Haag's Theorem: Importance & Implications in QFT

[rkastner]

I disagree with that too. CI, as an instrumentalist interpretation, does address issues raised by the Haag's theorem. The Haag's theorem is a consequence of the infinite number of degrees of feedom in QFT, especially the IR ones. CI has developed practical instrumentalists methods of dealing with such systems, by methods of regularization and renormalization. In this way, from a practical instrumentalist point of view, the problems raised by the Haag's theorem are avoided.

Good point. If one thinks that the quantum world doesn't exist, then I suppose it doesn't matter if a theorem shows that the interaction picture of fields doesn't exist. :) Evasion of fundamental questions about reality is a good pragmatic tactic for getting on with one's life I suppose. But in my view it is inconsistent with the spirit of science. And I argue in both my books that this sort of evasion is wholly unnecessary -- indeed it is based on specific metaphysical and epistemological assumptions which are not necessarily true at all. Just as Kant's view that Euclidean spacetime had to be a basic feature of knowable reality was shown to be wrong.

 

[atyy]

And concerning your first question, it has long been known that the direct action theory is empirically equivalent to QFT given the appropriate boundary conditions. This fact is discussed at some length in my paper.

At some point the direct action theory, if successful, should probably diverge from standard QED. If standard QED is not asymptotically safe, then it will blow up at high energies (Landau pole), and fail to make predictions for those experiments. Another way to see it is that QED is a lattice theory in finite volume and finite lattice spacing. If the direct action theory is successful, then it has to work in infinite volume and at arbitrarily high energy, so it should diverge from standard QED. Does the direct action theory do this?

 

[rkastner]

These sorts of infinities in QFT are artifacts of the need to renormalize, which is another aspect of the consistency problems inherent in QFT. They only appear because of the assumed infinite degrees of freedom of the putative mediating fields, which are denied in the direct action picture. The direct action theory does not require renormalization, so it's immune to these problems. It is empirically equivalent to QFT to the extent that the latter makes non-divergent empirical predictions. (See p. 7 of my preprint which discusses the Rohrlich theory). Caveat: there may be a slight deviation from QED in exotic systems such as heavy He-like ions which I've briefly explored in qualitative terms (see http://arxiv.org/abs/1312.4007)

 

[atyy]

These sorts of infinities in QFT are artifacts of the need to renormalize, which is another aspect of the consistency problems inherent in QFT. They only appear because of the assumed infinite degrees of freedom of the putative mediating fields, which are denied in the direct action picture. The direct action theory does not require renormalization, so it's immune to these problems. It is empirically equivalent to QFT to the extent that the latter makes non-divergent empirical predictions. (See p. 7 of my preprint which discusses the Rohrlich theory). Caveat: there may be a slight deviation from QED in exotic systems such as heavy He-like ions which I've briefly explored in qualitative terms (see http://arxiv.org/abs/1312.4007)

So direct action theory holds in infinite volume and at arbitrary energy? In other words, direct action theory is claimed to be a UV completion of standard QED?

 

[rkastner]

I have not seen it stated in those terms, but there is no self-energy divergence in the direct action theory. Rohrlich makes this point on p. 351 of this paper: http://philpapers.org/rec/ROHTED (It's a chapter in an edited collection by J. Mehra. You may be able to find the book excerpt online.)

 

[atyy]

I have not seen it stated in those terms, but there is no self-energy divergence in the direct action theory. Rohrlich makes this point on p. 351 of this paper: http://philpapers.org/rec/ROHTED (It's a chapter in an edited collection by J. Mehra. You may be able to find the book excerpt online.)

I think the self-energy divergence usually means a high energy cut-off is needed, so that doesn't seem to address the infinite volume requirement. Is the direct action theory also claimed to formulate QED in infinite volume?

 

[rkastner]

I am not aware of any problem facing the direct-action theory for the case of infinite volume. Let me know if you see anything that might suggest otherwise.
And thanks for your interest in this topic.

 

[Demystifier]

Concerning the alleged error, I think you misunderstand. The term "vacuum" in this context is the state with zero quanta, |0>, not zero energy. The ground state is indeed annihilated by the Hamiltonian defined as proportional to the number operator a(dag)a, since the eigenvalue of the number operator for |0> is zero. (See Wiki, http://en.wikipedia.org/wiki/Canonical_quantization for details on this definition of the Hamiltonian). Here's a relevant passage from Earman and Fraser (2005):

"...And suppose that the vacuum state is the ground state in that it is an eigenstate of the Hamiltonian with eigenvalue 0.."

