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.
Applying the functional Schrodinger picture to the Holstein-Primakoff boson theory, we obtain a dispersion energy relation for an antiferromagnetic spin polaron or magnon in two-dimensional systems of strongly correlated electrons. We show that our magnon ground-state wave function is equivalent to that of a retraceable path approximation by Reiter in the long-wavelength limit. Further, we demonstrate that the derived dispersion relation of a quasiparticle leads to the self-consistent Born approximation of the Green's function approach only in the large Heisenberg coupling J limit, showing that this new approach is general enough to contain more physical information than the Green's function approach in dealing with the antiferroinagnetic spin polaron of interest.