Invariant subspace

One speaks of Eqs. (9-144) and (9-145) as a representation of the operators a and o satisfying the commutation rules (9-128), (9-124), and (9-125). The states 1, - , ) = 0,1,2,- are the basis vectors spanning the Hilbert space in which the operators a and oj operate. The representation (9-144) and (9-145) is characterized by the fact that a no-particle state 0> exists which is annihilated by a, furthermore this representation is irreducible since in this representation a(a ) operating upon an n-particle state, results in an n — 1 ( + 1) particle state so that there are no invariant subspaces. Besides the above representation there exist other inequivalent irreducible representations of the commutation rules for which neither a no-particle state nor a number operator exists.8... 

The phase space is formally decomposed into one-dimensional invariant subspaces. This is only a formal decomposition because the coordinates Qj... 

This is exactly the autonomous linearized Hamiltonian (7), the dynamics of which was discussed in detail in Section II. One therefore finds the TS dividing surface and the full set of invariant manifolds described earlier one-dimensional stable and unstable manifolds corresponding to the dynamics of the variables A<2i and APt, respectively, and a central manifold of dimension 2N — 2 that itself decomposes into two-dimensional invariant subspaces spanned by APj and AQj. However, all these manifolds are now moving manifolds that are attached to the TS trajectory. Their actual location in phase space at any given time is obtained from their description in terms of relative coordinates by the time-dependent shift of origin, Eq. (42). 

The vector space Ln, which is invariant under G, may contain a proper subspace which is also invariant under G. In such a case, Lm is an invariant subspace of Ln under G, and the space Ln is said to be reducible under G. 

Since Lm is invariant under G, any operator A G transforms each vector >n Lm into another vector in Lm. Hence, the operation AM results in a matrix of the same form as T(A). It should be clear that the two sets of matrices I) 1) and D > give two new representations of dimensions m and n — m respectively for the group G. For there exists a set of basis vectors l, n] for rX2 The representation T is said to be reducible. It follows that the reducibility of a representation is linked to the existence of a proper invariant subspace in the full space. Only the subspace of the first m components is... 

In this section we show how to construct a new representation from an old one hy restricting the domain of the linear transformations. One cannot restrict the domain to any old subspace, only to invariant subspaces. 

Note that elements of an invariant subspace W are not necessarily fixed by the linear operators in the image of the representation. In other words, it is not necessary to have p(g)w = w for every group element g and every w e W. However, elements of W cannot be moved out of W by the representation ... 

In the proof of Proposition 6.3 we will use linear operators to identify invariant subspaces with the help of the following proposition. Recall the notion of an eigenspace of a linear operator (Exercise 2.26). 

We can use an invariant subspace W to construct a restriction of the representation by restricting the linear transformation p (g) to the subspace W for each group element g. Note that for each g e G the restriction /)(g) is a function from W to W. 

Proposition 5.3 Suppose G, V, p) is a unitary representation. Suppose VP is an invariant subspace. Then VP" - is also an invariant subspace. IfV is finite dimensional, then the characters satisfy the relation... 

Orthogonal projection (Definition 3.11) onto an invariant subspace is a ho-... 

Proposition 5.4 Suppose W is an invariant subspace for a unitary representation (G, V, p). Suppose that there is an orthogonal projection flyv V V onto a subspace VP. Then FI w is a homomorphism of representations. 

Invariant subspaces are the only physically natural subspaces. Recall from... 

Any physically natural, spherically symmetric set of states corresponds to an invariant subspace and a subrepresentation. For this reason the concepts in this section are fundamental to our analysis of the hydrogen atom. The various shells of the hydrogen atom correspond to subrepresentations of the natural representation of SO 13) on (R ). In particular, the subspaces and Z play a role in the analysis. 

Exercise 5.1 Suppose G, V, p) is a representation. Show that both the trivial subspace 0 and the entire subspace V are invariant subspaces for the representation. 

Exercise 5.2 Show that the intersection of any two invariant subspaces is an invariant subspace. 

Find a one-dimensional invariant subspace Wi ofC. Show that the representation S3, VPi, p) is trivial. 

Find a complementary two-dimensional invariant subspace Wj of C such that = VPi iy2-... 

In this section we will use the idea of invariant subspaces of a representation (see Definition 5.1) to define irreducible representations. Then we will prove Schur s lemma, which tells us that irreducible representations are indeed good building blocks. 

