  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.
