  We consider the Fermi polaron problem at zero temperature, where a single
impurity interacts with non-interacting host fermions. We approach the problem
starting with a Frohlich-like Hamiltonian where the impurity is described with
canonical position and momentum operators. We apply the Lee-Low-Pine (LLP)
transformation to change the fermionic Frohlich Hamiltonian into the fermionic
LLP Hamiltonian which describes a many-body system containing host fermions
only. We adapt the self-consistent Hartree-Fock (HF) approach, first proposed
by Edwards, to the fermionic LLP Hamiltonian in which a pair of host fermions
with momenta $\mathbf{k}$ and $\mathbf{k}'$ interact with a potential
proportional to $\mathbf{k}\cdot\mathbf{k}'$. We apply the HF theory, which has
the advantage of not restricting the number of particle-hole pairs, to
repulsive Fermi polarons in one dimension. When the impurity and host fermion
masses are equal our variational ansatz, where HF orbitals are expanded in
terms of free-particle states, produces results in excellent agreement with
McGuire's exact analytical results based on the Bethe ansatz. This work raises
the prospect of using the HF ansatz and its time-dependent generalization as
building blocks for developing all-coupling theories for both equilibrium and
nonequilibrium Fermi polarons in higher dimensions
