# Energy conservation of Virtual Particles - Quantum Fluctuation?

I (as a middle-school student) was wondering how virtual particles even conserve energy of the entire system? I don't mean just the particle's energy, but conservation with respect to the surroundings?

Let me explain more, Einstein proved that $E = mc^2$ and equivalently, we can say $m = \frac{E}{c^2}$ . We however know mass causes curvature in space-time (gravity well). Now that statement being completely valid we can use Quantum Fluctuation, implying energy and time can be related by the relation: $\Delta{E}\Delta{t}\approx\frac{\hbar}{2}$ and by that simple relation we can make the following points:

• As the energy increases, the time allowed to be "borrowed" decreases, before it gives it back to the universe.
• As time increases, the energy allowed to be "borrowed" decreases.

These are completely satisfactory points I've made, I believe.

Now, I have read that the vacuum is filled with constantly bubbling virtual particles that get created from the energy "borrowed" by the above relation and is destroyed again, which satisfies the energy conservation laws.

This seems to imply that in the short time the virtual particle exists, they must have a very small mass which is: $m_{1} = \frac{\Delta{E}}{c^2}$ and using basic classical mechanics rather than GR (for sake of simplicity) we calculate that the gravitational force mediated by this short-lived virtual particle;

$F_{g}= G\frac{m_1 * m_x}{r^2}$

where $m_x$ is an particle\mass in question. Nevertheless since we know the force set upon an object moves the object in a direction, we can say it would increase its kinetic energy by this factor: $F_g = \frac{1}{2}mv^2$ which is equal to $G\frac{m_1 * m_x}{r^2} = \frac{1}{2}mv^2$.

Surely, this implies that every time quantum fluctuations happen, energy of real-particles\masses increases and therefore violates energy conservation, but what prevents this violation from taking place?

If virtual particles are not really present and interact with objects like I thought, how can Hawking radiation (currently a large concept in black hole physics) then be accepted in physics community?

## 1 Answer

The whole point of virtual particles is that they do not obey the on-shell physical laws, they are just computational crutches in Feynman diagrams. You should not take the idea of them being exchanged or "filling the vacuum" too seriously, there is no reality to them (hence the name).

The energy-time uncertainty relation is one of the most misused results of quantum mechanics, for attempts at its proper interpretation, see this question. It may be used to explain something about virtual particles, but if so, it is not obvious and not uniquely so.

• Thanks! Can you please edit your answer to my more recent question update. Thanks – LogicProgrammer Oct 25 '14 at 13:46
• @RohanVijhalwar: Hawking radiation is a result in semiclassical gravity. The popular science way of saying it comes from virtual particles "falling in" is grossly oversimplified. If you look at the actual derivation, you'll see no mention of virtual particles at all. – ACuriousMind Oct 25 '14 at 13:51
• Oh, thank you you made it really easy for a Middle school student to understand such a difficult concept, I hope to dwell deeper. Thanks! – LogicProgrammer Oct 25 '14 at 13:52
• Quick Question: It does say particles that "pop out of the local acceleration horizon" does this mean virtual particles or something vastly different? – LogicProgrammer Oct 25 '14 at 13:54
• @RohanVijjhalwar: General rule: Any particle that can be observed is not virtual. The particles emitted (and sometimes not reabsorbed) by the event horizon are not virtual, they are really being created. This is known as the Unruh effect. – ACuriousMind Oct 25 '14 at 14:01