Physical Symmetries

Figure 2-8 Physical Symmetry of a Unidirectionally Reinforced Lamina...

The notion of a group is a natural mathematical abstraction of physical symmetry. Because quantum mechanical state spaces are linear, symmetries in quantum mechanics have the additional structure of group representations. Formally, a group is a set with a binary operation that satisfies certain criteria, and a representation is a natural function from a group to a set of linear operators. 

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. 

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]. 

A. symmetry of a quantum system, also known as a physical symmetry, is a function from P(V) to P(V) that preserves the absolute bracket K-l-. ... 

It follows easily from the dehnition that the composition of two physical symmetries is a physical symmetry and that every physical symmetry is injective (see Exercise 10,24). 

Complex conjugation is another physical symmetry of the qubit." We will find the following nomenclature useful. 

So the function r preserves the absolute value of the bracket and hence complex conjugation is a physical symmetry of the state space. This transformation corresponds to reflecting the sphere in Figure 10.6 in the xz-plane (Exercise 10.16). Complex conjugation does not descend from a (complex) linear transformation however, we have... 

Proposition 10.7 Suppose S is a physical symmetry of the qubit such that... 

The next proposition classifies the physical symmetries of the qubit. As promised in the introduction to this section, these symmetries consist of the projective unitary symmetries (rotations) and compositions of projective unitary symmetries with complex conjugation (reflections). It follows easily from Proposition 10.1 that for any unitary operator T eU (C ) both [u] [Tv] and [u] i- — fSu)] are well-defined physical symmetries of... 

P(C2). In fact, every physical symmetry of the qubit is of this form. 

Proof. First we show that any physical symmetry fixing the north pole [0 1] is of the desired form. Then we extend the result to arhitrary physical symmetries. 

By Proposition 10.7, the physical symmetry function S must be the identity. Hence 5 = [T]. 

The last task in the proof of the first statement is to generalize to an arbitrary physical symmetry S. Set [co t i 1 = [Pg.328]

Finally, we must show that k is unique and T is unique up to multiplication by a scalar of modulus one. Suppose Ti, /ci and 72, a 2 both satisfy the requirements of the proposition. We must show that /ci = k 2 and there is a real number d such that Ti = e" T2. know that for any element v e we have [T lKiCv)] = [T2/e2(r )]- Applying the physical symmetry to both sides we find that... 

To extend this result to projective space of arbitrary finite dimension we will need the technical proposition below. Since addition does not descend to projective space, it makes no sense to talk of linear maps from one projective space to another. Yet something of linearity survives in projective space subspaces, as we saw in Proposition 10.1. The next step toward our classification is to show that physical symmetries preserve finite-dimensional linear subspaces and their dimensions. 

With Proposition 10.8 and the technical result Proposition 10.9 in hand, we are ready to classify the physical symmetries of complex projective spaces of arbitrary finite dimension. 

Proposition 10.10 Suppose n is a natural number and ( , ] is the standard complex scalar product on C . Suppose S P(C") —> P(C") is a physical symmetry. Then there is a unitary operator T C" C" a function k, equal to either the identity or the conjugation fi me lion, such that... 

Now we must prove the general case. Let uq denote a length-one vector in <5([en]). Because Uq is of unit length, it is possible to construct a unitary transformation 7b on such that 7buo = c . Then [7b] o S is a physical symmetry fixing [e ], Hence [7b] o 5 also takes points in [ ] to (possibly different) points in [E ]. By induction, there is a unitary operator -> En and a function /c , equal to the identity or to conjugation, such that for every [v] e we have... 

In this section we have classified the physical symmetries of a finite-dimensional quantum system. Half of these symmetries are projective unitary transformations the other half are projective unitary transformations preceded hy complex conjugation. This result means that by studying projective unitary transformations and complex conjugation, one can understand all physical symmetries. 

Find an explicit formula for the physical symmetry of P(C2) that corre-... 

Exercise 10.22 Find a group isomorphism between S O (3) and a subgroup of the physical symmetries of the qubit. Use Proposition 10.1 to find a nontrivial group homomorphism from SU (2) into the group of physical symmetries of the qubit. Finally, express the group homomorphism SU(2) —> 50(3) from Section 4.3 in terms of these functions. 

Exercise 10.25 Show that the group of physical symmetries of the qubit is isomorphic to the group 0(3). 

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