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I've just watched one of Feynman's lectures on the character of physical law where he was talking about conservation laws. In that particular part he was reasoning why a mass can't "jump" from one place to another (link) which he elegantly proved would be a violation of angular momentum. While I get this, is there no other way to prove this? It seems odd that all the laws of motion and conservation of momentum laws can't reason against a simple mass jump.

Does a jump violate the continuity equation? And maybe some other, more fundamental law? Also the angular momentum reasoning can't be used against the jump of a charge.

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I think the more direct way to show this is to appeal to conservation of linear momentum: I can't move or "jump" the mass without some commensurate opposite momentum. Feynman effectively mentions this later with the rocket. Linear momentum conservation really comes directly from the invariance of all known laws of physics under spatial translation (see Noether's Theorem). I also think he's oversimplifying (or overcomplicating?) when he says the same argument can't be used for a charge, since all known (electromagnetically-)charged particles are massive, and subject to the same restriction of linear momentum conservation. What takes over for the momentum in that case, however, are the (massless) photons of whatever electromagnetic field the charge is interacting with.

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Feynman showed only that jump in the position of the particle would change mechanical angular momentum due to that particle. If the jump is due to other body or medium capable of carrying angular momentum, total angular momentum of the system could still be conserved.

Continuous mass conservation is a basic idea in physics, inferred directly from experience - things move continuously even if they are fast. It makes much more sense to derive the law of conservation of mechanical angular momentum from this idea and the laws of motion.

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