A Can Bohmian Mechanics Effectively Address Relativistic Quantum Field Theories?

[A. Neumaier]

Well, this you have also in the traditional approach working with counter-terms order by order in perturbation theory. At any step you work with finite parameters in a manifestly covariant formalism.

But you still have the counterterms, which are ill-defined, physically meaningless constants just introduced to ultimately arrive at a finite result.

Moreover, the traditional approach does not produce field the operators, except in the free case.

 

[Tendex]

Of course from a mathematical point of view this is necessary to have a rigorous foundation of perturbative QFT.

But we are back to how it is basically irrelevant or at least moot to talk about rigor in perturbative QFT in the absence of a theory that it supposedly approximating(I know for A. Neumaier that is too much to ask to a theory and only purists care about this minutia but others find it an important detail), or what you said:

I still don't see the real merit of even this "rigor" since it also has not yet lead to a construction of a non-perturbative interacting mathematically rigorous QED. So what's the point to use this overcomplicated formalism in the first place?

I must anyway admit that I fail to see in what way a lattice QED could improve the theoretical situation.

 

[A. Neumaier]

Gosh, I mean usually we do talk about the Dirac equation as a field equation.

Who is 'we'?? Wikipedia discusses in all but 3 lines of its article on the Dirac equation properties of the single-particle Dirac equation. This is the standard usage.

 

[Tendex]

Wikipedia discusses in all but 3 lines of its article on the Dirac equation properties of the single-particle Dirac equation. This is the standard usage.

All QFT textbooks I've seen devote some paragraphs at the beginning to explain why the one-particle approach doesn't work, precisely because it violates relativistic causality, so the use of a reduced one-particle Fock space is even in principle not a good idea, and insisting on it seems to even doom a so called "causal" approach to QED.

 

[A. Neumaier]

All QFT textbooks I've seen devote some paragraphs at the beginning to explain why the one-particle approach doesn't work, precisely because it violates relativistic causality, so the use of a reduced one-particle Fock space is even in principle not a good idea, and insisting on it seems to even doom a so called "causal" approach to QED.

Half the Hilbert space of the Dirac equation works perfectly. It describes exactly a free relativistic electron. This is sufficient to serve as a counterexample for atyy's claim that lack of microcausality implies lack of being relativistic.

If you dismiss the specific example of the Dirac equation as unphysical, I can point you instead to a survey by Keister and Polyzou. They discuss at length covariant models for few particle systems matching experimental data. Each of these serves as another, more physical counterexample.

 

[Tendex]

Half the Hilbert space of the Dirac equation works perfectly. It describes exactly a free relativistic electron. This is sufficient to serve as a counterexample for atyy's claim that lack of microcausality implies lack of being relativistic.

If you dismiss the specific example of the Dirac equation as unphysical, I can point you instead to a survey by Keister and Polyzou, who discusses at length covariant models for few particle systems matching experimental data. Each of these serves as another, more physical counterexample.

We were discussing QED that I believe is still a field theory. The particle quantum mechanics was dropped as a serious theory some time ago which doesn't prevent it from being used as toy model to obtain perturbative physical approximative results. Non-relativistic quantum mechanics or even Newtonian mechanics is still also valid for those purposes.

Whether to include or not microcausality in the definition of relativistic is more of a semantic problem. Certainly for quantum fields, "in my book", it is included.

 

[A. Neumaier]

We were discussing QED that I believe is still a field theory. The particle quantum mechanics was dropped as a serious theory some time ago which doesn't prevent it from being used as toy model to obtain perturbative physical approximative results. Non-relativistic quantum mechanics or even Newtonian mechanics is still also valid for those purposes.

Whether to include or not microcausality in the definition of relativistic is more of a semantic problem. Certainly for quantum fields, "in my book", it is included.

In this narrow sense, relativistic quantum field theory is currently restricted to quasifree theories and to spacetimes of dimension less than 4. This means that according to "your book" the material beyond the introductory chapters in all textbooks on relativistic quantum field theory is not relativistic - a strange choice of semantics.

The Wikipedia article on quantum field theory discusses almost exclusively the relativistic case, reflecting the typical content of the textbooks on relativistic quantum fields. It does not even mention the property of microcausality, showing how peripheral this notion is to the theory and practice of relativistic quantum fields.

