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Quantum systems of three identical particles on a plane are analyzed from the viewpoint of symmetry. Upon reduction by rotation, such systems are described in the space of sections of a line bundle over a three-dimensional shape space whose origin represents triple collision. It is shown that if the total angular momentum is nonzero, then the wave section must vanish at the origin, while if it is zero, then the wave section can be finite at the origin. Since the particles are assumed to be identical, the quantum system admits the action of the symmetric group S3 as well, which stands for the group of particle exchanges and is commutative with rotation. Hence the reduced system still admits the S3 action, so that Bose and Fermi states can be discussed in the space of sections of the line bundle. A detailed analysis of a system of three free particles on a plane is presented in the latter part of the article.

The reduction of a quantum system of three identical particles on a plane