Title: Closed hierarchy of Heisenberg equations in the quantum transverse-field Ising chain

Abstract: Dynamics of a quantum system can be described by coupled Heisenberg equations. In a generic many-body system these equations form an exponentially large hierarchy that is intractable without approximations. In contrast, in an integrable system a small subset of operators can be closed with respect to commutation with the Hamiltonian. As a result, the Heisenberg equations for these operators form a smaller closed system amenable to an analytical treatment. We solve a system of Heisenberg equations for the Onsager algebra of string operators in the transverse field Ising chain. This allows one to describe the dynamics of the corresponding observables for an arbitrary initial state. As a particular example, we calculate time-dependent expectation values $\langle \sigma^z_j \rangle_t$, $\langle \sigma^x_j \sigma^x_{j+1} \rangle_t$, $\langle \sigma^y_j \sigma^y_{j+1} \rangle_t$ and $\langle \sigma^x_j \sigma^y_{j+1} \rangle_t$ for a translationally invariant product initial state with an arbitrarily directed polarization.