Basic Representation Theory

This chapter discusses representation theory, which formalizes the notion of an object that transforms in a certain way under a given transformation. We begin with a heuristic introduction that shows how representations naturally arise in quantum mechanics, and how the panoply of such representations calls for organizing principles, which we develop in the rest of the chapter. We then define a representation of a group as a vector space on which that group acts, and we give many examples, using the vector spaces we met in Chap. 2 and the groups we met in Chap. 4. We then discuss how to take tensor products of representations, and we see how this reproduces the additivity of quantum numbers in Quantum Mechanics. We then define irreducible representations, which are in a sense the “smallest” ones we can work with, and we compute these representations for SU(2). These just end up being the familiar spin j representations, where j is a half-integer. We then use these results to compute the irreducible representations of the Lorentz group as well.