Introduction

In another post we gave an exposition of irreducible representations of $SO(3)$, where we find ourselves studying harmonic polynomials on a sphere. In this post, we study another category of representations of $SO(3)$ that have its own significance in physics: projective representation. The result will be written as direct sums of irreducible representations of $SU(2)$ so the reader is advised to review the corresponding post. We recall that

Every irreducible unitary irreducible representation of $SU(2)$ is of the form $V_n$, where

$$V_n=\operatorname{Sym}^n \mathbb{C}^2=\{\text{The space of homogeneous polynomials of degree $n$ in two variables}\}$$

Representation theory has a billion applications in physics. The group $SO(3)$ acts as the group of orientation-preserving orthogonal symmetries in $\mathbb{R}^3$ in an obvious way. The invariance under this action justifies the principle that physical reactions such as those between elementary particles should not depend on the observer’s vantage point.

Nevertheless, applications of representation theory in physics do not end at finite dimensional vector spaces. Put infinite dimensional vector spaces aside, we sometimes also need a class of vectors, in lieu of a single vector. For example, given a wavefunction $\psi$, we know $|\psi|^2$ has an interpretation of probability density. But then for any $\lambda...