Usually, a system or theory is called relativistic if it is based on a space with a Lorentzian metric satisfying local Lorentz covariance. A quantum field is an distribution valued operator. Combining the two, a relativistic quantum field theory is about quantum fields transforming covariantly. The examples given in post #43 are relativistic quantum fields in this sense.

Adding the requirement of microcausality gives the special class of local relativistic quantum field theories, of which only free ones are known in 4 spacetime dimensions. Thus microcausality may be considered to be a property at which relativistic QFTs may aim at, but hardly as a requirement for a quantum field to be relativistic.

 

[atyy]

On the other hand, Scharf constructs interacting fields perturbatively on the asymptotic Fock space. This cannot work nonperturbatively since Haag's theorem forbids it, but it works perturbatively order by order, and produces covariant field operators satisfying microcausality up to any desired order. For QED and order 6 the violation of microcausality is less than today's experimentally achievable accuracy. (Divergence of the asymptotic series is expected to begin only at an order of around $\alpha^{-1}\approx 137$, which will never be probed by experiment since it would require resources larger than the whole universe.)

So if you truncate at any order, do you always get a quantum theory with well defined Hilbert space and Hamiltonian dynamics?

 

[A. Neumaier]

So if you truncate at any order, do you always get a quantum theory with well defined Hilbert space and Hamiltonian dynamics?

Yes. In perturbation theory (only), the Hilbert space remains a Fock space - the asymptotic Fock space. This works fine in QED since there are no stable composite particles. The Hamiltonian is given by the generator of time translations in the standard representation of the Poincare group on Fock space. The field operators are deformations of the free field operators to the required order transforming covariantly under this representation. This accounts for the nontrivial interactions.

 

[atyy]

Yes. In perturbation theory (only), the Hilbert space remains a Fock space - the asymptotic Fock space. This works fine in QED since there are no stable composite particles. The Hamiltonian is given by the generator of time translations in the standard representation of the Poincare group on Fock space. The field operators are deformations of the free field operators to the required order transforming covariantly under this representation. This accounts for the nontrivial interactions.

Let's suppose the series starts to diverge after about 137 terms. Does the ability to construct a quantum theory at a given truncation still hold even if we truncate at sat 300 terms?

Does Haag's theorem not apply because microcausality is not satisfied for truncated series?

 

[A. Neumaier]

Let's suppose the series starts to diverge after about 137 terms. Does the ability to construct a quantum theory at a given truncation still hold even if we truncate at say 300 terms?

Yes, at order 300, causal perturbation QED still is a family of covariant quantum field theories, and for a fine structure constant $\alpha\ll 1/300$ it would still produce very accurate approximations to the putative local covariant QED with these values of $\alpha$. But for the physical value of the fine structure constant, the perturbative error will probably be already very large, so that the resulting theory no longer resembles QED. On the other hand, partially resummed versions might perform more adequately, since resummation is a partially nonpertubative process.

It is like the asymptotic series for the exponential integral, which approximates the exponential integral well at any order for sufficiently small $z^{-1}$ (the smaller the higher the order). But for fixed $z$, the series has larger and larger approximation errors when the order grows beyond some $z$-dependent threshold.

Does Haag's theorem not apply because microcausality is not satisfied for truncated series?

Yes. Assuming covariance, locality is equivalent to microcausality, and Haag assumes a local covariant quantum field theory to derive his theorem.

 

[vanhees71]

But you still have the counterterms, which are ill-defined, physically meaningless constants just introduced to ultimately arrive at a finite result.

Moreover, the traditional approach does not produce field the operators, except in the free case.

In the counterterm approach you work with finite quantities all the time. The divergences are subtracted, given a renormalization scheme. Using direct subtraction of the perturbative integrals over loop momenta as in the BPHZ approach you don't even need an intermediate regularization, though the use of an as symmetries-conserving regulator as possible like dim. reg. is very convenient to organize the calculation.

The finite parameters of the theory, coupling constants and masses, have to be fitted to observations. I don't think that you can predict the finite couplings and masses from any QFT only using the regulators of "causal perturbation theory", or the Epstein-Glaser approach, but they are phenomenological constants. In QED it's the mass of the electron/positron and the coupling constant.

I don't know, what you mean by your last sentence. Do you mean you define field operators of the interacting theory rigorously? If so, what else can you do with them than calculating the N-point functions?

