A Massive gauge bosons in QFT in/out states

A. Neumaier

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Resonances are indeed states in the continuous-spectrum part of the Hamiltonian. [...]

In QFT it's an internal line in a Feynman diagram (in the case that the corresponding "particles" are described as an elementary field in the Lagrangian). As such it's as "virtual" as any internal line in a Feynman diagram.
No. Resonances are definitely not internal lines in a Feynmn diagram, since the latter have arbitrary real 4-momentum and no associated states, while resonances have complex 4-momentum and unnormalized states.

I just want to understand this better. So if we take a reasonably long lived particle like a free neutron, ultimately it is still a resonance. What's going on during those ~15 minutes? Surely there is a state representing what is occurring in some manner. Or do you mean there is no state directly corresponding to the resonance pole on the second sheet of the S-matrix's analytic continuation.
There is no normalizable state directly corresponding to the resonance pole on the second sheet of the S-matrix's analytic continuation. But there are wave packets made from the energy eigenstates with energies in the region that approximate the resonance state. This shows in a modified short time behavior - unlike normalizable states, resonances decay purely exponentially at all times (which is impossible at times before preparation).
 

vanhees71

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I didn't claim that there's a normalizable state for a resonance. Of course with "internal line" for a resonance you have to take a dressed line, i.e., you have to sum the Dyson equation for it, because otherwise you don't get a pole in the complex energy plane to begin with. Sorry that I forgot this detail ;-).
 
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I agree that's how it appears in the perturbative formalism. My point to @Michael Price was more that this is simply a perturbative representation and you shouldn't think of virtual particles actually flying about.
Well, that is exactly how I do think of a virtual particle, namely as something flying about. If seems to me that it is the asymptotic in/out states that are fictions. Everything happens inside Feynman diagrams, not on the fictional asymptotic edges.
 

vanhees71

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Feynman diagrams are mathematical formulae in an ingeniously efficient notation, and their meaning in usual "vacuum QFT" is that they provide transition-matrix elements between asymptotic free in and out states.

Then there are also parts of such diagrams, which refer not directly to observable quantities but are building blocks to calculate systematically the transition-matrix elements. These are the proper vertex functions (one-particle-irreducible diagrams) which are evaluated, including renormalization and all that and then used finally in the diagrams describing S-matrix elements.

A "particle" interpretation however only the asymptotic free states have, and already in QED these are in fact quite complicated beasts due to the the long-range em. interaction!
 

DarMM

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Well, that is exactly how I do think of a virtual particle, namely as something flying about. If seems to me that it is the asymptotic in/out states that are fictions. Everything happens inside Feynman diagrams, not on the fictional asymptotic edges.
This is untenable to me because the asymptotic particles are what actually registers in our detectors and virtual particles don't even appear in a non-perturbative formalism and depending on how you perform the perturbative expansion you get virtual particles with different properties.
 
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This is untenable to me because the asymptotic particles are what actually registers in our detectors and virtual particles don't even appear in a non-perturbative formalism and depending on how you perform the perturbative expansion you get virtual particles with different properties.
Particles also change with gauge transformations, but we have learnt to get used to that. But to return to the asymptotic business. I can pick up a U238 atom and mail to someone on the other side of the world - clearly there is more to reality than asymptotic in/out states.
 

DarMM

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Particles also change with gauge transformations, but we have learnt to get used to that. But to return to the asymptotic business. I can pick up a U238 atom and mail to someone on the other side of the world - clearly there is more to reality than asymptotic in/out states.
I never said there wasn't, I said:
  1. The asymptotic states are real, i.e. they're not fictions.

  2. Virtual particles are not part of reality as they only appear in the perturbative formalism.
What particles change with gauge transformations?
 

vanhees71

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Particles also change with gauge transformations, but we have learnt to get used to that. But to return to the asymptotic business. I can pick up a U238 atom and mail to someone on the other side of the world - clearly there is more to reality than asymptotic in/out states.
No! Nothing changes with gauge transformations. To the contrary gauge transformations simply transform from one description of a situation to another complete equivalent one. That's why there's no spontaneous symmetry breaking possible for a local gauge symmetry, and that's why the ground state is not degenerate, and that's why in breaking a local gauge symmetry doesn't imply the existence of Goldstone modes.

What, in your opinion is a U238 atom else, in the quantum description, than an asymptotic free state?
 
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No! Nothing changes with gauge transformations. To the contrary gauge transformations simply transform from one description of a situation to another complete equivalent one. That's why there's no spontaneous symmetry breaking possible for a local gauge symmetry, and that's why the ground state is not degenerate, and that's why in breaking a local gauge symmetry doesn't imply the existence of Goldstone modes.

