Projective unitary representation

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. 

Definition 10.7 Suppose G is a group and V is a complex scalar product space. Then the triple G, V, p) is called a projective unitary representation if and only if p is a group homomorphism from G to PIT (V). 

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. 

The results of this section are another confirmation of the philosophy spelled out in Section 6.2. We expect that the irreducible representations of the symmetry group determined by equivalent observers should correspond to the elementary systems. In fact, the experimentally observed spin properties of elementary particles correspond to irreducible projective unitary representations of the Lie group SO(3). Once again, we see that representation theory makes a testable physical prediction. 

In Section 10.4 we studied projective unitary representations, important because they are symmetries of quantum systems. It is natural to wonder whether projective unitary symmetries are the only symmetries of quantum systems. In this section, we will show that complex conjugation, while not projective unitary, is a physical symmetry, i.e., it preserves all the physically relevant quantities. The good news is that complex conjugation is essentially the only physical symmetry we missed. More precisely, each physical symmetry is either projective unitary or it is the composition of a projective unitary symmetry with complex conjugation. This result (Proposition 10.10) is known as Wigner s theorem on quantum mechanical symmetries. The original proof can be found in the appendix to Chapter 20 in Wigner s book [Wi]. 

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

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 begin by reiterating the definition of a PR and listing some conventions regarding PFs. A projective unitary representation of a group G = g, of dimension g is a set of matrices that satisfy the relations... 

Weyl s proposal was a first step to derive the Heisenberg relations from the basic properties of projective unitary representations. He indicated how an observable H given in terms of the conjugate observable g and h could be characterized [18] by H = H(g,h), using the wave function expanded over the eigenvalues s and k ... 

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. 

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... 

The representation pi is called the spin-1/2 representation. It arises from the rotation of three-dimensional physical space and its effect on the qubit. In other words, experiments show that if two observers differ by a rotation g, then their observations of states of the qubit differ by a projective unitary transformation [[/] such that [Pg.320]

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). 

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. 

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... 

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). 

Equation (26) shows that (T(q, R) forms a unitary projective (or multiplier) representation of R j - P(q). Only for non-symmorphic groups with b different from zero (that is, when q lies on the surface of the BZ) are the projective factors exp [ib,.w,] in eq. (26) different from unity. 

The introduction of reciprocal space allows for both the characterization by unitary projection (planes are represented by dots) and completes projection (both information of orientation and interplanar distance are present) for the crystallographic planes of direct space as well as the possibility of their quantum representation by the characterization of Brillouin Zones of wave vectors associated to the crystal s eigen-states (Pettifor, 1995). 

The representations provided in the basis of degenerate eigenfunctions are usually irreducible and can be chosen to be unitary matrices (in fact usually orthogonal matrices if the functions are real functions). In practice, what one is usually faced with is a collection of functions which have arbitrary (but known) transformation properties and what one actually wants to do is to adapt these functions so that they actually transform like the true eigenfunctions of the problem. This can be done by means of the group theoretical projection operator. 

If the unitary matrix irreducible representation of the point group is known and denoted by (R), where labels the irreducible representation, then the projection operator is... 

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