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If virtual particles don’t exist, then what are, for example, the electron positron pairs that appear and disappear? One drifts beyond the event horizon, and one becomes a real particle.

This stuff people talk about seems so tangible, then they go and say virtual particles don’t exist at all, in any sense of the word.

Isn’t it at least possible that they exist? But even if they don’t, how could you possibly attribute that example to anything other than a particle? Or is that example completely incorrect too?

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In the early years of QFT a lot of physicists thought about this and many leaned quite strongly this way. As time has gone on however it has fallen out of favour. The problem is that the mathematical expressions in terms of virtual particles are in fact only an approximation for the correct QFT expression.

This is extremely useful as the full QFT treatment is almost always impossible to calculate exactly and we have found that in many cases the approximation using virtual particles works extremely well. Nevertheless we have over the years slowly found cases where we can do the full exact calculation and have also found other approximation methods that we can compare to the virtual particles calculation. In doing this we have found more and more cases where the virtual particles method gives the wrong answer. These cases go under the general heading of "non-perturpative effects".

When explaining things from QFT to the general public physicists will normally fall back on a virtual particles picture, even if this is not how they actually arrived at the answer (see the comments about Hawking's radiation). This is because quantum fields are very very strange things and we don't have many options to explain them without a lot of maths.

By Symmetry
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