The Three-Body Problem: the Ramsey Approach and Symmetry Considerations in the Classical and Quantum Field Theories

The graph theory based approach to the three-body problem is introduced. Vectors of linear and angular momenta of the particles form the vertices of the graph. Scalar products of the vectors of the linear and angular momenta define the colors of the links connecting the vertices. The bi-colored, complete graph emerges. This graph is called the “momenta graph”. According to the Ramsey theorem this graph contains at least one mono-chromatic triangle. The momenta graph contains at least one mono-chromatic triangle even for a chaotic motion of the particles. Coloring of the graph is independent on the rotation of frames; however, it is sensitive to Galilean transformations. The coloring of the momenta graph remains the same for general linear transformations of vectors with a positive-definite matrix. For a given motion, changing the order of the vertices does not change the number and distribution of monochromatic triangles. Symmetry of the momenta graph is addressed. The symmetry group remains the same for general linear transformation of vectors of the linear and angular momenta with a positive-definite matrix. Conditions defining conservation of the coloring of the momenta graph are addressed. The notion of the stereographic momenta graph is introduced. The suggested approach is generalized for the quantum field theory with the Pauli-Lubanski pseudo-vector. The suggested coloring procedure is Lorenz invariant.