If we have a many-body sistem and we study the limit with infinite particles (we may decide for example to have a fixed constant density), I am trying to understand how classical mechanics and quantum mechanics justify, from both a mathematical and a physical point of view, the existence of the total momentum defined as lim⁡N→+∞∑i=1Npi\lim_{N \to + \infty} \sum_{i = 1}^{N} p_i , where pip_i is the momentum (let's say over 3 dimensions) of i-th particle. My first guess is that usually the system considered is a Gibbs state, and the exponential damping of the total kinetic energy allows to have things well-defined even at this limit, but what about a state which is not a Gibbs state? Despite the fact that I am more interested on hearing how the total momentum operator in quantum mechanics is described, I would like to know also how the problem is fixed in classical mechanics. Edit There are two cases that in particular I don't understand how to justify the existence of total momentum operator in the limit with infinite number of particles: Let's assume that the system (either classical or quantum) is not isolated, but there is an external force acting on it such that the total momentum of the system is not constant. In this case, which I understand it is not trivial, I just would like to know if it has ever been discussed or if it has been found how to justify mathematically or physically the existence of the total momentum. The second one is again common to both quantum and classical mechanics. In this case, each particle ii has the same momentum qi=qq_i = q and with q≠0q \neq 0 . In quantum mechanics, this would mean that the state of the system is ∣ϕ>=⊗i=1N∣q>|\phi> = \otimes_{i=1}^{N} |q> . So the system has a well defined total momentum which is simply Nq. Maybe this case is simpler than the previous one, but it is however not fully clear to me how to explain the existence of the total momentum here as well except in the trivial case where q=0q=0 .