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I was wondering about whether quantum fluctuations can really violate energy conservation. For example, as far as I know, virtual particles are a result of the Heisenberg uncertainty principle, and according to this site,

"Quantum mechanics allows, and indeed requires, temporary violations of conservation of energy".

Now, I know the virtual particles are there for a very (insanely) short time period before mutually annihilating, but energy conservation was still violated, right?

On the other hand, I've seen this post, this post, this post, and this post, and searched online, and I read that virtual particles are mainly derived from the mathematical aspect of physics and Feynman Diagrams. So many people say they don't "exist", so to speak.

I know many scientists are still speculating, debating, and investigating whether virtual particles really exist or not, but if they did, would they be an exception to the law of conservation of energy? Thanks.

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    $\begingroup$ How is this question not a duplicate of the questions you've already linked? Can you be more explicit about what you're asking that's not already covered in these other questions? $\endgroup$
    – ACuriousMind
    Jun 10 at 19:35
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I had a physics teacher who used to say energy conservation will never be refuted because we'd just add an extra term that absorbs the apparent violation. That sounds like an accusation of unfalsifiability, but it makes sense because energy really does come in multiple forms that are detected in different ways, with varying degrees of difficulty. "Let's posit neutrinos to explain apparent energy loss in beta decay" is no worse than "let's posit GPE to explain why things speed up as they fall".

Energy is typically stored in particles, waves or fields, and you can mentally add all the particle-wave duality, field quantization etc. footnotes and asterisks you like to that. Let's think through what happens when an electron repels another electron by sending it a virtual photon. Your argument is that the first electron experiences a four-momentum change that temporarily isn't balanced by a concomitant change in the second electron's four-momentum. You're absolutely right. But if I ask you where that four-momentum went, it didn't just disappear; it got stored in the virtual particle. You can work out for yourself what happens when the transmitted four-momentum is received.

It needn't be within the scope of this answer to say whether "virtual particles exist, they're just unusual for their species in their value of $E^2/c^4-p^2/c^2$" or "no they don't, it's just a calculation aid before I write an integral over $3$-or $4$-momentum space". The most pedantic possible comment to the latter effect would just mean the energy temporarily in neither electron is in the electromagnetic ("photonic") field. So from that perspective, everything we need to understand to track energy conservation here is exactly as one would expect when one considers the problem classically. "One electron repels another by transmitting energy & momentum through the electromagnetic field", Maxwell might have said. The only information added by quantum field theory (strictly speaking, your question is within the province of QFT rather than e.g. non-relativistic QM) is different ways of thinking about how this field, now quantized, does what it does.

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whether virtual particles really exist or not, but if they did, would they be an exception to the law of conservation of energy?

virtual

This is a simple Feynman diagram where a virtual particle is needed, in order to calculate the crossection to first order of electron-electron scattering.

What exists are the two electrons interacting on the bottom, they exist because they have on mass shell four vectors, i.e. their energy and momentum are bounded by their mass, that is what is called real in particle physics. The "length " of the four vector is the mass.

Also existing and real are the two outgoing electrons on the top. What is virtual is the photon exchanged to transfer the interaction quantum numbers, except its four vector is not on mass shall( zero mass for photons) . In general exchanged particles are virtual, because they have all the quantum numbers of the named particle, charge, spin, etc, except they do not have the mass.

The vacuum loops you have in mind, cannot be detected, unless there is an energy source that allows momentum and energy in the loop, but the virtual particles will be off mass shell, and in that sense not real. They exist, because their presence modifies the calculations and gives higher order corrections that have been validated.

Energy and momentum and angular momentum conservation are absolute laws in quantum mechanics, and energy out of the vacuum cannot be within the current understanding of physics.

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