# Where does the energy in a fundamental interaction come from?

When you have two electrons, they in the most likely possibility will exchange a photon, and it will cause them to repel each other. When they repel, their momentum increases, right? This momentum increase is due to the photon exchange, but where does the photon come from in the first place? For it to appear, doesn't it need to be an excitation of the photon field, which needs energy? I understand how the energy from the photon gets converted into the electrons momenta increase, I don't understand how the photon gets that energy from nothing? Yes I am aware this is one of the many interactions possible, but this is just an example.

The fundamental question here is: A force needs energy to do anything(sorry for my lack of scientific language, I have no idea how to phrase it otherwise), where do fundamental forces get their energy?

Please, explain both intuitively and with math, don't worry about the math being too complex, I can work around that and keep in mind that my only drive to write this question is my curiosity about this topic, I have not enough formal education to discuss it. I tried really hard to come up with an answer using the intuition I got from YouTube videos about QFT, but I couldn't, obviously because said videos, although very good, entertaining and informative, were only simplifications of this model.

• The total system momentum does not change. Commented Jul 18 at 23:29
• Can you please explain how? The velocities increase, so the kinetic energy also increases, that energy must come from somewhere? If the velocities increase doesn't it mean the momentum increases? Thanks for answering! Commented Jul 18 at 23:34
• the kinetic energy comes from the reduction in potential energy, physics 101 Commented Jul 18 at 23:35
• If kinetic energy is energy stored in motion, then potential energy is energy stored in a particular position. Due to different fields (electromagnetic, gravitational, even strong and weak at quantum scales) there are certain quantities of energy inherent in certain locations, often inversely proportional to the square of the distance from the field's source. Commented Jul 18 at 23:48
• That two isolated charges are "exchanging photons" is one of those mental models that just don't want to go away, no matter how false they are. The virtual quanta in Feynman diagrams are mathematical terms. They are NOT actual particle exchanged. The only time an actual photon appears in quantum mechanics is when the electromagnetic field gains or loses a quantum of energy by interacting with an external system IRREVERSIBLY during an emission or absorption process. Commented Jul 19 at 6:00

In your electron example, there are two problems:

1. Although independently both electrons gain energy, you would never find two electrons close to one another to begin with, because such repulsion occurs. In order to bring them close together so that the force can do its job, you need to put energy into both electrons - in fact, in order to bring two electrons from infinity to a certain distance apart, you need to put in exactly as much energy as would be imparted on the electrons by the force between them as they move from that certain distance back to infinity. "Infinity" is sort of the ground state, where all forces have zero strength (since they are usually proportional to an inverse power of distance; the strong force is doing its own thing).

2. Even if that somewhat-circular logic is deemed unsatisfiable, note that the total energy of the system under consideration - the two electrons - does not change at all. Note that the total momentum is constant: if electron A imparts any force on electron B to change B's momentum, you will find that in all configurations of the system, B imparts an opposite and exactly-equal force on A - thus, the total force (derivative of momentum w.r.t. time) on the system is exactly zero. Therefore, no work is being done, and no energy is required for the interaction to take place. Trivially you can see that this also means the total energy does not change either, since the mass and total momentum of the system remain constant.

In terms of mediating the interaction, the virtual photon is generated via random quantum fluctuations. A photon appearing from nowhere might seem like it takes energy, but in reality no energy is lost in the entire interaction between the electrons: between two repelling electrons A and B, the interaction looks like this:

1. Electron A creates a virtual photon (or an extremely-large practically-infinite number of virtual photons) carrying momentum $$p$$, some of which get absorbed by electron B. (Since the area of a sphere is proportional to $$r^2$$, the number of photons that are absorbed by B is inversely proportional to the square of the distance between the two electrons - hence, Coulomb's law.)

2. These virtual photon(s) carry momentum $$p$$ away from A; to conserve momentum, A gains momentum $$-p$$, moving it away from B (hence, repulsion).

3. The virtual photon(s) are absorbed by B, which transfers momentum to B. That momentum is directed away from A.

4. Ultimately, momentum is conserved, and the energy taken from the vacuum to create the virtual photons is subsequently returned to the vacuum. No energy is lost or gained anywhere.

More generally, things have potential energy when they're deep in some field. Fields generally are considered to be of quantum-mechanical origin and involve infinite such virtual-particle interactions like was just describe, and in reality are much more complicated. But that's the gist of it.

But real quick: the vacuum has a lot of energy stored in it. The "vacuum state", completely devoid of particles and fields, still has some energy stored in it. This is primarily because we can't actually be certain exactly how much energy the vacuum has; it might be zero, and usually is, but due to the nature of the quantum vacuum, the energy density can spontaneously fluctuate and become high enough to produce virtual particles. Hence, the observed fundamental forces.

Exactly how much energy is stored in the vacuum has been predicted two different ways that yield values different by 120 orders of magnitude. I won't try to answer that question quite yet.

The final part to this answer is answering the question "but what if enough virtual particles spontaneously appear and pull all of the energy from the vacuum?" It makes sense to ask that question because if we're talking about using vacuum energy to do fundamental interactions then logically it's also good to talk about exhausting that energy. That event would be called vacuum decay: since such a "true vacuum" that's actually devoid of energy would be more stable than a "false vacuum", the surrounding false vacuum would also release all its energy now that it's able to fall into a true vacuum state. Then, the false vacuum surrounding that would fall, and so on and so forth until all vacuum energy has been released. Such an ta quick, clean and efficient way of wiping out the Universe. If you triggered a sufficiently-large vacuum decay event to release all vacuum energy, the creation of virtual particles would be impossible - you wouldn't just physically destroy everything in the Universe, but you would also render all fundamental forces - and thus all forces in general - completely inert. In other words, you would permanently kill the Universe.

Fortunately, it's not thought possible to exhaust the vacuum's energy in a physical way, so we can dodge that bullet easily.

• The vacuum state does not "have energy in it". Energy is the ability of a system to perform work on another system. The vacuum can not and does not perform work on anything. Neither does the vacuum state fluctuate because that would cause energy to be released. Commented Jul 19 at 6:02
• How is it that we don't know the energy of vacuum? Can't we just meassure its mass? Commented Jul 19 at 18:01
• @Alienfromfuture measuring the mass of something that has a mass of zero is difficult. The vacuum energy is technically the zero-point energy of a particular (vacuum) quantum state which can be nonzero when some fields are accounted for, but that's much harder to explain than just saying that there is energy in the vacuum. Commented Jul 19 at 18:02
• Didn't cosmological arguments refute this idea of vacuum having mass? A lot of vacuum energy would create a lot of gravity. Didn't such predictions fail misserably? Commented Jul 19 at 18:14
• @Alienfromfuture yes, but cosmology also predicts a vacuum energy - just a much smaller one. Both frameworks predict a nonzero vacuum energy, just the predictions differ by 120 orders of magnitude. What this suggests is that there's something that physics is missing - specifically, in the form of quantum gravity, Commented Jul 19 at 18:26