How does the spontaneous creation and destruction of matter not violate conservation of energy?

It's possible that I've grossly oversimplified this and I don't understand it properly. I have seen other questions here along similar but not identical lines.

In a near vacuum, quantum effects can cause short-lived particles to appear. When they disappear, on average no extra energy remains in the universe, so on average, the total amount of energy in the universe is conserved.

However, in the short time they exist, it should affect the universe in very minor ways, e.g. by having a minuscule amount of gravity and minutely accelerating other matter. Why does this not violate the conservation of energy?

• Fluctuations are seen as possible sources of the density perturbations that helped gave rise to the large scale structure of the universe, there isn't anything saying vacuum fluctuations don't have a more wide spread effect – Triatticus May 30 '18 at 3:00
• Sorry, you were told a lie. There is no such thing as virtual particles popping into existence. Such a weird phenomenon would break all of physics, such as energy conservation, as you found yourself. – knzhou May 30 '18 at 8:35
• However a lot of popsci authors find this picture very convenient for explaining things to the public. So out of laziness it gets out there. – knzhou May 30 '18 at 8:36

Quantum fluctuations are part of quantum field theory, a method of calculating and predicting the behavior of elementary particles. At present it is the basic underlying level of nature, out of which classical field theories emerge in a mathematically continuous and provable manner.

These calculations involve summations of a series of integrals over the variables describing an elementary particle interaction or decay, or even a stable state. The organization of this series expansion was greatly facilitated by the invention of Feynman diagrams which gave an iconic representation with fixed rules that could be translated one to one into the necessary integrals of the diminishing power series for calculating interactions,

Here is a first order diagram, i.e. dominating the total value for electron electron scattering:

In this diagram, only the incoming and outgoing electrons are real, i.e. are described by a four vector whose "length" is the exact mass of the electron. Internal lines are off mass shell, and under the final integration implied by the diagram, take continuous values for the mass that are not connected with the mass of the exchanged particle, in this case the virtual photon can have a mass, but it is a gimmick of the mathematical representation. Energy and momentum are conserved by the incoming and outgoing real electrons.

To get better and better accuracy in the calculation, higher order Feynman graphs have to be added and calculated. In this higher order graphs loops of virtual particle antiparticle appear , and their inclusion in the sums was important in getting correct values for experimental numbers, like the measurable Lamb shift.

In this Feynman diagram:

a single loop is introduced of a particle antiparticle pair, and can be used to correct the crossection for better accuracy. The diagram is higher order because there are more electromagnetic vertices. The important thing is to note tha there is always a line for loops connecting with a real particle, which bring in energy and momentum, and the total input and output energy and momentum are conserved. Virtual particles between real ones do not affect this conservation law, because the integration takes care of it.

The moral of the story is that vacuum fluctuations in order exist at all need an incoming real energy and momentum vertex. This is demonstrated for the Casimir effect , showing that the "shorthand" of "vacuum fluctuations" are dependent on real energy input and output interactions:

When the plates were idealized as perfect conductors, assumptions were made about the properties of the materials and the strength of the QED coupling α , that obscure the fact that the Casimir force originates in the forces between charged particles in the metal plates.

So your:

In a near vacuum, quantum effects can cause short-lived particles to appear. When they disappear, on average no extra energy remains in the universe, so on average, the total amount of energy in the universe is conserved.

The word "near vacuum" is necessary to give an input energy and momentum conserving vertex to any vacuum fluctuations . In a complete vacuum there can be no loops affecting the four dimensional space.

In a near vacuum, quantum effects can cause short-lived particles to appear

This is a common misunderstanding. The virtual particles used in calculating the properties of the QFT vacuum are a computational device and do not exist in real life. I go into this in some detail in my answer to Are vacuum fluctuations really happening all the time? The vacuum is not fluctuating and there are no violations of energy conservation - not even short lived ones.

Having said this, you are quite correct that the properties of the QFT vacuum should have observable effects, and indeed a naive calculation suggests those effects should be extremely large. For example the calculation predicts the cosmological constant should be $10^{120}$ times larger than is observed. This is sometimes referred to in the popular science press as the worst prediction in physics. At present we simply don't understand why this discrepancy exists.

You are OK up to the last sentence. The vacuum contains particle - antiparticle pairs, though whether they "exist" is semantics. They do affect the rest of the world. But that doesn't violate energy conservation. The Casimir effect is a nice example. Virtual particles produce a force between two conducting plates. So if we have a dynamics problem to solve with those plates there is one more force to consider. (Not a fifth force- it is electromagnetic but it's not from Coulomb's law). But that force is well behaved, it obeys Newton's laws, it conserves energy and momentum.

It does. Many phenomena involve involve violating energy conservation for short times, within the limits of the uncertainty principle. Light-by-light scattering, the Casimir effect, and understanding the value of the cosmological constant, for instances.

• This does not answer my question. I know that energy conservation can be violated for short times as long as there is no net increase or decrease afterwards, which I stated in different words in my question. – CJ Dennis May 30 '18 at 3:55
• "Why does this not violate the conservation of energy?" It does.. (?). In light-by-light scattering exactly such an acceleration by an energy violation occurs. – CriglCragl May 30 '18 at 4:12
• However, you qualified it with "for short times". Either it has a net effect or it doesn't. – CJ Dennis May 30 '18 at 4:14
• This answer is incorrect. For many phenomena there is a fake explanation involving energy nonconservation. But all of them are wrong. QM conserves energy exactly. – knzhou May 30 '18 at 8:38
• "e.g. by having a minuscule amount of gravity and minutely accelerating other matter" These are included in delta.Edelta.t<hbar.. The 'realness' of the 'impossible' violations is recorded by the different ordering of particles, that is the net effecg, not any net energy effects. – CriglCragl May 30 '18 at 12:03