If virtual particles sometimes add more mass/energy to a system then was inputed or comes out in the output, how do they not violate conservation of mass/energy.

  1. Virtual particles are not real. Though sounding like a tautology, it is an important one - they are not actual states in the asymptotic Hilbert spaces of a quantum field theory, where particles usually live. They are a name given to internal lines of Feynman diagrams, which, in turn, are mere computational tools in a perturbative approach to QFT. Nothing in the formalism itself justifies imbuing these lines or diagrams with any more meaning, so it is unclear what it means to say "virtual particles sometimes add more mass/energy to a system" as you do in the question.

  2. Conservation of energy holds, quantumly, only inside correlation functions and up to contact terms - the quantum version of Noether's theorem, which classically guarantees conservation of energy, are the Ward-Takahashi identities. Even if the internal lines of Feynman diagrams represent particles in any sense, conservation of energy/momentum is only guaranteed to hold as a statement about expectation values, so that individual states may well, from a classical viewpoint "violate conservation of energy". (Note, though, that "energy of a state" may even be an ill-defined thing to say)

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    $\begingroup$ You said "Virtual particles are not real", being internal lines of Feynman diagrams. But the electron (for instance) that is detected can then be considered as an internal line of a bigger Feynman diagram including the detector. I know that this debate has been open many times, but I don't find satisfying answer to this point. (the usual answer consists in saying that a single Feynman diagram is meaningless, all the higher orders contributing to the process. yes. But it just shifts the problem: what we usually call an electron would be simply be the global contribution). $\endgroup$ – Paganini Mar 7 '15 at 9:00
  • $\begingroup$ (con't) in addition, even in orthodox quantum mechanics, an electron can be slightly off-shell (living a finite time between its creation and its destruction), which is considered usually as a degree of virtuality. $\endgroup$ – Paganini Mar 7 '15 at 9:04
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    $\begingroup$ @Paganini: Feynman diagrams are a way of organizing perturbation theory because we are too dumb to solve the exact theory. It is meaningless to say that a real particle "can be considered as an internal line, because the internal line is not a quantum state, while a real particle is. The possible slight off-shellness of real particles is explained by my point 2 - the quantum theory obeys classical laws only on average. $\endgroup$ – ACuriousMind Mar 7 '15 at 10:05
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    $\begingroup$ yes, but considering external lines as pure quantum state is also an approximation no? It assumes pure free particles without interaction at all, which is only approximative too. $\endgroup$ – Paganini Mar 7 '15 at 10:36
  • $\begingroup$ @Paganini: Of course. Unfortunately, the Hilbert space of interacting QFTs is unknown except for very special cases. $\endgroup$ – ACuriousMind Mar 7 '15 at 10:46

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