Consider an ensemble of $n$ relativistic particles of fixed masses $m_i \geq 0$, $i=1,\ldots,n$ with four momenta $p_i$ such that $p_i^2=m_i^2$. In center of mass frame they sum up to $$P=p_1+\cdots+p_n=(E,0,0,0),$$ where $E>m_1+...+m_n$ is fixed number (last inequality is to exlude trivial case in which all particles are at rest).

The set of all possible configurations $(p_1,...,p_n)$ satisfying these two constraints, i.e. $n$-tuples of vectors such that each is on mass shell and they sum up to our fixed vector is a compact subspace of $\mathbb R^{4n}$. If all masses are strictly greater than zero then the implicit function theorem implies that it is actually a smooth submanifold of dimension $3n-4$, but if some masses are nonzero I'm not sure if one can exclude existence of cusps.

For example for $n=2$ three spatial components of equation $P=p_1+p_2$ and mass-shell condition imply that they are of the form $p_1=(\sqrt{m_1^2+\vec p ^2}, \vec p)$, $p_2=(\sqrt{m_2^2+\vec p^2}, - \vec p)$. Energy equation $E=\sqrt{m_1^2+\vec p ^2} + \sqrt{m_2^2+\vec p ^2}$ fix length of $\vec p$ uniquely. However angles of $\vec p$ remain unspecified. Therefore set of all solutions is a sphere.

This post imported from StackExchange Physics at 2016-10-30 15:22 (UTC), posted by SE-user Blazej