Heisenberg model has long been an interesting and intensive research topic owing to its intrinsic theoretical interest and potential application in many areas. Understanding non-equilibrium system of the Heisenberg antiferromagnet might promote our understanding of magnetic system and give us theoretical background on quantum signal manipulation. In this thesis, we consider the Heisenberg antiferromagnet in time periodic magnetic field using the Classical Spin Wave Theory and the Schwinger Boson Theory. The spin-spin response functions for the Heisenberg antiferromagnet in 2 cases of time periodic field were obtained, including case (1) B⃗0(t) = Bo(t)ẑ, where B0(t) is a periodic magnetic field in time, and (2) B⃗0(t) = Bo(cos(Ωt)x̑+sin(Ωt)ŷ). In the Classical Spin Wave Theory, the spin-spin response function and absorption spectrum of spin wave are obtained. In the Schwinger Boson Theory, the spin-spin response calculation gives same result to its classical analog. The 1/S quantum correction factor is shown to be identical to that of case when time dependent magnetic field is absent.
Some collective behaviors of boson assemblies are described in terms of the classical density operator and the classical velocity operator. We introduce these two classical quantities by taking average of the quantized operators with respect to Heisenberg wave packets. Equations of these two quantities are determined and we obtain the transport equations. Thus the assembly is described as a fluid having the quantum mechanical pressure tensor. The collective behavior of the assembly in question is analysed applying the theory of sound waves. Such an approximate treatment enables us to regard the assembly in question as the assembly of oscillation quanta. When a particle is projected into the assembly with the velocity v0 which is larger than the sound speed of the medium, suffering the collective response of the remainder of the bosons, the particle is decelerated and emits sound quanta. These processes are analysed by a classical treatment. Further we consider the interaction of every particle in the assembly with the field of the collective oscillation representing behaviors of the remainder of the assembly. The self-energy of the particle and the Møller interaction between particles are obtained according to the interaction representation of Tomonaga and Schwinger as well as the canonical transformation of Bloch and Nordsieck.