And concerning your first question, it has long been known that the direct action theory is empirically equivalent to QFT given the appropriate boundary conditions. This fact is discussed at some length in my paper.

Thanks for the reply, now I am becoming more interested.

First something trivial. I have noted a typo in your Ref. [12]; the volume should be 4, not 6.

Now some non-trivial questions:
1. If the two formulations are empirically equivalent, then why the Wheeler-Feynman (WF) one is much less popular?
2. In particular why Feynman himself abandoned it?
3. Is perhaps WF more complicated in practical applications?
4. Can WF be generalized to Yang-Mills theory?
5. Do infinities appear in a similar way as in standard formulation, and can they be cured by an appropriate renormalization theory?
6. How would you comment the following statement at wikipedia?
http://en.wikipedia.org/wiki/Wheeler–Feynman_absorber_theory
"Finally, the main drawback of the theory turned out to be the result that particles are not self-interacting. Indeed, as demonstrated by Hans Bethe, the Lamb shift necessitated a self-energy term to be explained. Feynman and Bethe had an intense discussion over that issue and eventually Feynman himself stated that self-interaction is needed to correctly account for this effect."

 

[atyy]

I am not aware of any problem facing the direct-action theory for the case of infinite volume. Let me know if you see anything that might suggest otherwise.
And thanks for your interest in this topic.

I took a quick look at the Davies papers in J Phys A, and he mentions that the system has to be in a light tight box. At least naively, that seems to require finite volume.

 

[rkastner]

Thanks for the reply, now I am becoming more interested.

Now some non-trivial questions:
1. If the two formulations are empirically equivalent, then why the Wheeler-Feynman (WF) one is much less popular?
2. In particular why Feynman himself abandoned it?
3. Is perhaps WF more complicated in practical applications?
4. Can WF be generalized to Yang-Mills theory?
5. Do infinities appear in a similar way as in standard formulation, and can they be cured by an appropriate renormalization theory?
6. How would you comment the following statement at wikipedia?
http://en.wikipedia.org/wiki/Wheeler–Feynman_absorber_theory
"Finally, the main drawback of the theory turned out to be the result that particles are not self-interacting. Indeed, as demonstrated by Hans Bethe, the Lamb shift necessitated a self-energy term to be explained. Feynman and Bethe had an intense discussion over that issue and eventually Feynman himself stated that self-interaction is needed to correctly account for this effect."

Thanks, I'll check into the typo.
1. I can't find a good reason for the general lack of interest in direct-action picture, especially given that Wheeler was still advocating it in 2005 as noted in my paper. I'm trying to remedy that with my current work.
2. This is related to your #6--see below.
3. It does seem easier to use quantized fields as stand-ins for unknown charge configurations, so probably yes, although Wheeler didn't seem to think so, as I note in my paper on Haag's thm.
4. I don't see why not. Worth exploring.
5. No, since infinities result from the assumption that there are Fock space states for all interactions, which is denied in direct-action picture.
6. Apparently Feynman was mistaken. You don't need to omit all self-interaction to use the direct action picture successfully. Davies showed how to do this in QED (Davies 1971 and 1972 papers). I think Feynman was overly committed to the assumption of zero self-interaction when that is not necessary. Perhaps he didn't notice that once you include quantum indistinguishability of currents, there is no real fact of that matter as to what constitutes self-interaction and what doesn't. So of course at the quantum level you are naturally going to have to allow for some self-interaction, which is just the right kind to explain such things as the Lamb Shift. As I've noted, Wheeler in 2005 saw direct-action picture as perfectly viable. So apparently he disagreed with Feynman's abandonment of it.

 

[bhobba]

2. In particular why Feynman himself abandoned it?

I thought it was because he could never figure out a quantum version of it.

Regarding renormalisation I thought Wilson sorted it out ages ago. Its simply we pushed our theories into regions where they break down eg the Landau pole. But well before that the electroweak theory takes over so the theory is wrong anyway. The same with gravity:
http://arxiv.org/pdf/1209.3511v1.pdf

We, at this stage, only have effective theories.

