I have heard virtual particles pop in and out of existence all the time, most notable being the pairs that pop out beside black holes and while one gets pulled away. But wouldn't this actually violate the conservation of energy?
Quantum Field Theory – Do Virtual Particles Actually Physically Exist?
energy-conservationfeynman-diagramsquantum-field-theoryvacuumvirtual-particles
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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.
Virtual particles are not real. Though sounding like a tautology, it is an important one - they are not actual states in the asymptotic Hilbert spaces of a quantum field theory, where particles usually live. They are a name given to internal lines of Feynman diagrams, which, in turn, are mere computational tools in a perturbative approach to QFT. Nothing in the formalism itself justifies imbuing these lines or diagrams with any more meaning, so it is unclear what it means to say "virtual particles sometimes add more mass/energy to a system" as you do in the question.
Conservation of energy holds, quantumly, only inside correlation functions and up to contact terms - the quantum version of Noether's theorem, which classically guarantees conservation of energy, are the Ward-Takahashi identities. Even if the internal lines of Feynman diagrams represent particles in any sense, conservation of energy/momentum is only guaranteed to hold as a statement about expectation values, so that individual states may well, from a classical viewpoint "violate conservation of energy". (Note, though, that "energy of a state" may even be an ill-defined thing to say)
Best Answer
Ever since Newton and the use of mathematics in physics, physics can be defined as a discipline where nature is modeled by mathematics. One should have clear in mind what nature means and what mathematics is.
Nature we know by measurements and observations. Mathematics is a self consistent discipline with axioms, theorems and statements having absolute proofs, mathematically deduced from the axioms. "Existence" for physics means "measurable", for mathematics "possible to be included in the self consistent theory.
Modern physics has used mathematical models to describe the measurements and observations in the microcosm of atoms, molecules, elementary particles, adding postulates that connect the mathematical calculations with the physical observables
The dominant mathematical model is the field theoretical model that simplifies the mathematics using Feynman diagrams
These diagrams represent terms in an expansion of the desired solution, each term has a diminishing contribution to the cross section of the interaction. The diagram below would be the dominant term, as the next one would be more complicated and therefore smaller by orders of magnitude.
To each component of the diagram there corresponds one to one a mathematical formula twhich integrated properly will give a prediction for a measurable quantity. In this case the probability of repulsion when one electron scatters on another.
This diagram for example, has as measurable quantities the incoming energy and momentum of the electrons ( four vectors) and of outgoing four vectors . The line in between is not measurable, because it represents a mathematical term that is integrated over the limits of integration, and within the integral energy and momentum are independent variables. The line has the quantum numbers of the photon though not its mass , and so it is called a "virtual photon". It does not obey the energy momentum rule which says that :
$$\sqrt{P\cdot P} = \sqrt{E^2 - (pc)^2} = m_0 c^2$$
The photon has mass zero.
Through the above relation which connects energy and momentum through the rest mass, the un-physical mass of the virtual line depends on one variable, which will be integrated over the diagram; it is often taken as the momentum transfer.
Quantum number conservation is a strong rule and is the only rule that virtual particles have to obey.
There are innumerable Feynman diagrams one can write, and the internal lines considered as particles would not conserve energy and momentum rules if they were on mass shell. These diagrams include vacuum fluctuations that you are asking about, where by construction there are no outgoing measurable lines in the Feynman diagrams describing them. They are useful/necessary in summing up higher order calculations in order to get the final numbers that will predict a measurable value for some interaction.
Thus virtual particles exist only in the mathematics of the model used to describe the measurements of real particles . To coin a word virtual particles are particlemorphic ( :) ), having a form like particle but not a particle.