The paper discusses the existence and properties of unpaired Majorana fermions in one-dimensional quantum wires. These fermions are described by Majorana operators, which are linear combinations of creation and annihilation operators that commute with each other. The authors propose a Hamiltonian for a quantum wire that can generate Majorana fermions as effective low-energy degrees of freedom, even in the absence of strong interactions. The Hamiltonian is designed to have a gap in the bulk excitation spectrum and to break the $U(1)$ symmetry to a $\mathbf{Z}_2$ symmetry, which is necessary for the presence of unpaired Majorana fermions at the ends of the wire. The paper presents a toy model to illustrate the phenomenon, where the Hamiltonian is derived from a general translationally invariant one-dimensional Hamiltonian with short-range interactions. The model exhibits two phases: one with trivial pairing and another with unpaired Majorana fermions. The existence of unpaired Majorana fermions is characterized by a "Majorana number" $\mathcal{M} = -1$, which can be computed for any non-interacting electron system. The authors also discuss the physical realization of such a quantum wire, suggesting that it could be achieved through spin-orbit interactions or charge density waves in two-dimensional $p$-wave superconductors. They propose experiments to test the presence of Majorana fermions, such as measuring the Josephson current in a quantum wire bridge and observing spontaneous phase slips in a superconducting island connected to a quantum wire.The paper discusses the existence and properties of unpaired Majorana fermions in one-dimensional quantum wires. These fermions are described by Majorana operators, which are linear combinations of creation and annihilation operators that commute with each other. The authors propose a Hamiltonian for a quantum wire that can generate Majorana fermions as effective low-energy degrees of freedom, even in the absence of strong interactions. The Hamiltonian is designed to have a gap in the bulk excitation spectrum and to break the $U(1)$ symmetry to a $\mathbf{Z}_2$ symmetry, which is necessary for the presence of unpaired Majorana fermions at the ends of the wire. The paper presents a toy model to illustrate the phenomenon, where the Hamiltonian is derived from a general translationally invariant one-dimensional Hamiltonian with short-range interactions. The model exhibits two phases: one with trivial pairing and another with unpaired Majorana fermions. The existence of unpaired Majorana fermions is characterized by a "Majorana number" $\mathcal{M} = -1$, which can be computed for any non-interacting electron system. The authors also discuss the physical realization of such a quantum wire, suggesting that it could be achieved through spin-orbit interactions or charge density waves in two-dimensional $p$-wave superconductors. They propose experiments to test the presence of Majorana fermions, such as measuring the Josephson current in a quantum wire bridge and observing spontaneous phase slips in a superconducting island connected to a quantum wire.