By Peter Woit
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Additional resources for Quantum Theory, Groups and Representations: An Introduction
2 Exponentials of Pauli matrices: unitary transformations of the two-state system We saw in chapter 2 that in the U (1) case, knowing the observable operator Q on H determined the representation of U (1), with the representation matrices found by exponentiating iθQ. Here we will find the representation corresponding to the two-state system observables by exponentiating the observables in a similar way. Taking the identity matrix first, multiplication by iθ and exponentiation gives the diagonal unitary matrix eiθ1 = eiθ 0 0 eiθ This is exactly the case studied in chapter 2, for a U (1) group acting on H = C2 , with 1 0 Q= 0 1 This matrix commutes with any other 2 by 2 matrix, so we can treat its action on H independently of the action of the σj .
Note that we have so far restricted attention to finite dimensional representations. 1 we will consider an important infinite dimensional case, a representation on functions on the circle which is essentially the theory of Fourier series. 3. 6 For further reading I’ve had trouble finding another source that covers the material here. Most quantum mechanics books consider it somehow too trivial to mention, starting their discussion of group actions and symmetries with more complicated examples. 23 24 Chapter 3 Two-state Systems and SU (2) The simplest truly non-trivial quantum systems have state spaces that are inherently two-complex dimensional.
We will see in this chapter how the general picture described in chapter 1 works out in this simple case. State spaces will be unitary representations of the group U (1), and we will see that any such representation decomposes into a sum of one dimensional representations. These one dimensional representations will be characterized by an integer q, and such integers are the eigenvalues of a self-adjoint operator we will call Q, which is an observable of the quantum theory. One motivation for the notation Q is that this is the conventional physics notation for electric charge, and this is one of the places where a U (1) group occurs in physics.