the picture of virtual particle pairs is categorically not the right way to think about Hawking radiation. Quite obviously it must be wrong, because it is a loop level effect, and loops in QFT have to close, which they don't in this heuristic picture. (in the comment by Edward Hughes from What are the virtual particles generated during the Hawking radiation?)
Can anyone explain how virtual particle explanation of Hawking radiation goes against consistent loop description - why loop is not being closed?
Answer
It is true that vacuum loops of particle antiparticle generation can have a measurable effect on crossections and other measurements , for interactions that already occur,in higher order feynman diagrams.
For example:
Two example eighth-order Feynman diagrams that contribute to the electron self-interaction. The horizontal line with an arrow represents the electron while the wavy lines are virtual photons, and the circles represent virtual electron–positron pairs.
The loops are particle antiparticle pairs, so they have the quantum numbers of the vacuum. The energy for their mathematical existence comes from the line of the real electron, which has an energy and momentum vector on mass shell.
It is a heuristic picture to talk about vacuum loops generating e+e- pairs at the black hole horizon, because the leaving partner (e+ or e-) takes away energy and momentum, which must be given by the ambient particles about the horizon, eventually the energy lost from the black hole. This is a better diagram
Feynman diagram of the Hawking process for particle creation by a black hole. The heavy, zigzag solid lines represent the worldlines of electrons ( − e ) and of a positron (+ e ). The heavy, vertical dashed line on the left represents the event horizon of the blackhole, and the light, vertical dashed line on the right represents an effective emissive surface, for the case of electron emission. Points A and B represent pair creation and pair annihilation, respectively [12]. The “Compton layer” has a thickness on the order of the Compton wavelength, h/mc . The coordinate system being used here is that of a freely falling observer orbiting around the black hole just outside of its horizon.
Here is how Hawking% describes it :
Just outside the event horizon [of a black hole] there will be virtual pairs of particles, one with negative energy and one with positive energy. The negative particle is in a region which is classically forbidden but it can tunnel [emphasis added] through the event horizon to the region inside the black hole where the Killing vector which represents time translations is spacelike. In this region the particle can exist as a real particle with a timelike momentum vector even though its energy relative to infinity as measured by the time translation Killing vector is negative. The other particle of the pair, having a positive energy, can escape to infinity where it constitutes a part of the thermal emission described above.”
The vacuum loops , assumed to exist due to the uncertainty principle, are an easy graphic for the appearance of particle pairs, but not really true, since in order for a real particle to appear energy and momentum must be supplied and there will no longer be a loop.
%page 202 of refered as ref. 1 here
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