Variational wave functions and Green's functions are two important paradigms for solving quantum Hamiltonians, each having their own advantages. Here we detail the Variational Discrete Action Theory (VDAT), which exploits the advantages of both paradigms in order to approximately solve the ground state of quantum Hamiltonians. VDAT consists of two central components: the sequential product density matrix (SPD) ansatz and a discrete action associated with the SPD. The SPD is a variational ansatz inspired by the Trotter decomposition and characterized by an integer $\mathcal{N}$, recovering many well known variational wave functions, in addition to the exact solution for $\mathcal{N}=\infty$. The discrete action describes all dynamical information of an effective integer time evolution with respect to the SPD. We generalize the path integral to our integer time formalism, which converts a dynamic correlation function in integer time to a static correlation function in a compound space. We also generalize the usual many-body Green's function formalism to integer time, which results in analogous but distinct mathematical structures, yielding integer time versions of the generating functional, Dyson equation, and Bethe-Salpeter equation. We prove that the SPD can be exactly evaluated in the multi-band Anderson impurity model (AIM) by summing a finite number of diagrams. For the multi-band Hubbard model, we prove that the self-consistent canonical discrete action approximation (SCDA), which is the integer time analogue of the dynamical mean-field theory, exactly evaluates the SPD for $d=\infty$. VDAT within the SCDA provides an efficient yet reliable method for capturing the local physics of quantum lattice models, which will have broad applications for strongly correlated electron materials. More generally, VDAT should find applications in various many-body problems in physics.

30 pages, 9 figures. Fixed the typo of Eq (125) in PRB version (i.e the order of applying two creation operators has been reversed in PRB version.)