Consider the Heisenberg picture of Quantum Mechanics. For a two state system we have the Pauli matrices evolving according to the relation $$\sigma_i(t)=U^+\sigma_i(0)U$$ where $U=e^{-iHt/\hbar}$ and $i=x,y,z.$
But in a particular research paper I saw the evolution written as $$\vec\sigma(t)=U^+\vec\sigma(0)U$$ where $\vec\sigma=(\sigma_x,\sigma_y,\sigma_z)$. Thinking about the matrix representation of this equation, it is a 2×2 matrix $U^+$ multiplied by a 3×1 matrix $\vec\sigma$ multiplied by a 2×2 matrix $U$ which is obviously wrong. Where am I making mistakes in interpreting the above equation?
Best Answer
Take your first relation for the 3 Pauli matrices individually:
$$\sigma_1(t)=U^\dagger\sigma_1(0)U$$
$$\sigma_1(t)=U^\dagger\sigma_2(0)U$$
$$\sigma_3(t)=U^\dagger\sigma_3(0)U$$
Now you define a "vector" for notational convenience like the OP says in the question. I will choose to rewrite it as a column vector to visually show the relation of the above 3 relations: \begin{equation} \vec\sigma= \begin{pmatrix} \sigma_x\\ \sigma_y\\ \sigma_z \end{pmatrix} \end{equation}
Now we can write the above 3 notations as: $$\vec\sigma(t)=U^\dagger\vec\sigma(0)U$$
What we have really done here is define a new product that acts on the vectors defined for notational convenience. It acts as follows:
\begin{equation} U \begin{pmatrix} \sigma_x\\ \sigma_y\\ \sigma_z \end{pmatrix} := \begin{pmatrix} U \sigma_x\\ U \sigma_y\\ U \sigma_z \end{pmatrix} \end{equation}
Note this is a definition for notational convenience and there is nothing physical about it.