For some representations, the largest and smallest subspaces are the only invariant ones. Consider, for example, the natural representation of the group G = 50(3) on the three-dimensional vector space C . Suppose W is an invariant subspace with at least one nonzero element. We will show that W = C-. In other words, we will show that only itself (all) and the trivial subspace 0 (nothing) are invariant subspaces of this representation. It will suffice to show that the vector (1, 0, 0) lies in W, since W would then have to contain both... 

So if V = 0 then, by the argument above, we have W =. Onthe other hand, if V 7 0, then again by Exercise 4.11 we can choose a rotation M e SO(3) such that Mv = (ryOyOy for some nonzero real number r. Thus the invariant subspace VT contains the vector Mw = a + b, cY for some real numbers a, b and c. It follows that the subspace W also contains the vector... 

Note that because r is nonzero, so is 2a + 2ir. So in this case as well we have (1, 0, 0) e VE and hence, as argued above, W — C. This shows that the only nonzero invariant subspace of the representation is the whole space... 

Definition 6.1 A representation (G, V, p) is irreducible z/to only invariant subspaces are V itself and the trivial subspace 0. Representations that are not irreducible are called reducible. 

Definition 6.2 Suppose (G, V, p) is a representation and (G. VE, pyy) is a subrepresentation. Suppose that (G, VE, pw) is an irreducible representation. Then we call VE an irreducible subspace or an irreducible invariant subspace... 

Proof. Suppose every linear transformation 7 V —> V that commutes with p is a scalar multiple of the identity. Suppose also that W is an invariant subspace for (G, V, p. must show that IV = V. By Proposition 3.5, because V is finite dimensional there is an orthogonal projection fliy V V whose image is W. Since p is unitary, we can apply Proposition 5.4 to show that the linear transformation flyy is a homomorphism of representations. So, by... 

We saw in Section 4.5 that a quantum mechanical system with symmetry determines a unitary representation of the symmetry group. It is natural then to ask about the physical meaning of representation-theoretic concepts. In this section, we consider the meaning of invariant subspaces and irreducible representations. 

Consider a complex scalar product space V that models the states of a quantum system. Suppose G is the symmetry group and (G, V, p) is the natural representation. By the argument in Section 5.1, the only physically natural subspaces are invariant subspaces. Suppose there are invariant subspaces Gi, U2, W c V such that W = U U2. Now consider a state w of the quantum system such that w e W, but w Uy and w U2. Then there is a nonzero mi e Gi and a nonzero M2 e U2 such that w = ui + U2. This means that the state w is a superposition of states ui and U2. It follows that w is not an elementary state of the system — by the principle of superposition, anything we want to know about w we can deduce by studying mi and M2. 

We know from numerous experiments that every quantum system has elementary states. An elementary state of a quantum system should be observer-independent. In other words, any observer should be able (in theory) to recognize that state experimentally, and the observations should all agree. Second, an elementary state should be indivisible. That is. one should not be able to think of the elementary state as a superposition of two or more more elementary states. If we accept the model that every recognizable state corresponds to a vector subspace of the state space of the system, then we can conclude that elementary states correspond to irreducible representations. The independence of the choice of observer compels the subspace to be invariant under the representahon. The indivisible nature of the subspace requires the subspace to be irreducible. So elementary states correspond to irreducible representations. More specifically, if a vector w represents an elementary state, then w should lie in an irreducible invariant subspace W, that is, a subspace whose only invariant subspaces are itself and 0. In fact, every vector in W represents a state indistinguishable from w, as a consequence of Exercise 6.6. 

Proof. Let w denote a weight vector of weight n. Let W denote the smallest invariant subspace containing w. Since w 0 by the definition of a weight vector, we have W 7 0. Let W be a nontrivial irreducible invariant subspace of W and note that w = 7 0, because otherwise W- - would contain w... 

Recall from Proposition 5.4 that orthogonal projection onto an invariant subspace of a unitary representation is a homomorphism of representations. Hence for any we have... 

Let us show injectivity. Note that is an injective homomorphism of representations. Hence the subspace j Qf -pt jg invariant subspace and... 

We know that is an irreducible invariant subspace of P by Proposition 7.2. By Proposition 6.5 and Proposition 7.1 we know that is not isomorphic to any subrepresentation of the Cartesian sum... 

Proposition 7.6 Suppose that V is a nontrivial irreducible invariant subspace of the natural representation of S 0(f) on Lf(S f Then there is a nonnegative integer such that V = y. ... 

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