Does $[P_j,B_k]=i(Mc^2)\delta_{jk}$ imply particle number conservation? From reading Weinberg's Quantum Theory of Fields, Vol. 1, I learnt that for the Galilean group $[P_j,B_k]=i(Mc^2)\delta_{jk}$, and for the Poincare group $[P_j,B_k]=iH\delta_{jk}$ where $P_j$ and $B_k$ respectively denote the generators of translation and boost, and $H$ denotes the Hamiltonian.
I have been told by one of my friends that $[P_j,B_k]=i(Mc^2)\delta_{jk}$, conserves the particle number in non-relativistic physics. I just want to ensure whether that is correct and if yes, how? In a way, I'm also asking what is the physical content/implication of this commutation relation.
 A: I now this is a little late, but I might be able to answer. Look in this paper. As stated in the comments, non-zero mass is consequence of the Bargmann extension of the Galilean Lie algebra. This leads to the Bargmann superselection rule, which forbids the superposition of two particles with unequal masses. On page 4 of the paper I link, the authors state that

The mass appears as  a  central  element  of the  extended  Lie algebra... that is, it  commutes with  all the  other  elements, in  particular  with  the  Hamiltonian  H.  It  is thus  a  conserved  quantity.  It  is  even  a  ''super-conserved"  quantity, giving  rise to a superselection  rule  ...This is due to the fact that the  extension...is  nontrivial  so that  a  physically  trivial  transformation may  result  in  a  definite  modification of  the  phase  of  the  state  vector, depending  on the  mass  of the  system. In  order  not  to  alter  the  physical properties  of  the  system  by  such  a  transformation, we  must  forbid  any superposition  of states with  different  masses, thus  obtaining  Bargmanns superselection  rule.  It  has  the  effect  of  breaking  the  Hilbert  space into mutually  incoherent eigenspaces  of the  mass  operator  M.  The consequences  of this  superselection  rule  are rather  far-reaching  with  respect to the  possible types  of particle reactions allowed by  Galilean invariance. Suppose,  for  instance,  that  we  treat  a  theory  with  only  one  kind  of particle,  with  definite  mass  m.  In  this  case,  conservation  of  the  mass implies  conservation of the  number of particles, thus forbidding  any  type of  production  process.

Hence, for one kind of particle with one kind of mass, the Bargmann superselection rule directly implies particle number conservation.