Thanks
Bill

 

[rkastner]

There's some evidence for that in the historical record, although it seems quite straightforward (see Davies).
Haag's thm points to a more serious consistency problem with QFT than simply pushing it beyond its range. The interaction picture does not exist, period, at any range.
This is remedied in the direct action picture.

 

[king vitamin]

These sorts of infinities in QFT are artifacts of the need to renormalize, which is another aspect of the consistency problems inherent in QFT. They only appear because of the assumed infinite degrees of freedom of the putative mediating fields, which are denied in the direct action picture. The direct action theory does not require renormalization, so it's immune to these problems. It is empirically equivalent to QFT to the extent that the latter makes non-divergent empirical predictions. (See p. 7 of my preprint which discusses the Rohrlich theory). Caveat: there may be a slight deviation from QED in exotic systems such as heavy He-like ions which I've briefly explored in qualitative terms (see http://arxiv.org/abs/1312.4007)

I'm not happy with this dismissal of atyy's point. What atyy is referring to is the triviality problem of QED: that QED formulated as a continuum theory probably doesn't exist. If you're claiming that it does, you need to address this. Is the direct-action approach more general, and QED is an effective theory with a cutoff inherited from it?

 

[rkastner]

The quoted reply was not a response to atyy, whose post I had not yet seen. I will reply to that separately.
I'm certainly not claiming that QFT formulated as a continuum theory exists. Quite the opposite.
Concerning the direct-action theory, I explain how it evades these divergence problems, without the need for any cutoff, in my paper: http://arxiv.org/abs/1502.03814
The direct-action theory is not a generalization of quantized field theory. It is a different model that is empirically equivalent.

 

[rkastner]

I took a quick look at the Davies papers in J Phys A, and he mentions that the system has to be in a light tight box. At least naively, that seems to require finite volume.

Sorry for overlooking this post earlier. You could perhaps argue that this condition suggests a finite volume. However, in my possibilist approach to the transactional picture, absorbers need not be actualized spacetime objects. In that ontology, the existence of absorbers sufficient to satisfy this boundary condition would not automatically translate into a spacetime condition. I'm aware that to many physicists these ontological considerations sound speculative, but I am certainly not the only researcher considering an emergent spacetime -- i.e. one that arises from a quantum level that is not contained in the spacetime manifold. See, e.g., Sorkin's work on Causal Set theory, and Daniele Oriti's work in quantum gravity.

 

[atyy]

Sorry for overlooking this post earlier. You could perhaps argue that this condition suggests a finite volume. However, in my possibilist approach to the transactional picture, absorbers need not be actualized spacetime objects. In that ontology, the existence of absorbers sufficient to satisfy this boundary condition would not automatically translate into a spacetime condition. I'm aware that to many physicists these ontological considerations sound speculative, but I am certainly not the only researcher considering an emergent spacetime -- i.e. one that arises from a quantum level that is not contained in the spacetime manifold. See, e.g., Sorkin's work on Causal Set theory, and Daniele Oriti's work in quantum gravity.

The finite volume is a key point. If it is still speculative as to whether it works or not in infinite volume then how can it be claimed to solve the "problem" of Haag's theorem?

I think it is ok for spacetime and QED to be emergent from a theory that is mathematically complete and non-perturbatively defined. But to solve the "problem" of Haag's theorem seems to require that the emergent spacetime be infinite volume Minkowski spacetime.

 

[rkastner]

I noted that the ontology 'sounds' speculative to some researchers, but it is a specific physical model that does work.
It is also not established that full absorption is equivalent to a finite spacetime volume even if you reject the ontology I've proposed.

 

[atyy]

I noted that the ontology 'sounds' speculative to some researchers, but it is a specific physical model that does work.
It is also not established that full absorption is equivalent to a finite spacetime volume even if you reject the ontology I've proposed.

Yes, but can it be established that the theory does give rise to QED in infinite volume Minskowski spacetime?

 

[rkastner]

Yes this has been established by the work referenced in my paper, e.g. Narlikar and Davies. Similarly Rohrlich's partial direct-action version is not restricted to finite volume.

 

[Demystifier]

5. No, since infinities result from the assumption that there are Fock space states for all interactions, which is denied in direct-action picture.

I don't think that this is how the most problematic infinities in QFT arise. They arise in loop Feynman diagrams, and I don't see how they are related to the assumption that there are Fock space states for all interactions. Indeed, one of the Davies papers contains some loop diagrams, and he does not claim that these diagrams are finite.

