A Quantum Theory of Boson Assemblies, II : A Classical Approach to Many-Boson Problems
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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 assmbly of oscillation quanta. When a particle is projected into the assembly with the velocity ν_0 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 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 Mφller interaction between particles are obtained according to the interaction representation of Tomonaga and Schwinger as well as the canonical transformation of Bloch and Nordsieck.
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