Representation Theory and Quantum Conservation Laws

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 ∈ G. A morphism of G-representations ϕ : V → W is a C-linear map satisfying ϕ(g · v) = g · ϕ(v) for all v ∈ 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 ∈ W , we have g · w ∈ 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 ∈ 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 ∈ 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 ∈ G and v, w ∈ 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 ⊥ := {v ∈ V | hv, wi = 0} 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 {vi} 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. g∈G 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. h∈G hg∈G

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 homomorphism 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 ϕ : V → W is a C-linear map which intertwines the g-actions: x · ϕ(v) = ϕ(x · v)

for all x ∈ g and v ∈ 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 ∈ 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. In other words, if at some time the system is in a state such that the observable A has a deﬁnite value, then the system will remain in that state for all time.

Proposition 2.2.1. An observable A : H → H is conserved if and only if it commutes with the Hamiltonian operator H, i.e. [H,A] = 0. Proof. According the Schr¨odingerequation dψ i = Hψ, ~ dt being a steady state is equivalent to being an eigenstate of H. On the other hand, by basic linear algebra the diagonalizable linear operators H and A commute if and only if they are simultaneously diagonalizable, i.e. any eigenstate of one is an eigenstate of the other.

4 Now we begin to discuss the role of symmetry in quantum mechanics. Suppose that H is the state space of a quantum system. Then we interpret a structure of unitary G-representation on H as symmetries of the physical system, provided that the action of G commutes with the Hamiltonian operator H, i.e. H(g · v) = g · H(v) for all g ∈ G and v ∈ V .

Proposition 2.2.2. If H is a unitary representation of G, then g acts on H by skew-Hermitian operators.

Proof. Choosing an orthonormal basis Cn =∼ H , the unitarity condition says that the image of the action 0 homomorphism ρ : G → GLn(C) is contained in Un. Thus the image of ρ : g → gln(C) is contained in the Lie algebra un of Un, which we have seen consists of skew-Hermitian operators. More concisely: diﬀerentiating the equation

hg · v, g · wi = hv, wi for g ∈ G and v, w ∈ V yields the equation

hx · v, wi + hv, x · wi = 0

for x ∈ g.

In particular, any element x ∈ g gives rise to a self-adjoint operator iρ0(x) on H . The following is the quantum-mechanical analogue of Nother’s theorem. Proposition 2.2.3. If the action of G commutes with the Hamiltonian, then for any x ∈ g, the observable iρ0(x) is a conserved quantity. Proof. It is not hard to show that if the action of G commutes with H, so does the induced action of g. The claim follows immediately from Proposition 2.2.1.

2 1 For example, let H = C be the state space of a ﬁxed spin 2 particle in a magnetic ﬁeld B = (0, 0,B3). 1 Then H = B3σ3 up to a positive scalar, so the observable A = 2 σ3 corresponding to measurement of spin along the z-axis is conserved, as we have seen. On the other hand, if A ∈ i · su2 is a self-adjoint operator on 2 C which is not proportional to σ3, then [A, H] 6= 0 and the observable quantity corresponding to A is not conserved.