 

[bhobba]

I don't think that this is how the most problematic infinities in QFT arise.

Nor do I.

I think they arise from a cruddy choice of perturbation parameter:
http://arxiv.org/pdf/hep-th/0212049.pdf

That after a better parameter is chosen that some things like the unrenormalised coupling constant goes to infinity with the cutoff is a problem - but only if you believe its valid without a cutoff.

Thanks
Bill

 

[rkastner]

I think if you read Rohrlich's review (the Mehra reference I gave earlier), you will recall that a non-quantized self-interaction of this type does not lead to divergences.
It's all there in the literature. In terms of the transactional picture, all such interactions are truly 'virtual' in that no real energy is exchanged in such a loop. There is no real photon involved (ie no Fock space state, and therefore no real energy). This is where the real vs virtual distinction becomes important (see e.g. http://arxiv.org/abs/1312.4007)

 

[atyy]

I think if you read Rohrlich's review (the Mehra reference I gave earlier), you will recall that a non-quantized self-interaction of this type does not lead to divergences.
It's all there in the literature. In terms of the transactional picture, all such interactions are truly 'virtual' in that no real energy is exchanged in such a loop. There is no real photon involved (ie no Fock space state, and therefore no real energy). This is where the real vs virtual distinction becomes important (see e.g. http://arxiv.org/abs/1312.4007)

The 1995 Rev Mod Phys article by Hoyle and Narlikar states (p150) "Recall that the classical self-energy problem is solved in this theory by the use of advanced reaction from the rest of the universe. The problem appears in quantum field theory from the ultraviolet divergence ... In the action-at-a-distance version, the self-energy problem appears in principle ... However action-at-a-distance requires a lower cutoff ........ Neither of these cutoffs, however, reflect the global nature of the problem ... There we found that because of the event horizon in the future absorber the response is limited to frequencies up to those satisfying ..."

So in the Hoyle and Narlikar version, it is unclear if the ultraviolet cutoff can be removed, and also it is also unclear if the theory works in infinite volume flat spacetime because of the boundary conditions needed.

 

[rkastner]

The 1995 Rev Mod Phys article by Hoyle and Narlikar states (p150) "Recall that the classical self-energy problem is solved in this theory by the use of advanced reaction from the rest of the universe. The problem appears in quantum field theory from the ultraviolet divergence ... In the action-at-a-distance version, the self-energy problem appears in principle ... However action-at-a-distance requires a lower cutoff ........ Neither of these cutoffs, however, reflect the global nature of the problem ... There we found that because of the event horizon in the future absorber the response is limited to frequencies up to those satisfying ..."

So in the Hoyle and Narlikar version, it is unclear if the ultraviolet cutoff can be removed, and also it is also unclear if the theory works in infinite volume flat spacetime because of the boundary conditions needed.

I looked at the HN paper, and I don't see them using the time-symmetric propagator to characterize the self-interaction (as Davies does). They seem to be assuming that only positive energies characterize this interaction (eq 5.1). In that approach, you would still get divergences. But this is unnecessary, and I think inappropriate for the direct action approach. That is, they appear to be assuming that there is a response of the universe in such self-interaction. This assumption in my view should be questioned. In Davies' theory (which I think is the most straightfoward application of the direct-action theory to QED) the self-interaction is only via the time-symmetric propagator; there is no 'response of the universe'. That is why no energy is conveyed in the self-interaction. Again see my paper on the distinction between interactions involving absorber reponse, which gives rise to real photons (Fock space states), and those, the true virtual photons, which are only via the time-symmetric propagator, and which do not lead to exchanges of real positive energy. There is much confusion about this point in the literature, and I attempt to clarify the issues in this paper (http://arxiv.org/abs/1312.4007)

Regarding the full absorption boundary condition, the existence of charges is not equivalent to a condition on the volume of spacetime. These are two separate issues.
Thanks again for your interest in these ideas.

