We present functional Schrödinger representations of Holstein-Primakoff boson and slave-boson theories for the Heisenberg Hamiltonian and the
t−J Hamiltonian respectively. Based on these representations we obtain the dispersion relations of magnons for two dimensional antiferromagnets. By applying the functional Schrödinger representation of the Holstein-Primakoff boson theory to the Heisenberg Hamiltonian, the exchange energy is correctly predicted and the self-energy of quasi-hole is obtained. From the use of the functional Schrödinger representation of the
t−J Hamiltonian it is shown that at half-filling the dispersion relation obtained from the slave-boson theory leads to that obtained from the Holstein-Primakoff boson approach.
An alternative formalism that does not require the assumption of the deconfinement phase of a U(1) gauge field is proposed for the slave particle mean field theory. Starting form the spin-fermion model, a spinon field, which is either fermion or boson, is introduced to represent the localized spin moment. We find a d-wave superconductive state in the mean field theory in the case of the fermion representation of the localized spin moment that corresponds to the slave boson mean field theory of the t-J model, whereas the d-wave superconductive state is absent in case of the Schwinger boson representation of the localized spin moments.