Product of Representations and Further Physical Applications

Abstract

Let $\Phi = \hat A\Psi$, where Ψ is a wave function and $\hat A$ some operator. Then, acting with a unitary operator R, one finds $R\Phi = \hat A'R\Psi = R\hat AR^\dag R\Psi$, where the transformed operator is $\hat A' = R\hat AR^\dag$. Thus, functions are transformed according to Ψ → RΨ while operators are transformed according to $\hat A \to R\hat AR^\dag$; actually the two rules differ by a matter of notation. R acts on everything on its right, so in the case of operators the last factor R^† = R⁻¹ is there just to ensure that the action of R is limited to $\hat A$, while functions are at the extreme right and there is no need for that. We can consider (x₁, x₂, x₃) as a set of functions or as the components of an operator. As functions that transform as a basis of a representation of some symmetry Group, they transform according to the rule

$$ (x_1 ,x_2 ,x_3 ) \to (Rx_1 ,Rx_2 ,Rx_3 ) = (D(R)_{k1} x_k ,D(R)_{k2} x_k ,D(R)_{k3} x_k ); $$

this is the vector representation, which is irreducible in cubic and higher Groups. If we treat them as a set of operators, we write

$$ (x,y,z) \to (RxR^\dag ,RyR^\dag ,RzR^\dag ). $$

This defines a vector operator, but the linear combinations that result are the same. A tensor is a set of operators T_i (the components) that are mapped into linear combinations by every R, that is,

$$ T_i \mathop \to \limits^S ST_i S^\dag = \sum\limits_j {T_j D_{ji} (S);} $$

((9.1))

the multiplication Table is followed since

$$ T_i \mathop \to \limits^{RS} RST_i S^\dag R^\dag = \sum\limits_j {T_j D_{ji} (RS).} $$

((9.2))