 

[atyy]

I looked at the HN paper, and I don't see them using the time-symmetric propagator to characterize the self-interaction (as Davies does). They seem to be assuming that only positive energies characterize this interaction (eq 5.1). In that approach, you would still get divergences. But this is unnecessary, and I think inappropriate for the direct action approach. That is, they appear to be assuming that there is a response of the universe in such self-interaction. This assumption in my view should be questioned. In Davies' theory (which I think is the most straightfoward application of the direct-action theory to QED) the self-interaction is only via the time-symmetric propagator; there is no 'response of the universe'. That is why no energy is conveyed in the self-interaction. Again see my paper on the distinction between interactions involving absorber reponse, which gives rise to real photons (Fock space states), and those, the true virtual photons, which are only via the time-symmetric propagator, and which do not lead to exchanges of real positive energy. There is much confusion about this point in the literature, and I attempt to clarify the issues in this paper (http://arxiv.org/abs/1312.4007)

Regarding the full absorption boundary condition, the existence of charges is not equivalent to a condition on the volume of spacetime. These are two separate issues.
Thanks again for your interest in these ideas.

If we are going with the Davies version, he clearly states a light tight box. At least in the classical theory, this does seem to be a finite volume requirement. I think an argument needs to be clearly presented why this is not a finite volume condition.

I haven't read Rohrlich, which is not accessible to me. But so far the more accessible versions like Narlikar and Hoyle, and Davies, state conditions like an ultraviolet cutoff and/or a light tight box.

 

[rkastner]

If we are going with the Davies version, he clearly states a light tight box. At least in the classical theory, this does seem to be a finite volume requirement. I think an argument needs to be clearly presented why this is not a finite volume condition.

I haven't read Rohrlich, which is not accessible to me. But so far the more accessible versions like Narlikar and Hoyle, and Davies, state conditions like an ultraviolet cutoff and/or a light tight box.

But we're not talking about a classical theory here. This is strictly a quantum mechanical theory with important departures from classical theory.
Also, if the universe is not a complete light tight box, then the theory is not completely equivalent to standard QED. We do not know whether the universe is a light-tight box or not. It may be 'almost' light tight, which could lead to very good empirical equivalence to the standard theory even if not identical. And if it's almost light tight but not completely, there is no concern about the boundary condition leading to a finite volume requirement.
But I understand your concern about whether the exact light-tight box condition leads to a finite volume requirement, and will look further into it.

 

[atyy]

But we're not talking about a classical theory here. This is strictly a quantum mechanical theory with important departures from classical theory.
Also, if the universe is not a complete light tight box, then the theory is not completely equivalent to standard QED. We do not know whether the universe is a light-tight box or not. It may be 'almost' light tight, which could lead to very good empirical equivalence to the standard theory even if not identical. And if it's almost light tight but not completely, there is no concern about the boundary condition leading to a finite volume requirement.
But I understand your concern about whether the exact light-tight box condition leads to a finite volume requirement, and will look further into it.

Thanks. Anyway, if I understand correctly, the claim is: Wheeler-Feynman theory provides a relativistic quantum mechanical theory that is a UV completion of standard perturbative QED (since the UV divergences are resolved), and is valid in infinite volume (Haag's theorem assumes infinite volume).

I think one reason the light box condition might have to be exact is that Davies states the theory is not unitary unless some condition like the light box condition is imposed.

 

[strangerep]

Concerning the alleged error, I think you misunderstand. The term "vacuum" in this context is the state with zero quanta, |0>, not zero energy. The ground state is indeed annihilated by the Hamiltonian defined as proportional to the number operator a(dag)a, since the eigenvalue of the number operator for |0> is zero. [...]

Like Demystifier, I was also puzzled by the paragraph near the top of p2 in your paper. You wrote:

Given the representability of H in terms of number operators, it is clear that the associated
vacuum state |0> will be annihilated by H (the eigenvalue of the state |0> with zero occupancy being zero).

Did you perhaps mean $H_F$, not $H$, here?

If not, then... why do you think the full Hamiltonian H is representable in terms of number operators? Did you perhaps mean "If we assume that H is representable in terms of number operators ..." ?

 

[bhobba]

but only if you believe its valid without a cutoff.

Also QED is just a low energy approximation to the electroweak theory so, even if you somehow avoid the infinities in QED, from the modern vantage point its not really an issue anyway.

Thanks
Bill

 

[atyy]

Also QED is just a low energy approximation to the electroweak theory so, even if you somehow avoid the infinities in QED, from the modern vantage point its not really an issue anyway.

Only for rich people. Otherwise, a non-perturbative formulation of Yang-Mills that is UV complete in infinite volume might get you some pocket money.

(See question 4 of posts #68 and #70)