A sound-wave description of fermion assemblies

Sin‐itiro Tomonaga

B. Jancovici, Daniel Schiff

S.-i. Tomonaga

The fact implied by Bloch several years ago that in some approximate sense the behavior of an assembly of Fermi particles can be described by a quantized field of sound waves in the Fermi gas, where the sound field obeys Bose statistics, is proved in the one-dimensional case. This fact provides us with a new possibility of treating an assembly of Fermi particles in terms of the equivalent assembly of Bose particles, namely, the assembly of sound quanta. The field equation for the sound wave is found to be linear irrespective of the absence or presence of mutual interaction between particles, so that this method is a very useful means of dealing with many-Fermion problems. It is also applicable to the case where the interparticle force is not weak. In the case of force of too short a range this method fails.

Sin‐itiro Tomonaga

The fact implied by Bloch several years ago that in some approximate sense the behavior of an assembly of Fermi particles can be described by a quantized field of sound waves in the Fermi gas, where the sound field obeys Bose statistics, is proved in the one-dimensional case. This fact provides us with a new possibility of treating an assembly of Fermi particles in terms of the equivalent assembly of Bose particles, namely, the assembly of sound quanta. The field equation for the sound wave is found to be linear irrespective of the absence or presence of mutual interaction between particles, so that this method is a very useful means of dealing with many-Fermion problems. It is also applicable to the case where the interparticle force is not weak. In the case of force of too short a range this method fails.

Parth Patel, Biswaroop Mukherjee, Zhenjie Yan +6 more

Toshiyuki Nishiyama

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.