HUP in QFT and QM:virtual particles

[haushofer]

This is wrong. The summation of this Dirichlet series was known as the Basel problem and Euler showed that the sum is $\pi^2/6$, which is also $\zeta(2)$, the Riemann zeta function.

Ah, so then I'm misstaking it for some other series, but you still get the point i assume ;)

 

[haushofer]

No; the Coulomb interaction in QED can be formulated w/o virtual photons at all, using Coulomb or Weyl-gauge; so virtual photons are not gauge invariant concepts and are therefore unphysical!

That sounds interesting, tom. Could you give some references

 

[Dickfore]

Ah, so then I'm misstaking it for some other series, but you still get the point i assume ;)

No, I don't, since I only paid attention to this wrong equation and discarded the rest.

 

[tom.stoer]

@tom
Could you send me a link or something explaining that?
And why do you say that you fix the gauge in a certain way to show this and then say that virtual photons are not gauge invariant? I'm confused.

That sounds interesting, tom. Could you give some references

To find the Coulomb term w/o virtual photons simply use Coulomb gauge; have a look at the (outline of a) derivation here https://www.physicsforums.com/showthread.php?t=553666

Deriving the Feynman rules in the path integral formalism requires gauge fixing. But different gauge choices produce different sets of Feynman rules, so the propagators and vertices depend on the gauge. Only physical matrix elements are gauge invariant. The most prominent example is the comparison of a non-abelian gauge theory like QCD formulated a) in Lorentz gauge (as an example) and b) in axial gauge (as an example). In the Lorentz gauge the gauge fixing procedure "creates a new species of particles", so-called Fadeev-Popov ghosts which have their own propagators and vertices. These ghosts are absent in physical gauges like the axial gauge. So the on the level of Feynman diagrams one has different diagrams, different terms, even different particles (!) but on the level of matrix elements both gauges are of course equivalent (this is expressed in the Slavnov-Taylor identities expressing gauge invariance in the QFT formalism). The ghosts do never appear as external particles, only as internal lines i.e. "virtual particels". But that means that the particle content as defined by the Feynman diagrams of the theory differs between these gauges, whereas the particle content on the level of the physical Hilbert space (which requires studying BRST symmetry in the ghost case) is identical.

For a reference any QFT book discussing non-abelian gauge theory will do. Try Srednidzki b/c it's free for download. Or try Ryder as an introduction.

 

[andrien]

Deriving the Feynman rules in the path integral formalism requires gauge fixing.

and I was thinking that feynman rules are derived first,then the gauge is fixed (which is supposed to be the key advantage of path integral formalism)

 

[Dickfore]

and I was thinking that feynman rules are derived first,then the gauge is fixed (which is supposed to be the key advantage of path integral formalism)

you were thinking wrong.

 

[haushofer]

and I was thinking that feynman rules are derived first,then the gauge is fixed (which is supposed to be the key advantage of path integral formalism)

In the path integral one has to plug in the Fadeev Popov determinant to fix the gauge. If I remember correctly, one after that the Feynman rules are derived.

I think I see Tom's point, but I've never really made the connection with the gauge-variance of the virtual states. I'll certainly read those references :)

 

[tom.stoer]

and I was thinking that feynman rules are derived first,then the gauge is fixed (which is supposed to be the key advantage of path integral formalism)

You have to be careful; there are the so-called $R_\xi$ gauges which are a generalization of the Lorentz gauge $\partial_\mu A^\mu = 0$; in the $R_\xi$ gauges one adds a gauge breaking term to the Lagrangian in the path integral

$$\delta\mathcal{L}_\xi = -\frac{(\partial_\mu A^\mu)^2}{2\xi}$$

which is a Gaussian located at $\partial_\mu A^\mu = 0$ with width $\xi$ in the "gauge field space". Via this mechanism one has a "family of gauge fixings" labelled by the continuous parameter $\xi$; for $\xi\to 0$ the gauge breaking term in the action reduces to a delta functional in the PI fixing the theory to ordinary Lorentz gauge.

Another possibility is to introduce the axial gauge condition $n_\mu A^\mu = 0$ where the global direction $n_\mu$ remains a s a free parameter in the theory on the level of Feynman diagrams.

It is true that via this mechanism one introduces a free parameter into the Feynman rules and that chosing a specific gauge (i.e. a specific value for $\xi$, $n^\mu$, ...) is done afer deriving the Feynman rules.

But this is not what I refer to. What I mean is that one first fixes a family auf gauges, which may depend on a free parameter and then derives the Feynman rules for this family.

 

[andrien]

you were thinking wrong.

No,I am not and it is illustrated by tom.However I did not explained that i.e. the so called landau gauge and feynman gauge to which I was really referring.

 

[tom.stoer]

No,I am not and it is illustrated by tom.However I did not explained that i.e. the so called landau gauge and feynman gauge to which I was really referring.

Dickfore is right, and you were indeed thinking wrong!

What I explained is that one does not fix a specific gauge but a class of gauges. But this IS essentially gauge fixing in the sense that the ∂_μA^μ family excludes other gauges like Coulomb gauge, axial gauge, Weyl gauge etc. So one could say that
1) one fixes a family of gauges labelled by a free parameter
2) derives the Feynman rules
3) fixes the parameter
Step 1) is gauge fixing!

 

[andrien]

Dickfore is right, and you were indeed thinking wrong!

What I explained is that one does not fix a specific gauge but a class of gauges. But this IS essentially gauge fixing in the sense that the ∂_μA^μ family excludes other gauges like Coulomb gauge, axial gauge, Weyl gauge etc. So one could say that
1) one fixes a family of gauges labelled by a free parameter
2) derives the Feynman rules
3) fixes the parameter
Step 1) is gauge fixing!

o.k. so maybe I thought about some specific parameter as gauge fixing and not the whole family of gauges which you have inherited in your mind.

 

[tom.stoer]

... which you have inherited in your mind

?

 

[andrien]

I think the problem was with two gauges ,i have seen for photon propagators .one in landau gauge,other in feynman gauge which just arises because we choose different parameters after deriving the propagator by path integral formalism.But perhaps introducing the term like -(∂_μA^μ)²/2ε should rather be called gauge fixing.

 

[tom.stoer]

So let's come back to the main issue regarding "physical interpretaion of virtual particles"

Deriving the Feynman rules in the path integral formalism requires gauge fixing. But different gauge choices produce different sets of Feynman rules, so the propagators and vertices depend on the gauge. ... The most prominent example is the comparison of a non-abelian gauge theory like QCD ... In the Lorentz gauge the gauge fixing procedure "creates a new species of particles", so-called Fadeev-Popov ghosts which have their own propagators and vertices. These ghosts are absent in physical gauges like the axial gauge. ... that means that the particle content as defined by the Feynman diagrams of the theory differs between these gauges, whereas the particle content on the level of the physical Hilbert space ... is identical