 

[A. Neumaier]

I don't think that you can predict the finite couplings and masses from any QFT only using the regulators of "causal perturbation theory", or the Epstein-Glaser approach, but they are phenomenological constants. In QED it's the mass of the electron/positron and the coupling constant.

Yes, they are numerical parameters of the theory that can take any positive value. The physically realized case is just one of these.

I don't know, what you mean by your last sentence. Do you mean you define field operators of the interacting theory rigorously? If so, what else can you do with them than calculating the N-point functions?

Causal perturbation theory rigorously constructs field operators that agree to a fixed order with those of the putative local interacting theory. This is like rigorously constructing an asymptotic series for the exponential integral. It is not a full construction but it is what perturbation theory can do. For QED at experimentally accessible accuracies, this is more than sufficient.

what else can you do with them than calculating the N-point functions?

BPHZ and dimensional regularization are restricted to scattering theory and only calculate time-ordered vacuum N-point functions, not even the Wightman N-point functions.

With field operators one could also calculate the latter, and thus gets more information. Moreover, one can calculate in principle N-point functions in arbitrary states, not only the vacuum state.

More generally, one can do with field operators whatever one can do with operators in a quantum theory. Though at present people don't utilize these possibilities, there is no theoretical obstruction for doing so. Calculating expectations of products of position operators at various times - the analogue of N-point functions - is not the major use of ordinary quantum theory; many other interesting things are investigated there. Thus there appears to be much scope for expanding the present boundary of QFT in an analogous way. It might be more rewarding than pursuing the stalled quest for extending QFT to a theory of quantum gravity.

 

[vanhees71]

Ok, I was talking about "vacuum QFT" only, but as you well know, one can extend the formalism to many-body theory. The most simple case is equilibrium theory, where you can use all the (non-rigorous) formalism of vacuum QFT in the Matsubara (imaginary-time) formalism of thermal QFT. The only addition are the KMS conditions (periodicity/antiperiodicity of bosonic/fermionic field operators). There you can show that perturbative renormalization is possible with only vacuum counterterms.

For non-equilibrium we have the real-time contour (Schwinger-Keldysh) technique. Here it's not so clear to me, whether one can show that you can always renormalize everything with vacuum counterterms only.

Does the more rigorous Epstein-Glaser approach help there somehow?

 

[A. Neumaier]

Ok, I was talking about "vacuum QFT" only, but as you well know, one can extend the formalism to many-body theory. The most simple case is equilibrium theory, where you can use all the (non-rigorous) formalism of vacuum QFT in the Matsubara (imaginary-time) formalism of thermal QFT. [...]
For non-equilibrium we have the real-time contour (Schwinger-Keldysh) technique. Here it's not so clear to me, whether one can show that you can always renormalize everything with vacuum counterterms only.
Does the more rigorous Epstein-Glaser approach help there somehow?

This is an open question. There are very few papers on causal perturbation theory at nonzero temperature and beyond; the following might be the complete list.

• https://arxiv.org/pdf/1906.04098.pdf
• https://inis.iaea.org/collection/NCLCollectionStore/_Public/44/096/44096564.pdf
• https://etd.library.vanderbilt.edu/available/etd-03222019-221902/unrestricted/parvizi.pdf
• https://arxiv.org/pdf/1809.08557.pdf
• https://arxiv.org/pdf/1609.01124.pdf
• https://arxiv.org/pdf/1306.6519.pdf
• https://arxiv.org/pdf/0710.5652.pdf
• https://cds.cern.ch/record/624397/files/0306169.pdf

Some of them look quite interesting, but I haven't studied these papers in depth, so cannot comment on their quality.

 

[atyy]

Yes, at order 300, causal perturbation QED still is a family of covariant quantum field theories, and for a fine structure constant $\alpha\ll 1/300$ it would still produce very accurate approximations to the putative local covariant QED with these values of $\alpha$. But for the physical value of the fine structure constant, the perturbative error will probably be already very large, so that the resulting theory no longer resembles QED. On the other hand, partially resummed versions might perform more adequately, since resummation is a partially nonpertubative process.

So let's take $\alpha \approx 1/137$ in the "true" theory to which we think causal perturbation theory produces an approximation. The difference between causal perturbation theory and the true theory gets smaller and smaller up to around 137 terms, then it gets bigger and bigger. How about microcausality of the theory we get by successively including more and more terms - does it also get better and better, then worse and worse (internal to the theory, not by comparison to the "true" theory)?

