndC(V ) makes it a Lie algebra over C, and we can then restrict scalars and view it as a Lie algebra over R. Alternatively, one can extend scalars and construct the vector space g ⊗R C = g ⊕ ig over C, which has a unique C-bilinear Lie bracket which restricts to the given R-bilinear bracket on g. The Lie algebra g ⊗R C over C is called the complexification of g. A representation of g on V is equivalently a homomorphism g ⊗R C −! gl(V ) of Lie algebras over C. A morphism of g-representations ' : V ! W is a C-linear map which intertwines the g-actions: x · '(v) = '(x · v) for all x 2 g and v 2 V . Other notions relating to representations of groups, such as irreducibility, complete decomposability, etc., are easily modified to the case of Lie algebras. Now suppose g is the Lie algebra attached to the real algebraic group G. Then a group representation ρ : G ! GL(V ) gives rise to a Lie algebra representation ρ0 : g ! gl(V ) by differention, as follows. Given x 2 g, one can choose a path γ :(−, ) ! G such that γ(0) = I and γ0(0) = x. Then we define 0 d ρ (x) := ρ(γ(t)) : dt t=0 2 For example, the tautological representation of SU2 on C differentiates to the tautological representation 2 of su2 on C . We claim that the corresponding homomorphism 0 ρ : su2 ⊗R C −! gl2(C) 0 0 is an isomorphism onto sl2(C). Since su2 ⊂ sl2(R), the image of ρ is contained in sl2(C). But ρ is clearly injective, and since su2 ⊗R C and sl2(C) are both 3-dimensional over C the claim follows. More explicitly, recall that iσ1, iσ2, iσ3 is an R-basis of su2, where σ1, σ2, σ3 are the Pauli matrices. The standard C-basis of sl2(C) is 0 1 0 0 1 0 e = ; f = ; h = : 0 0 1 0 0 −1 Then we have 1 1 e = 2 (σ1 − iσ2); f = 2 (σ1 + iσ2); h = σ3: 2.2 An observable quantity, corresponding to a self-adjoint operator A : H ! H , is called a conserved quantity if its eigenstates are steady states.