  After a brief review of the definition and properties of the quantum
effective Hamiltonian action we describe its renormalization flow by a
functional RG equation. This equation can be used for a non-perturbative
quantization and study also of theories with bare Hamiltonians which are not
quadratic in the momenta. As an example the vacuum energy and gap of quantum
mechanical models are computed. Extensions of this framework to quantum field
theories are discussed. In particular one possible Lorentz covariant approach
for simple scalar field theories is developed. Fermionic degrees of freedom,
being naturally described by a first order formulation, can be easily
accommodated in this approach.
