We present a new boson representation for systems of fermions. This representation allows a direct quantization of the mean field equations. The relation to existing boson representations is discussed.
Essential contents on quantization of fields and second quantization, which are necessary for undergraduate and graduate level, are presented in this chapter. An innovative approach is used for the presentations and the discussions on field quantization and second quantization for systems of Bosons and Fermions. The chapter covers field operators, the procedure of second quantization and its use for the Schrödinger equation for both Bosons and Fermions, matrix representation of Fermionic operators, number operators for both Bosons and Fermions, and systems of weakly interacting Bosons and Fermions. A good number of solved examples and unsolved exercises clarify the basic concepts behind quantization of fields and second quantization.
The report discusses the slave-fermion representations of the t-J model and describes another representation, in which fermions and bosons are completely commuting and in which the properties of fermions are directly related to the properties of physical holes. For a study of the system in the new representation at half-filling, interaction of fermions with two magnons is treated in mean-field theory. The obtained effective model, in comparison to that of the usual slave-fermion representation, has an additional bare hole dispersion due to the hole moving by using quantum spin fluctuations present in the undoped antiferromagnetic ground state. The single-hole Green's function at half-filling is then found numerically using the self-consistent Born approximation. For all studied quantities good or excellent agreement with numerical data is observed in the entire parameter range, noticeably better than in the studies with the slave-fermion representation. Using the same effective model, the two-hole problem is also studied by solving numerically the Bethe-Salpeter equation with noncrossing diagrams.