Linear unitary representations

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). 

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). 

Exercise 10.20 Show that the projective unitary representation px of SO (3) does not descend from any linear unitary representation of SO (3. ... 

Conversely, suppose that (SO(3), P( V), cr) is a finite-dimensional projective rmitary representation. We want to show that cr is the pushforward of the projectivization of a linear unitary representation p of SO (2). In other words, we must show that there is a Lie group representation p that makes the diagram in Figure B.2 commutative, and that this p is a Lie group representation. 

Proposition 6.6 Suppose V is a finite-dimensional complex vector space with a complex scalar product. Suppose G, V, p) is a unitary representation. Suppose that every linear operator 7 V V that commutes with p is a scalar multiple of the identity. Then G, V, p is irreducible. 

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. 

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. 

Figure B.2. Commutative diagram for proof that every projective unitary representation of 50(3) comes from a linear representation of 5(7(2).

We have used for the row vectors of the respective entities, while we denote by ( ) and O the orbitals and many-electron functions, and by O and T(0) the two corresponding linear transformations, respectively. Various types of many-electron space for which such transformations may be carried out have been described by Malmqvist [34], In general, O may be non-unitary, possibly with subsidiary conditions imposed for ensuring that the corresponding transformation of the V-electron space exists e.g. a block-diagonal form according to orbital subsets or irreducible representations). 

Proof. By Proposition 3.5, since V2 is finite dimensional we know that there is an orthogonal projection 112 with range V2. Because p is unitary, the linear transformation 112 is a homomorphism of representations by Proposition 5.4. Thus by Exercise 5.15 the restriction of 112 to Vi is a homomorphism of representations. By hypothesis, this homomorphism cannot be injective. Hence Schur s lemma (Proposition 6.2) implies that since Vi is irreducible, fl2[Vi] is the trivial subspace. In other words, Vi is perpendicidar to V2. ... 

Proof, (of Proposition 10.6) First we suppose that (S(/(2), V, p) is a linear irreducible unitary Lie group representation. By Proposition 6.14 we know that p is isomorphic to the representation R for some n. By Proposition 10.5 we know that R can be pushed forward to an irreducible projective representation of SO(3). Hence p can be pushed forward to an irreducible projective Lie group representation of SO(3). 

The consequence of these relations is that every proper 2n rotation on S + — in the present instance the Poincare sphere—corresponds to precisely two unitary spin rotations. As every rotation on the Poincare sphere corresponds to a polarization/rotation modulation, then every proper 2n polarization/rotation modulation corresponds to precisely two unitary spin rotations. The vector K in Fig. lb corresponds to two vectorial components one is the negative of the other. As every unitary spin transformation corresponds to a unique proper rotation of S +, then any static (unipolarized, e.g., linearly, circularly or ellipti-cally polarized, as opposed to polarization-modulated) representation on S + (Poincare sphere) corresponds to a trisphere representation (Fig. 3a). Therefore... 

The Hamiltonian in Eq. [26] is usually referred to as the diabatic representation, employing the diabatic basis set <1), Hamiltonian matrix is not diagonal. There is, of course, no unique diabatic basis as any pair obtained from (]), by a unitary transformation can define a new basis. A unitary transformation defines a linear combination of cj) and < >b which, for a two-state system, can be represented as a rotation of the (]), basis on the angle /... 

In Section 2.3 below we are going to show to what extend the anticommutation relations determine the properties of the Dirac matrices. Here we just note that these relations do not define the Dirac matrices uniquely. If (/ , 0, ) is a set of Hermitian matrices satisfying (5), then 0 = S0S and a). = SakS with some unitary matrix S is another set of Hermitian matrices obeying the same relations. Any specific set is said to define a representation of Dirac matrices. With respect to a given representation, the Dirac equation is a system of coupled linear partial differential equations. It is of first order in space and time derivatives. 

Spatial symmetry operations are linear transformations of a coordinate function space. When choosing the space in orthonormal form, symmetry operations will conserve orthonormality, and hence all transformations will be carried out by unitary matrices. This will be the case for all spatial representation matrices in this book. When all elements of a unitary matrix are real, it is called an orthogonal matrix. As unitary matrices, orthogonal matrices have the same properties except that complex conjugation leaves them unchanged. The determinant of an orthogonal matrix wiU thus be equal to 1. The rotation matrices in Chap. 1 are all orthogonal and have determinant -I-1. 

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