  Renormalization is a powerful concept in the many-body problem. Inspired by
the highly successful density matrix renormalization group (DMRG) algorithm,
and the quantum chemical graphical representation of configuration space, we
introduce a new theoretical tool: Hilbert space renormalization, to describe
many-electron correlations. While in DMRG, the many-body states in nested Fock
subspaces are successively renormalized, in Hilbert space renormalization,
many-body states in nested Hilbert subspaces undergo renormalization. This
provides a new way to classify and combine configurations. The underlying
wavefunction ansatz, namely the Hilbert space matrix product state (HS-MPS),
has a very rich and flexible mathematical structure. It provides low-rank
tensor approximations to any configuration interaction (CI) space through
restricting either the 'physical indices' or the coupling rules in the HS-MPS.
Alternatively, simply truncating the 'virtual dimension' of the HS-MPS leads to
a family of size-extensive wave function ansaetze that can be used efficiently
in variational calculations. We make formal and numerical comparisons between
the HS-MPS, the traditional Fock-space MPS used in DMRG, and traditional CI
approximations. The analysis and results shed light on fundamental aspects of
the efficient representation of many-electron wavefunctions through the
renormalization of many-body states.
