  The composite fermion (CF) theory gives both a phenomenological description
for many fractional quantum Hall (FQH) states, as well as a microscopic
construction for large scale numerical calculation of these topological phases.
The fundamental postulate of mapping FQH states of electrons to integer quantum
Hall (IQH) states of CFs, however, was not formally established. The Landau
level (LL) projection needed for the microscopic calculations is in some sense
uncontrolled and unpredictable. We rigorously derive the unitary relationship
between electrons and the CFs, showing the latter naturally emerge from special
interactions within a single LL, without resorting to any projection by hand.
In this framework, all FQH states topologically equivalent to those described
by the conventional CF theory (e.g. the Jain series) have exact model
Hamiltonians that can be explicitly derived, and we can easily generalise to
FQH states from interacting CFs. Our derivations reveal fundamental connections
between the CF theory and the pseudopotential/Jack polynomial constructions,
and argue that all Abelian CF states are physically equivalent to the IQH
states, while a plethora of non-Abelian CF states can be systematically
constructed and classified. We also discuss about implications to experiments
and effective field theory descriptions based on the descriptions with CFs as
elementary particles.
