According to Noether's theorem, all conservation laws originate from invariance of a system to shifts in a certain space. For example conservation of energy stems from invariance to time translation.
What kind of symmetry creates the conservation of mass?
Answer
Mass is only conserved in the low-energy limit of relativistic systems. In relativistic systems, mass can be converted into energy, and you can have processes like massive electron-positron pairs annhillating to form massless photons.
What is conserved (in theories obeying special relativity, at least) is mass energy--this conservation is enforced by the time and space translation invariance of the theory. Since the amount of energy in the mass dominates the amount of energy in kinetic energy ($mc^{2}$ means a lot of energy is stored even in a small mass) for nonrelativistic motion, you get a very good approximation of mass conservation. out of the energy conservation.
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