### Question:

quantum field theory - Some questions about virtual particles
• When reading about quantum mechanics, I've stumbled upon a description of the force between two electrons as a result of virtual particles being "exchanged" by the two electrons.

So far I've understood that the virtual particles are a "prediction" of the uncertainty principle, and that their lifespan is inversely proportional with their mass.

Is an electron (or real particles in general) somehow stimulating the creation of virtual particles? If so, why?

If these guys pop in and out of existence, then won't they interfere with real particles? If so, then I guess the average effect on a particle is zero, as the creation is completely random along with the momenta of the particles. Is this correct?

• First of all, virtual particles are indeed a consequence of the uncertainty principle – without any quotation marks. Virtual particles are those that don't satisfy the correct dispersion relation $$E = \sqrt{m^2 c^4 +p^2 c^2}$$ because they have a different value of energy by $\Delta E$. For such a "wrong" value of energy, they have to borrow (or lend) $\Delta E$ from the rest of the Universe. This is possible for a limited amount of time $\Delta t$ as long as the "negated" time-energy uncertainty relationship $$\Delta t \cdot \Delta E \leq \hbar / 2$$ is obeyed. One simply can't measure energy $E$ during too short an interval $\Delta t$ more accurately than with the error $\Delta E$ given by the formula above which makes it possible to borrow/lend this much energy for such a short time.

Pretty much by definition, virtual particles are effects that look like a temporary existence of a real particle which is bounded in time by the inequality above. The more virtual the particle is – the greater the deviation of the energy $\Delta E$ is – the shorter is the timescale over which the virtual particles may operate. In the limit $\Delta E\to 0$, the virtual particles become "real" which means that they may also be observed. For a nonzero value, they can't be observed and they're just "intermediate effects in between the measurements" that modify the behavior of other particles. Most explicitly, virtual particles appear as propagators (internal lines) of a Feynman diagram.

The electron is not necessarily "simulating" anyone, whatever "simulating" was supposed to mean. Instead, the electron may "emit" a virtual particle such as a photon. The emission of a real photon is impossible by the energy/momentum conservation: in the initial electron's rest frame, the energy is just $m_e c^2$ but it would get increased both by the extra kinetic energy of the final moving electron and by the positive photon's energy, thus violating the energy conservation law. But the electron may emit a virtual photon for which the energy conservation law is effectively violated (or the photon has a different energy, perhaps negative one, than it should have) which is OK for the time $\Delta t$ described above. As long as the photon disappears before this $\Delta t$ deadline arrives – it is absorbed by another charged particle, everything is fine and this intermediate history contributes to the probability amplitudes. That's why charged particles influence each other due to electromagnetism; this is how the virtual photons operate.

Concerning the last question, yes, virtual particles may interfere with the real ones. For example, if we study processes in an external electric field create by many coherent long-wavelength photons, there will still be Feynman diagrams with virtual photons in them. The amplitudes from these diagrams have to be added to the amplitudes with the real classical electric field, and only the result (sum) is squared in absolute value. That's what we mean by interference.

And yes, the effects of virtual particles on a isolated electron are equally likely in all directions and in this sense they "average out". An electron state with a sharply defined 3-momentum still remains an energy eigenstate and moves along a straight line. However, due to the constant emission and reabsorption of some virtual particles, the real electron-like energy eigenstate has a "cloud" of virtual photons around it. The symmetries of the theory such as the gauge symmetry and the Lorentz symmetry aren't broken by the virtual photons. After all, the virtual photons result from the theory whose Lagrangian does respect these symmetries and no anomaly breaks them.

• I have encountered this question elsewhere on the web, but I can't seem to remember. My answer assumes that you are referring to the virtual particles that mediate the forces.

The idea here is to understand the S-Matrix and scattering amplitudes. QFT is the study of the birth and death of particles between 2 observations. Every QFT scattering interaction has an 'in' and 'out' state, where the states are defined in the Hartree-Fock scheme. What we observe are the in and out states only.

I will give an example of the scattering of an electron in an electrical field. So the in state (eigenstate we created) consists of an electron in a certain state of momentum and spin and a photon (from the applied field) in a certain momentum and helicity.

We leave the system undisturbed and after a long time we observe it again. We now have the out state. There is a probability of getting all kinds of configurations of the electron and a photon (constrained only by the preservation of momentum, energy, charge etc.).

We want to find out the probability for obtaining all kinds of out states. We use series expansions to calculate this amplitude. Feynman and co-workers came up with tools to keep track of the terms of this series. We represent terms of each term of a series (where oen out state corresponds to a whole series) using Feynman diagrams. To facilitate the easy calculation, we use virtual particles in these diagrams.

As QFT deals with discontinuous observations, there are no actual photons being exchanged, just before and after states. To make the math easy we imagine that virtual particles were exchanged between the in and out states.

I recommend any standard QFT (Antony Zee for example) book to get a clear picture of how we use virtual particles as a tool. I hope this answer is satisfactory.

EDIT: I have taken the time interval to be infinite as I mentioned earlier. It is never really the case in real life, but it allows energy to be conserved in my scattering process.