Uncovering the Mystery of Kallen-Lehmann Spectral Representation

[AndreasC]

The Kallen-Lehmann representation is a (non perturbative) result in QFT that is proved with what seems to me like very minimal assumptions: https://en.m.wikipedia.org/wiki/Källén–Lehmann_spectral_representation

According to this wiki page, in gauge theories something goes wrong and you can no longer get a positive definite measure. What goes wrong exactly? The reference in the wiki page is rather obscure, source recommendations would be appreciated!

 

[Demystifier]

At one point of derivation one inserts a complete set of states. If this refers only to physical states, everything should be fine. But if one includes also the gauge ghosts, I can imagine that this can ruin positivity.

 

[Demystifier]

The reference in the wiki page is rather obscure, source recommendations would be appreciated!

Wikpedia cites Ref. [3], a short book by Sttrochi. For example, at page 103 it says: "Contrary to what sometimes stated in the literature, ghosts or states with "negative norm" do not imply a violation of the weak spectral condition IV, as can be checked in the known examples of free fields and in particular in the Gupta-Bleuler formulation of free QED." I haven't studied the book in detail, but maybe this will motivate you to do it by yourself.

 

[AndreasC]

Wikpedia cites Ref. [3], a short book by Sttrochi. For example, at page 103 it says: "Contrary to what sometimes stated in the literature, ghosts or states with "negative norm" do not imply a violation of the weak spectral condition IV, as can be checked in the known examples of free fields and in particular in the Gupta-Bleuler formulation of free QED." I haven't studied the book in detail, but maybe this will motivate you to do it by yourself.

Hmmm I found that book using shady means online but it was borderline illegible in the version I found, maybe I will look for a better one. I can sort of see how ghost states could ruin positivity, I will look into it a bit more. It is interesting (and kind of weird I guess) that the Kallen-Lehmann representation is used so commonly but in depth accounts are relatively rare...

 

[AndreasC]

At one point of derivation one inserts a complete set of states. If this refers only to physical states, everything should be fine. But if one includes also the gauge ghosts, I can imagine that this can ruin positivity

But wait, the expression derived in, say, the wiki article contains the measure squared of a braket. This is positive no matter what. So I can't see how including gauge ghosts would change anything... Furthermore, my understanding is that negative norm states corresponding to gauge ghosts do not belong in the Hilbert space, so I'm not sure how you would even begin to do that... The book referenced by wiki doesn't seem to get into any more detail so I can't quite figure out what's going on.

 

[Demystifier]

But wait, the expression derived in, say, the wiki article contains the measure squared of a braket. This is positive no matter what. So I can't see how including gauge ghosts would change anything... Furthermore, my understanding is that negative norm states corresponding to gauge ghosts do not belong in the Hilbert space, so I'm not sure how you would even begin to do that... The book referenced by wiki doesn't seem to get into any more detail so I can't quite figure out what's going on.

So, maybe wikipedia is just wrong about that?

 

[AndreasC]

So, maybe wikipedia is just wrong about that?

It also appears in other wiki pages though: https://en.wikipedia.org/wiki/Mass_gap

I also found it claimed here:
https://physics.stackexchange.com/questions/209383/källén-lehmann-spectral-representation-for-massless-particle

No references however. There seems to be a shortage of sources.... Very annoying.

 

[Demystifier]

No references however.

https://en.wikipedia.org/wiki/Urban_legends_and_myths

 

[AndreasC]

https://en.wikipedia.org/wiki/Urban_legends_and_myths

Maybe? Maybe there is something to it? Idk, I know very little about the subject so I can't really tell, and the way gauge fields are quantized is often a bit confusing. The wiki page asserts a relation can not be true in a gauge theory. Why? I don't know. It doesn't say. But then it says:

"Rather it must be proved that a Källén–Lehmann representation for the propagator holds also for this case."

This kind of implies that there IS such a form after all so I don't know what all that was about. Apparently Itzykson and Zuber has a derivation for the photon propagator (it being a gauge field). I looked into it a little bit and couldn't find it. Will look again. Who knows. I really wish there were better sources.

