  A local transformation from fermionic operators to spin matrices is proposed
and studied in this work. For this purpose, a system of fermions on a lattice
is considered and one applies the scheme to replace the fermionic variables
with spin matrices, while the transformation relates only those fermionic/spin
operators which are assigned to nearby lattice sites. In one dimension, this
proposal yields the same result as the well-known Jordan-Wigner transformation,
while not being restricted to $d=1$ dimension. To obtain the equivalent
description in the spin picture, one needs to impose constraints on the spin
space. Since finding the reduced spin Hilbert space constitutes a substantial
stage of the whole procedure, the constraints are paid particular attention.
The full set of necessary constraints is determined in both representations. To
approach the task to solve the constraints, a suitable basis is constructed.
The introduction of the basis in the spin representation along with the
construction of the constraints and the Hamiltonian in this basis show how the
transformation proposed in this work can be applied to obtain observables in
the spin picture. Explicit construction of the constraints in the basis allows
one to solve them and, once the basis vectors of the reduced spin Hilbert space
are found, the spin Hamiltonian is expressed in this basis and diagonalized.
The constraints are constructed in the basis as discussed above and analyzed
with the Wolfram Mathematica programs for lattice sizes $3\times3$, $4\times3$
and $4\times4$. Their mutual relations are determined and the reduced spin
Hilbert space is specified. The Hamiltonian is constructed in this
representation and diagonalized. It is verified that the eigenenergies obtained
in the spin picture agree with the analytic formulas from the fermionic
representation.
