Representations unitary

If we have a number of equivalent representations of a particular point group, it is useful to choose just one of them as a prototype for all the others. It makes sense that the one we choose for this role has matrices which are unitary, since unitary matrices are much easier to handle and manipulate than non-unitary matrices. The reader will recall (Appendix A.4-l(g)) that a unitary matrix is defined by A-1 = Af. Just as there are two ways of interpreting equivalent representations 

Equivalent representations of using the p-orbital function spacef 

If we choose our basis functions for a particular function space to be orthonormal (orthogonal and normalized) i.e. (/ /,) = J/, /, dr then, since the transformation operators are unitary ( 5-7), the representation created will consist of unitary matrices. This is proved in Appendix A.6-1. It should be stated that it is always possible to find an orthonormal basis and one way, the Schmidt orthogonalization process, is given in Appendix A.6-2. 

Alternatively, we can prove that there is always a similarity transformation which will transform simultaneously all of the matrices of a representation into unitary matrices. This is proved in Appendix A.6-3. 

From now on therefore it will be no restriction to consider, if we wish to, only unitary representations. 

---

To make these notions precise, the transformation properties of the wavefunction x under spatial and time translations as well as under spatial rotations and pure Lorentz transformations must be specified and it must be shown that the generators of these transformations form a unitary representation of the group of translations and proper Lorentz transformations. This can in fact be shown5 but will not be here. 

We shall only mention the fact, that a unitary representation of the inhomogeneous proper Lorentz group is exhibited in this Hilbert space through the following identification of the generators of the... 

Lorentz group, inhomogeneous proper, unitary representation in Hilbert space, 497... 

This w -algebra structure can be used to develop a representation theory of symmetry groups, taking H as a representation space for Lie algebras. As before let g be a Lie algebra specified by giOgj = C gu-A unitary representation of g in H is then given by... 

In this section we define representations and give examples. We also define homomorphisms and isomorphisms of representations, as well as unitary representations and isomorphisms. 

By Proposition 3.2, this is a unitary representation with respect to the natural... 

Isomorphisms of unitary representations ought to preserve the unitary structure. When they do, they are called unitary isomorphisms of representations. 

In addition, p is a unitary representation. Because complex inner prod-... 

The unitary representations R in this section turn out to be the building... 

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... 

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. 

The subspace VP is invariant under p by hypothesis since p is a unitary representation, it follows from Proposition 5.3 that W- - is also invariant under p. Thus we have p(g)n e VP and pCgjrtvifj-U VP- -, Hence... 

If both factors are unitary representations, then so is the tensor product. If both V and V have complex scalar products defined on them, then there is a natural complex scalar product on the tensor product V 0 V of vector spaces. Specifically, we define... 

Proposition 5.11 Suppose (G, V, p) is a finite-dimensional unitary representation with character /, Then the character of the dual representation G, V, p ) is X - (Recall that x denotes the complex conjugate of the C-valuedfunction xf Fitt thermore, (G, V, p is a unitary representation with respect to the natural complex scalar product on V. ... 

In Section 6.1 we define irreducible representations. Then we state, prove and illustrate Schur s lemma. Schur s lemma is the statement of the all-or-nothing personality of irreducible representations. In the Section 6.2 we discuss the physical importance of irreducible representations. In Section 6.3 we introduce invariant integration and apply it to show that characters of irreducible representations form an orthonormal set. In the optional Section 6.4 we use the technology we have developed to show that finite-dimensional unitary representations are no more than the sum of their irreducible parts. The remainder of the chapter is devoted to classifying the irreducible representations of 5(7(2) and 50(3). 

For unitary representations we have a converse to Proposition 6.3. Unitary irreducible representations are sometimes called unirreps for short. 

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. 

Next, fix a natural number n and suppose that the result is known for all natural numbers k < n. Because every li nite-dimensional representation contains at least one irreducible representation, we can choose one and call it Wo-Set Co = dim Home (Wo, V). Then by Proposition 6.10 we know that Wq° is isomorphic to a subrepresentation U of V. Since the representation V is unitary, we can consider the complementary unitary representation [/- -, whose dimension is strictly less than n. 

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... 

Exercise 6.13 Suppose (G, V, p) is a Lie group representation where G is a Lie group with a volume-one invariant integral and V is a complex scalar product space ( , ), Then there is a complex scalar product ( , p on V such that p is a unitary representation on V with respect to ( , p. (Hint define... 

In this section we have studied the shadow downstairs (in projective space) of the complex scalar product upstairs (in the linear space). We have found that although the scalar product itself does not descend, we can use it to define angles and orthogonality. Up to a phase factor, we can expand kets in orthogonal bases. We will use this projective unitary structure to define projective unitary representations and physical symmetries. 

We start by defining the projective unitary representations. Recall the unitary group ZT (V) of a complex scalar product space V from Definition 4.2. The following definition is an analog of Definition 4.11. 

Sometimes, to stress the distinction between unitary group representations as defined in Chapter 4 and projective unitary representations, we will call the former linear unitary representations. Any (linear) unitary representation descends to a projective unitary representation. More specifically, suppose G is a group, suppose V is complex scalar product space and suppose p G U (U) is a (linear) unitary representation. Then we can define a projective unitary representation p G P(V) by... 

So pi is a projective unitary representation of SO(3). In fact, pi is a bona fide projective Lie group representation, i.e,. it is a differentiable ftinction, as we will show in Proposition 10.5. However, pi does not descend from any linear unitary representation of St/(2) (Exercise 10.20). 

Definition 10.8 Suppose G is a group, V is a complex scalar product space and p. G PU (V) is a projective unitary representation. We say that p is irreducible if the only subspace W of V such that [VT] is invariant under p is V itself. 

It is natural to wonder whether we have missed any irreducible projective unitary representations of 50(3). Are there any others besides those that come from irreducible linear representations The answer is no. 

Proposition 10.6 The irreducible projective unitaty representations of the Lie group SO if) are in one-to-one correspondence with the irreducible (linear) unitary representations of the Lie group SU (2). 

---