 

[A. Neumaier]

The Kallen-Lehmann representation is a (non perturbative) result in QFT that is proved with what seems to me like very minimal assumptions: https://en.m.wikipedia.org/wiki/Källén–Lehmann_spectral_representation

According to this wiki page, in gauge theories something goes wrong and you can no longer get a positive definite measure. What goes wrong exactly? The reference in the wiki page is rather obscure, source recommendations would be appreciated!

In a local quantum field formulation of a gauge theory, one needs in place of a Hilbert space a Krein space with an indefinite inner product. (Many papers of Strocchi are on this topic, not only the book mentioned.) This makes any results inapplicable that are derived under the assumption of a positive definite inner product.

Hence no Kallen-Lehmann theorem. But the latter applies to the asymptotic states and hence to the S-matrix elements between physical multiparticle states.

 

[A. Neumaier]

The Krein space context was discussed at length in the recent thread A single quark at rest.

 

[AndreasC]

In a local quantum field formulation of a gauge theory, one needs in place of a Hilbert space a Krein space with an indefinite inner product. (Many papers of Strocchi are on this topic, not only the book mentioned.) This makes any results inapplicable that are derived under the assumption of a positive definite inner product.

Hence no Kallen-Lehmann theorem. But the latter applies to the asymptotic states and hence to the S-matrix elements between physical multiparticle states

But wait, what part of the theorem assumes the positive definiteness of the inner product? Is it the insertion of a complete set of eigenstates? Is it not actually legitimate in the Krein space?

 

[A. Neumaier]

But wait, what part of the theorem assumes the positive definiteness of the inner product? Is it the insertion of a complete set of eigenstates? Is it not actually legitimate in the Krein space?

In the derivation given by Wikipedia (for spin 0 fields), it is

all the intermediate states have $p^{2}\geq 0$ and $p_{0}>0$

... that you get a measure with zero weights at the unphysical spin 0 representations of the Poincare group.

In a Krein space you may have noncausal spin 0 representations with $p^{2}<0$.

For positive spin, you get a matrix-valued and $p$-dependent spectral density, with positve semidefinite matrices in the Hilbert space case. But quarks have spin 1/2 and are known to have a noncausal contribution.

 

[AndreasC]

In the derivation given by Wikipedia (for spin 0 fields), it is

... that you get a measure with zero weights at the unphysical spin 0 representations of the Poincare group.

In a Krein space you may have noncausal spin 0 representations with $p^{2}<0$.

For positive spin, you get a matrix-valued and $p$-dependent spectral density, with positve semidefinite matrices in the Hilbert space case. But quarks have spin 1/2 and are known to have a noncausal contribution.

Alright, so the first two equations in the wiki derivation are fine, but then you can't write out the measure like that because it doesn't cover the states with negative momentum norm, right? How do people go ahead and derive these forms for photons and QCD particles then despite these issues? Also, do you have any good references on these subjects I could look up?

 

[vanhees71]

At one point of derivation one inserts a complete set of states. If this refers only to physical states, everything should be fine. But if one includes also the gauge ghosts, I can imagine that this can ruin positivity.

The point is that these ghosts are good ghosts, i.e., you introduce the Faddeev-Popov ghosts to cancel the contributions of unphysical degrees of freedom of the gauge fields, using a gauge fixing. The Faddeev-Popov procedure makes you to effectively sum only over the physical states, and you guarantee with that the gauge invariance and unitarity of the S-matrix and positive definiteness of the norm within the "physical Hilbert space". A somewhat more complicated approach is the covariant operator quantization of gauge fields, which makes this a bit more explicit than the Faddeev-Popov procedure, which uses the path-integral approach.

 

[A. Neumaier]

Alright, so the first two equations in the wiki derivation are fine, but then you can't write out the measure like that because it doesn't cover the states with negative momentum norm, right?

... with negative $m^2$.

How do people go ahead and derive these forms for photons and QCD particles then despite these issues? Also, do you have any good references on these subjects I could look up?

This is not done on the level of mathematical theorems but at the level of rigor of theoretical physics. Read the work by Strocchi to see what can be done rigorously, and the references I gave in the other PF thread I had mentioned to see what is done more informally.