 

[A. Neumaier]

The difference between causal perturbation theory and the true theory gets smaller and smaller up to around 137 terms, then it gets bigger and bigger. How about microcausality of the theory we get by successively including more and more terms - does it also get better and better, then worse and worse (internal to the theory, not by comparison to the "true" theory)?

Yes because of Haag's theorem, it should diverge. That's why perturbation theory doesn't provide a full answer to the construction of a local QFT.

 

[atyy]

One of the reasons we discuss lattice QED is to provide a well-defined high (but not infinitely high) energy theory, to which the usual predictions of perturbative QED arise as excellent low energy approximations.

It seems we could simply take causal perturbation theory truncated at the 137th term, as the definition of the full quantum theory for that purpose? It would be non-relativistic (by my definition), but that would be fine.

Could we fit together the causal perturbation theory and effective field theory viewpoints?

 

[Tendex]

That's why perturbation theory doesn't provide a full answer to the construction of a local QFT.

And physical measurements require local theories. At least in principle. So perturbation theory can only give effective theories in the sense of condensed matter physics. This was already the case even with Dyson-Feynman perturbation theory.

 

[Auto-Didact]

It seems we could simply take causal perturbation theory truncated at the 137th term, as the definition of the full quantum theory for that purpose? It would be non-relativistic (by my definition), but that would be fine.

No, this would merely give a 'best approximation' to the full theory; these 'best approximations' can be relatively easily improved upon if one employs more sophisticated techniques from asymptotic analysis. This works in essentially the same sense of being able to achieve a nice statistical curve fit to some dataset ('nice' purely because the regression method is conventional) which from a mathematical viewpoint is actually hopelessly underfit, or worse, not even wrong.

To any experimentally-oriented physicist the whole argument that 'such an approximation should be taken as true, because in practice an $\alpha^{-1} \approx 137$ will never be probed by experiment since it would require resources larger than the whole universe' sounds very convincing and therefore is quite tempting. Unfortunately the argument is not only unscientific for trying to censor what can be determined experimentally, but actually self-defeating as well by conflating what might be true in practice for what must be true in principle, i.e. mistakes contingency for necessity.

I have said this before and I will say it again: an approximation is de facto not the same thing as the unapproximated thing; equivocating these two distinct things based on the indistinguishability at some level of precision between the two is just logically inconsistent reasoning. Some may say that this as an awefully harsh viewpoint to take, but it is merely remaining sober without resorting to exaggeration; more importantly, accepting the facts as they are seems to directly illuminate the right path forward.

Making the acknowledgment almost forces us into a position to make some analogy which tells us that the two can only become equivalent in some specific limit, based on our experience, familiarity and intuition of having dealt with similar problems before. As the history of mathematics and mathematical physics has shown countless times, if such a limit exists and is unique, actually finding it will revolutionize both physics - by suggesting experiments currently undreamed of - as well as revitalizing old withering branches of mathematics by bestowing upon them new fruits; it should go without saying that this is far too much to give up for essentially a 20th century version of epicycles.

 

[A. Neumaier]

One of the reasons we discuss lattice QED is to provide a well-defined high (but not infinitely high) energy theory, to which the usual predictions of perturbative QED arise as excellent low energy approximations.

It seems we could simply take causal perturbation theory truncated at the 137th term, as the definition of the full quantum theory for that purpose? It would be non-relativistic (by my definition), but that would be fine.

We can take causal perturbation theory for QED truncated at the 10th term ad being exact FAPP. All its predictions are equally valid in all Lorentz frames. It is fully covariant but still slightly nonlocal.

Calling it nonrelativistic is a gross misnomer, no matter what your definition is.

Could we fit together the causal perturbation theory and effective field theory viewpoints?

Nature does not follow QED exactly; deviation (due to the non-QED form factors of nuclei) are already relevant at order 4. Thus as far as nature is concerned, QED is only an effective theory. The same holds for all our QFTs as long as gravity is not included.

But QED is a well-defined (though at present only perturbatively constructed) local QFT. Is meaning is rigorously defined through the through the Bogoliubov axioms, the starting point of causal perturbation theory and stated in my Insight article on it. This definition is completely independent of its status as an effective field theory in nature.

