Escape velocity of electron/positron pair

[nikkkom]

so if you are saying that it only needs +2mc^2 of energy relative to the vacuum energy to create the pair and separate them to infinity, then where does the kinetic energy come from when they approach each other again (from infinity, or let's say "almost infinity", back to 0 and annihilation)?

The total energy of the system of two e+ e- particles approaching each other does NOT increase as they come closer and closer. It stays exactly 2m.

You can think of it this way, if it makes the picture easier to digest: "kinetic" part of the total energy increases at the expence of decreasing rest mass of the particles, which decreases because bound system of e+e- close together (positronium) has less mass than two unbound particles (as all other bound systems do, from solar system to hydrogen atoms).

 

[xortdsc]

mass is either constant (invariant mass) or increases with kinetic energy (relativistic mass)
check this article: http://hepth.hanyang.ac.kr/~sjs/physics/mass.html

so how could it possibly pay the kinetic energy toll with its mass ? It makes no sense.
most people here don't seem to understand what i want to do. I don't care how much kinetic energy a electron needs to escape a positron in proximity, but how much energy is needed to create them. Okay, then you keep repeating it is 2 times the (invariant) rest-mass of the electron, but if mass is invariant, where does the kinetic energy come from ? from the coulomb potential (so there is additional energy involved) ! But the classical coulomb potential breaks down at close separations, because it doesn't consider quantum effects.
And saying that "it is not allowed to make the separation very small up to zero" is surely true for the classical coulomb potential formula, but that's just because the classical coulomb potential is a crude approximation which does not reflect nature when separations are small. In nature you can see particle-pairs spawning from vacuum with finite energy.
I really don't know how to make my problem any more clear... :/

 

[mfb]

most people here don't seem to understand what i want to do.

I think we do, but that is not possible, and you don't understand our answers.
To consider the region where the classical description does not work any more, you need quantum field theory. If you use this, and write down several pages of calculations, you arrive at the result that gets repeated over and over again: you need at least 2m to create an electron/positron pair that escapes to infinity, you need a bit less to create a pair that stays in a bound state (positronium).

 

[xortdsc]

and the argument...
"it is 2 times the (invariant) rest-mass of the electron, but if mass is invariant, where does the kinetic energy come from ?"
...does not raise any fundamental questions in you ?

and again, I'm aware that classical physics will not be sufficient and the "bug" is in the coulomb potential function (which didn't find much acknowledging here).

i have the impression people just keep repeating the textbook solution and don't even consider the arguments i give against it. if you're right there should be a plausable answer to my argument, right ?

 

[ChrisVer]

http://cds.cern.ch/record/404321/files/9905076.pdf
one example where you can see the energy you need and how it appears, outside of the main framework of QFT...

 

[mfb]

As soon as you can assign a kinetic energy in a meaningful way, you also have a potential energy, and the sum of both is zero.

and the "bug" is in the coulomb potential function

No, it is the whole idea to describe the process with classical concepts.

 

[nikkkom]

mass is either constant (invariant mass) or increases with kinetic energy (relativistic mass)
check this article: http://hepth.hanyang.ac.kr/~sjs/physics/mass.html

so how could it possibly pay the kinetic energy toll with its mass ? It makes no sense.

Does it make sense to you that a hydrogen atom (a system of interacting proton and electorn) has LESS mass than a separate electron and separate proton? If yes, then what's your problem with interacting electron and positron having similarly reduced masses?

most people here don't seem to understand what i want to do. I don't care how much kinetic energy a electron needs to escape a positron in proximity, but how much energy is needed to create them. Okay, then you keep repeating it is 2 times the (invariant) rest-mass of the electron, but if mass is invariant, where does the kinetic energy come from ?

"Invariant rest mass of electron" refers to a free electron, one which does not interact with other particles.

When e+e- pair is created, these particles ARE interacting (at least at first when they are close to each other). Their rest masses ARE NOT equal to "Invariant rest mass of electron", they are less than that. The remainder is in their kinetic energy.

 

[xortdsc]

No, it is the whole idea to describe the process with classical concepts.

if by "classical concepts" you mean position and momentum, how can any theory get rid of these ? Sure they are only approximations as in reality there are fluctuations all over the place. So what I mean by terms like position/momentum would be rather an average (to smooth the noise present) position/momentum.
Particles are localized in space, a bit unsharp, but still on average you can determine its position, right ?

 

[xortdsc]

Does it make sense to you that a hydrogen atom (a system of interacting proton and electorn) has LESS mass than a separate electron and separate proton? If yes, then what's your problem with interacting electron and positron having similarly reduced masses?



"Invariant rest mass of electron" refers to a free electron, one which does not interact with other particles.

When e+e- pair is created, these particles ARE interacting (at least at first when they are close to each other). Their rest masses ARE NOT equal to "Invariant rest mass of electron", they are less than that. The remainder is in their kinetic energy.

okay that would make sense. so it really is the case that as they approach, mass energy goes down and kinetic energy goes up by equal amounts (so rest mass is converted to kinetic energy) ? Upon annihilation mass would become zero and all energy is stored as kinetic energy which then radiates off as (mass-less) photons...
That would totally explain where the kinetic energy comes from. I can live with that :) Thanks for the enlightenment

 

[mfb]

if by "classical concepts" you mean position and momentum, how can any theory get rid of these ? Sure they are only approximations as in reality there are fluctuations all over the place. So what I mean by terms like position/momentum would be rather an average (to smooth the noise present) position/momentum.
Particles are localized in space, a bit unsharp, but still on average you can determine its position, right ?

I mean concepts like "position and momentum of particles". Quantum mechanics uses position and momentum operators. They are not simple values.

So what I mean by terms like position/momentum would be rather an average

Those averages don't help for strong interactions, or they can even be ill-defined.