Heisenberg equation of motion in Liouville space

Question

Consider a quantum system with finite dimensional Hilbert space. Assume that on this Hilbert space, there exist a basis of operators $G_i$, so that any aribtrary operator $A$ can be expressed as a linear comination of the $G_i$ according to $A=\sum_ia_iG_i$, where the coefficients $a_i$ can be obtained by the Hilbert-Schmidt scalar product according to $a_i=\text{Tr}(G_i^\dagger A)$. Furthermore we assume $\left[G_i,G_j\right]=\sum_kc_k^{ij}G_k$. The space spanned by the operators $G_i$ is often called Liouville space.

Let us now consider some arbitrary operator $V(t)=\sum_iv_i(t)G_i$ with explicit time dependence in the Schrödinger picture. Let us furthermore assume that the dynamics of the systen is governed by a time-dependent Hamiltonian $H(t)=\sum_ih_i(t)G_i$ generating the time evolution operator $U(t)$. Then $V(t)$ in the Heisenberg picture takes the form $\tilde{V}(t)=U^\dagger(t)V(t)U(t)=\sum_i\tilde{v}_i(t)G_i$. Hereby the tilde denotes the Heisenberg picture. Obviously, $\tilde{V}(t)$ fulfills the Heisenberg equation of motion \begin{equation} \frac{\mathrm{d}}{\mathrm{d}t}\tilde{V}(t)=\frac{\mathrm{i}}{\hbar}\left[\tilde{H}(t),\tilde{V}(t)\right]+U^\dagger(t)\left(\frac{\mathrm{d}}{\mathrm{d}t}V(t)\right)U^\dagger(t). \label{eq:HE} \end{equation} My question is how I can find the Liouville representation of the Heisenberg equation of motion? To my understanding this would be a differential equation for $v_i(t)$ or $\tilde{v}_i(t)$, arising from the Heisenberg equation of motion. According to my calculations, as long, as $V(t)$ depends explicitely on time, on cannot find an equation depending either only on $v_i(t)$ or only on $\tilde{v}_i(t)$. Is that correct?

In this paper, in Appendix A, it is said the Liouville representation of the Heisenberg equation is given by \begin{equation} \frac{\mathrm{d}}{\mathrm{d}t}\vec{v}(t)=\left(\frac{\mathrm{i}}{\hbar}\left[H(t),\cdot\right]+\frac{\partial}{\partial t}\right)\vec{v}(t). \end{equation} I cannot really make any sense of this equation and I am not able to derive it from the basic assumptions given here despite several trials. Might it be that this equation is wrong? Any help would be really appreciated.

Answer 1

It seems to me that the authors, instead of keeping the basis $G_i$ and changing the decomposition vector $v_i(t) \rightarrow \tilde v_i(t)$, change the decomposition basis

$\tilde V(t) = \sum\limits_i \tilde v_i(t) G_i = \sum\limits_i v_i(t) \tilde G_i(t)$.

The operators $\tilde G_i(t)$ still build a basis, as the transformation $U(t)$ is unitary. From the evolution equation for $\tilde V(t)$ in Heisenberg picture one can derive then equation which confused you.