What, in your opinion is a U238 atom else, in the quantum description, than an asymptotic free state?
I agree, nothing measurable changes with a gauge transformation, but I don't want to get diverted off-topic here. To return to the U238 (or W boson) I got the distinct impression that some people are saying the U238 does not appear in an asymptotic state.
 

vanhees71

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Of course it does. Why shouldn't it? It's well observable after all!
 

DarMM

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Of course it does. Why shouldn't it? It's well observable after all!
Strictly speaking in eletroweak theory it is not an asymptotic state.

In other theories it is. That's why what is asymptotic is theory dependent.
 

DarMM

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I agree, nothing measurable changes with a gauge transformation, but I don't want to get diverted off-topic here. To return to the U238 (or W boson) I got the distinct impression that some people are saying the U238 does not appear in an asymptotic state.
Particle states on the physical Hilbert space (BRST cohomology) do not transform under gauge transformations. Fields acting on the entire Krein space do.
 

Vanadium 50

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Well, that is exactly how I do think of a virtual particle, namely as something flying about.
Well, that's wrong and you should stop doing that.

If seems to me that it is the asymptotic in/out states that are fictions.
At worst, they are approximations. If you are going to complain about this, you should start with massless pulleys, frictionless planes, spherical earth, etc. Furthermore, it seems a little like cutting off one's nose to spite one's face to reject some of the calculational successes: e.g. the muon magnetic moment to 12 decimal places - because there might be an error in the 20th.
 
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Well, that's wrong and you should stop doing that.
You should not be so dogmatic about interpretational issues. Virtual particles are off shell, and real particles are on shell. The on-shell relations are a sign of classicallity. Is any particle on the mass shell exactly? No. So you could equally argue that it is the virtual particles that exist and the real particles that are fiction.
 

DarMM

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You should not be so dogmatic about interpretational issues. Virtual particles are off shell, and real particles are on shell. The on-shell relations are a sign of classicallity. Is any particle on the mass shell exactly? No. So you could equally argue that it is the virtual particles that exist and the real particles that are fiction.
No state in the Hilbert space has energy-momentum relations like those of virtual particles. This can be proven directly from the Wightman axioms.
 
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No state in the Hilbert space has energy-momentum relations like those of virtual particles. This can be proven directly from the Wightman axioms.
Which begs the question a bit, and thus irrelevant to my point.
 

DarMM

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Which begs the question a bit, and thus irrelevant to my point.
I don't understand. All states of a quantum theory are elements of the Hilbert space. No elements of the Hilbert space have this relation.

What question is it begging?
 

Vanadium 50

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So you could equally argue that it is the virtual particles that exist and the real particles that are fiction.
Sorry. Thought you were asking a question. Didn't realize you were here to tell everyone they are doing it wrong.

Sure, you could look at it that way. Indeed, I just said you could. But it is, as I said, a failure to use a convenient approximation that cuts off one's nose to spite one's face.
 
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Sorry. Thought you were asking a question. Didn't realize you were here to tell everyone they are doing it wrong.
No, just pointing out there are two sides to some interpretational issues. Have a good day.
 

vanhees71

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These are no interpretational issues, but mathematical facts about QFT.
 
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These are no interpretational issues, but mathematical facts about QFT.
Whether virtual particles exist is not mathematics but interpretation.
 

vanhees71

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What do you mean by exist? "Virtual particles" are internal lines in Feynman diagrams with a clear mathematical meaning. Feynman diagrams are condensed notations for formulae to calculate perturbatively S-matrix elements and cross sections or decay rates. A propagator does not represent a state. So it's not clear what you think what should "exist" or being "real". For sure it's not representing a particle that is detectable in any way.
 

DarMM

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Whether virtual particles exist is not mathematics but interpretation.
No, I agree with @vanhees71 that this is not interpretational.

No states we detect are like virtual particles. No states in the Hilbert space are like virtual particles.

The latter fact is insurmountable if you want to consider them as real, because they're are not an element of QM's state space.

As @vanhees71 said they are condensed notations for perturbative calculations. If we consider parts of a step in a calculational method as real, then we'd equally think the updating in the MCMC algorithm of a lattice calculation was a real process.
 
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What do you mean by exist? "Virtual particles" are internal lines in Feynman diagrams with a clear mathematical meaning. Feynman diagrams are condensed notations for formulae to calculate perturbatively S-matrix elements and cross sections or decay rates. A propagator does not represent a state. So it's not clear what you think what should "exist" or being "real". For sure it's not representing a particle that is detectable in any way.
Can you conduct an experiment to say whether a virtual particle exists or not? We can certainly feel their effects; what more is needed? As I said, I think of them as something flying about, and see no reason to drop this viewpoint. YMMV.
 

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