Exploring Bohmian Quantum Field Attempts

[A. Neumaier]

Isn't lattice quantization of electrodynamics then not just a not suitable regularization to define QED?

Yes!

So what?

Each approximation method has its own limitations. Here it means that the usefulness of lattice methods is limited to those situations (like QCD) where one gets a nontrivial continuum limit. Other methods must (and can) be used in the remaining cases.

 

[atyy]

Isn't lattice quantization of electrodynamics then not just a not suitable regularization to define QED? So what? There are plenty of equivalent ways to define perturbative QED which is agreeing with observations. The modern understanding of QFT is anyway that it's an effective description for a limited range of energy scales. Whether or not there's a more fundamental description or not is not known yet.

Yes!

Each approximation method has its own limitations. Here it means that the usefulness of lattice methods is limited to those situations (like QCD) where one gets a nontrivial continuum limit. Other methods must (and can) be used in the remaining cases.

No, it doesn't. That is like saying a regularization must construct the continuum theory in the direction of being UV complete. But no regularization is currently known to do such a thing. The point of the lattice in QED is not to provide practical numerics, but to serve as an in principle non-perturbative definition of QED as a non-covariant quantum theory.

 

[vanhees71]

Sure, that's an open question, but obviously lattice QED is not a way to proceed.

 

[A. Neumaier]

No, it doesn't. That is like saying a regularization must construct the continuum theory in the direction of being UV complete. But no regularization is currently known to do such a thing.

Note that [URL='https://www.physicsforums.com/insights/causal-perturbation-theory/']causal perturbation theory[/URL] does not have a regulator; it works directly from the nonperturbative axioms without regularizing anything. Instead it pays detailed attention to the singularity structure to avoid any potentially faulty operation.

The construction of QED by causal perturbation theory, truncated at one loop, say, is a rigorously valid, covariant construction of a Hilbert space with field operators and a Hamiltonian dynamics.. While not completely accurate since it satisfies microcausality only to first order, it is a construction already highly successful.

The point of the lattice in QED is not to provide practical numerics, but to serve as an in principle non-perturbative definition

The principle is worth nothing since one cannot extract good predictions from it. There is no point in proposing a worthless principle.

definition of QED as a non-covariant quantum theory.

But as you can see from every textbook on QFT, QED was always defined as a covariant quantum theory.

What you propose is the definition of a demonstrably very poor lattice approximation to QED that does not even deserve the name QED.

 

[atyy]

Note that [URL='https://www.physicsforums.com/insights/causal-perturbation-theory/']causal perturbation theory[/URL] does not have a regulator; it works directly from the nonperturbative axioms without regularizing anything. Instead it pays detailed attention to the singularity structure to avoid any potentially faulty operation.

The construction of QED by causal perturbation theory, truncated at one loop, say, is a rigorously valid, covariant construction of a Hilbert space with field operators and a Hamiltonian dynamics.. While not completely accurate since it satisfies microcausality only to first order, it is a construction already highly successful.

So in the end you still don't have a covariant theory since microcausality is not satisfied.

The principle is worth nothing since one cannot extract good predictions from it. There is no point in proposing a worthless principle.

Sure, as long as you agree that quantum theory explains very little, since there are not explicit calculations for most observations.

But as you can see from every textbook on QFT, QED was always defined as a covariant quantum theory.

What you propose is the definition of a demonstrably very poor lattice approximation to QED that does not even deserve the name QED.

In fact, most modern books understand QED as an effective field theory. So it is not truly covariant.

Just to be clear, I do appreciate your criticism of lattice QED, and I do agree that QED may be asymptotically safe. What I don't agree with is that causal perturbation theory in any way solves what you criticize about lattice QED.

 

[A. Neumaier]

So in the end you still don't have a covariant theory since microcausality is not satisfied.

Causal perturbation theory is manifestly covariant. Microcausality is a condition different from covariance; It means that field operators at spacelike distance commute. It is only the rigorous combation of both covariance and microcausality that is unsolved. But lattices don't solve this either, so this cannot be counted against causal perturbation theory.

