Chapter 6
Groups and Representations
in Quantum Mechanics
The universe is an enormous direct product of representations of sym-
metry groups.
—Steven Weinberg
1
This chapter is devoted to applying the mathematical theory of groups
and representations which we have developed in the preceding chapters
to the quantum mechanical description of physical systems. The power
of applying group theory to quantum mechanics is that it provides a
framework for making exact statements about a physical system with
a knowledge only of the symmetry operations which leave its Hamil-
tonian invariant, the so-called “group of the Hamiltonian.” Moreover,
when we apply the machinery of groups to quantum mechanics, we find
that representations—and irreducible representations in particular—
arise quite naturally, as do related concepts such as the importance of
unitarity of representations and the connection between the symmetry
of a physical system and the degeneracy of its eigenstates. We will
follow the general sequence of the discussion in Sections 1.2 and 1.3,
1
Steven Weinberg, Sheldon Glashow, and Abdus Salam were awarded the 1979
Nobel Prize in Physics for their incorporation of the weak and electromagnetic
interactions into a single theory.
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