  Non-commutative algebras and entanglement are two of the most important
hallmarks of many-body quantum systems. Dynamical perturbation methods are the
most widely used approaches for quantum many-body systems. While study of
entanglement-based numerical methods are booming recently, the traditional
dynamical perturbation methods have not benefited from study of quantum
entanglement. In this work, we formulate an algebraic-dynamical theory (ADT) by
combining the power of quantum algebras and dynamical methods in which quantum
entanglement naturally emerges as the organizing principle. We start by
introducing a complete operator basis set (COBS), with which an arbitrary
state, either pure or mixed, can be represented by the expectation values of
COBS. Then we establish a complete mapping from a given state to a complete set
of dynamical correlation functions of the state through the Heisenberg- and
Schwinger-Dyson-equations-of-motion (SDEOM). The completeness of COBS and the
mapping ensures ADT to be a mathematically complete framework in principle.
Applying ADT to many-body systems on lattices, we find that the quantum
entanglement is represented by the cumulant structure of expectation values of
the many-body COBS. The cumulant structure of the state forms a hierarchy in
correlations. More importantly, such static correlational hierarchy is
inherited by the dynamical correlations and their SDEOM. We propose that the
dynamical hierarchy is also carried into any perturbative calculation on that
state. We demonstrate the validity of such perturbation hierarchy with an
explicit example, in which we show that a single-particle-type perturbative
calculation fails while a many-body perturbation following the hierarchy
succeeds. We also discuss the computation and approximation schemes of ADT and
its implications to other strong coupling theories like parton and slave
particle methods.
