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The law of the conservation of momentum has been established for hundred of years. Even in Quantum field theory every particle collision must be momentum-conserving if there is homogenity in space. Can this theorem still be violated?

If yes, what requirements must have a momentum-non-conserving theory? Is Heisenberg's uncertainity principle $\Delta x \Delta p \geq \frac{\hbar}{2}$ the possible answer? (when one considers physical Systems in which $\Delta x$ is very small)?

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  • $\begingroup$ momentum isn't conserved globally in GR $\endgroup$ – Jim Apr 18 '15 at 14:29
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If the theory is invariant under translations in space, then linear momentum is conserved by Noether's theorem. If the theory is quantum, conservation holds only on the level of the expectation values (because that's the only meaningful level where you can talk about momentum as a number that's conserved in time), but it still holds.

There is no way out. You must break homogeneity/translation invariance to break momentum conservation. Heisenberg's uncertainty principle has nothing to do with it, as it is just a statement about standard deviations, not expectation values, and hence has no influence on the quantum version of conservation.

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  • $\begingroup$ And how one can add inhomogenity of space in a theory without violating elementary consistence conditions? $\endgroup$ – kryomaxim Feb 26 '15 at 12:58
  • $\begingroup$ You say that it's only meaningful to talk about expectation values, but isn't the whole probability spectrum conserved? Conservation of the spectrum would be a much stronger condition. $\endgroup$ – kristjan Feb 26 '15 at 12:59
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    $\begingroup$ @kryomaxim: I did not claim that that is possible. $\endgroup$ – ACuriousMind Feb 26 '15 at 13:03
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    $\begingroup$ @kristjan: I didn't really say that it is the only meaningful level, it's the only meaningful level at which momentum is a number. The full quantum version of Noether's theorem are the Ward identities. I don't understand what you mean by "probability spectrum" or its conservation. $\endgroup$ – ACuriousMind Feb 26 '15 at 13:05
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    $\begingroup$ @kryomaxim: Nothing is an absolutely true fact. That does not change the fact that there is no way within current scientific knowledge to write down a consistent theory that does not obey Noether's theorem and/or has a non-conserved momentum, nor is there any experimental indication that we should look for such a theory. $\endgroup$ – ACuriousMind Feb 26 '15 at 13:23
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From my readings; the key to conservation of momentum appears to be based on defining a closed system to see if any mass crosses the boundaries of the system.

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