Symmetries in the many-body problems, a method to find its analytical solution, and Helium atom spectrum

Abstract: In this work it is shown that there are symmetries beyond the Euclidean group $E\left(3\right)$ in 3-body problem, and by extension in many-body problem, with inverse squared distance inter particle force. The symmetries in 3-body problem form a group: $SO\left(4\times3,2\times3\right)/\left(C\left(3\times2\right)\right)$, where $C\left(n\right)$ is the planar translation group in n dimensions, which forms its Spectrum-Generating group. Some of these quantities commute with the Hamiltonian. The existence of these conserved quantities was verified by calculating energy spectrum of the Helium atom. This method can also be used to find symmetries in many-body problem, and to calculate energy levels, and wave-functions of more complicated systems, which include every possible atomic and molecular systems in chemistry.