  We show how certain long-range models of interacting fermions in $d+1$
dimensions are equivalent to $U\left(1\right)$ gauge theories in $D+1$
dimensions, with the dimension $D$ in which gauge fields are defined larger
than the dimension $d$ of the fermionic theory to be simulated. For $d=1$ it is
possible to obtain an exact mapping, providing an expression of the fermionic
interaction potential in terms of half-integer powers of the Laplacian. An
analogous mapping can be applied to the kinetic term of the bosonized theory. A
diagrammatic representation of the theories obtained by dimensional mismatch is
presented, and consequences and applications of the established duality are
discussed. Finally, by using a perturbative approach, we address the canonical
quantization of fermionic theories presenting non-locality in the interaction
term to construct the Hamiltonians for the effective theories found by
dimensional reduction. We conclude by showing that one can engineer the gauge
fields and the dimensional mismatch in order to obtain long-range effective
Hamiltonians with $1/r$ potentials.