 

[A. Neumaier]

And physical measurements require local theories. At least in principle.

Why shoud they require locality?

N-particle quantum mechanics is nonlocal but successfully survived very thorough testing by a huge amount of physical measurements.

 

[atyy]

We can take causal perturbation theory for QED truncated at the 10th term ad being exact FAPP. All its predictions are equally valid in all Lorentz frames. It is fully covariant but still slightly nonlocal.

Calling it nonrelativistic is a gross misnomer, no matter what your definition is.

Where in Scharf's book would you recommend reading for understanding how it is that everu truncation of the series has a well-defined Hilbert space and Hamiltonian dynamics?

Nature does not follow QED exactly; deviation (due to the non-QED form factors of nuclei) are already relevant at order 4. Thus as far as nature is concerned, QED is only an effective theory. The same holds for all our QFTs as long as gravity is not included.

But QED is a well-defined (though at present only perturbatively constructed) local QFT. Is meaning is rigorously defined through the through the Bogoliubov axioms, the starting point of causal perturbation theory and stated in my Insight article on it. This definition is completely independent of its status as an effective field theory in nature.

How it can be well-defined if it is only perturbatively constructed - since the entire series does not have physical meaning - only truncations of the series have physical meaning. But presumably the Bogoliubov axioms would be consistent only with the whole series, not any truncation of the series.

 

[A. Neumaier]

How it can be well-defined if it is only perturbatively constructed - since the entire series does not have physical meaning - only truncations of the series have physical meaning.

In the same way as the Navier-Stokes equations are well-defined though the construction of global solutions is an open (millenium) problem. To rigorously define a concept doesn't entail having to construct it.

In the version used by Scharf and stated in my Insight article, the Bogoliubov axioms never refer to a series, only to nonperturbative objects.

Where in Scharf's book would you recommend reading for understanding how it is that every truncation of the series has a well-defined Hilbert space and Hamiltonian dynamics?

I'd suggest that you read my Insight article, in particular the just updated section. Then ask questions there, since the present thread should be about lattice QED, not about causal perturbation theory.

 

[Tendex]

Why shoud they require locality?

N-particle quantum mechanics is nonlocal but successfully survived very thorough testing by a huge amount of physical measurements.

Not referring to perturbative patches here, surely they give good results . I suggest you to take a look at "Local quantum physics" by R. Haag to understand why they require locality in the quantum field context.

 

[vanhees71]

Indeed, relativistic QFT is a local theory in principle. I still don't understand the argument, in which sense the bread-and-butter perturbative evaluation of the (mathematically not rigorously defined) theory should violate locality. At least the measurable consequences do not contradict locality: The S-matrix is Lorentz covariant covariant as it should be and it fulfills the linked cluster principle, which is a direct consequence of locality/microcausality assumption. At least to an overwhelmingly good approximation the usual physicists' treatment fulfills the locality assumption.

One should not think that this contradicts inseparability as described by entanglement of far-distant parts of a quantum system (e.g., entangled photon pairs produced in parametric downconversion and used for highly accurate "Bell tests"). This is of course also contained (and in fact an inevitable consequence) of any QT, and relativistic QFT is not an exception.

Already at the classical level in relativistic physics local field theories are the ones that work best. Even an apparently so simple a concept of a classical point particle has its (only partially solved!) problems, as the notorious trouble with radiation reaction already in electrodynamics shows.

Nonlocal theories seem not to be successful, at least I don't know any particular one that is used in contemporary physics.

 

[A. Neumaier]

Not referring to perturbative patches here, surely they give good results . I suggest you to take a look at "Local quantum physics" by R. Haag to understand why they require locality in the quantum field context.

I know the book quite well, but it covers in 4D so far only putative theories, not constructed ones. As a desirable rigorous goal, locality is interesting but not yet achieved in 4 dimensions, hence at present of very limited usefulness.

But by no means is it a requirement for doing successful modeling. Approximate locality is enough, and that's what is achieved by finite order perturbation theory.

I still don't understand the argument, in which sense the bread-and-butter perturbative evaluation of the (mathematically not rigorously defined) theory should violate locality.

Truncation at finite order makes locality also valid only to this order.

 

[vanhees71]

Well, that's the prize you have to pay as long as there's no applicable way to find exact solutions. AFAIK there's not even a complete understanding, whether such a thing exists at all.