Sure, as long as you agree that quantum theory explains very little, since there are not explicit calculations for most observations.

There are explicit high accuracy calculations with high accuracy for far more than in lattice QED, where not a single result has been calculated explicitly.

What I don't agree with is that causal perturbation theory in any way solves what you criticize about lattice QED.

?

What I criticize about lattice QED is that it has not produced a single result even approximately matching one of the many successes of (perturbative) QED. It is a sterile theory, and therefore attracts essentially no research.

 

[Tendex]

The construction of QED by causal perturbation theory, truncated at one loop, say, is a rigorously valid, covariant construction of a Hilbert space with field operators and a Hamiltonian dynamics.. While not completely accurate since it satisfies microcausality only to first order, it is a construction already highly successful.

I don't think Poincaré covariance can be rigurously valid just for certain order.

 

[A. Neumaier]

I don't think Poincaré covariance can be rigorously valid just for certain order.

You think incorrectly. Maybe you should read the second edition of Scharf's book on QED to be better informed.

Any unitary representation of the Poincare group provides a model for a field theory by picking a sequence of observables ##\Phi_j## and defining the field operators by a weak limit $$\Phi(x):=\lim_{j\to\infty}U(x)\Phi_jU(-x),$$ where ##U(x)## denotes translation by ##x##. The unsolved challenge is to find a representation for which this field satisfies microcausality.

Some of the field theories in the above construction are quite trivial. For example if the representation is irreducible it describes only a single particle. However, with a sufficiently reducible representation of similar complexity as that on the Fock space over a covariant 1-particle space, one can come quite close to microcausality. Since in the free case one reaches this goal exactly by a proper choice of the ##\Phi_j## (exercise), one can use perturbation theory to deform these at any order in perturbation theory.

 

[atyy]

Causal perturbation theory is manifestly covariant. Microcausality is a condition different from covariance; It means that field operators at spacelike distance commute. It is only the rigorous combation of both covariance and microcausality that is unsolved. But lattices don't solve this either, so this cannot be counted against causal perturbation theory.

Nonetheless, it means that causal perturbation theory is not relativistic. So causal perturbation theory is basically not a theory (does not construct a relativistic theory), or if it is (by truncation), it is not relativistic.

What I criticize about lattice QED is that it has not produced a single result even approximately matching one of the many successes of (perturbative) QED. It is a sterile theory, and therefore attracts essentially no research.

It is true it attracts no research, but that is because it is not useful for numerics and we already have a good perturbative way of getting the numbers. But it is not sterile, since it is conceptually accepted (at the non-rigorous level) as a non-perturbative definition of QED, and is the basis on which research is built.

 

[A. Neumaier]

Nonetheless, it means that causal perturbation theory is not relativistic.

According to your ideosyncratic notion of relativity, the Dirac equation would also not be relativistic.

Causal perturbation theory is relativistic because it is covariant under space-time translations and Lorentz transformations.

it is not sterile, since it is conceptually accepted (at the non-rigorous level) as a non-perturbative definition of QED, and is the basis on which research is built.

Which research? Just a few papers on triviality... Lattice QED is not a definition of QED but a mock definition of a poor approximation of QED. Only you and a few others accept it as a non-perturbative definition of QED.

But all textbooks on QFT define QED (at a nonrigorous level) in terms of a covariant description.

 

[atyy]

According to your ideosyncratic notion of relativity, the Dirac equation would also not be relativistic.

Causal perturbation theory is relativistic because it is covariant under space-time translations and Lorentz transformations.

How can one violate microcausality and still be relativistic?

Which research? Just a few papers on triviality... Lattice QED is not a definition of QED but a mock definition of a poor approximation of QED. Only you and a few others accept it as a non-perturbative definition of QED.

But all textbooks on QFT define QED (at a nonrigorous level) in terms of a covariant description.

There is research to try to build a lattice standard model. There are open problems, but fundamental conceptual problems with lattice QED are not among them.
https://saalburg.aei.mpg.de/wp-content/uploads/sites/25/2017/03/wiese.pdf
https://arxiv.org/abs/0912.2560
https://arxiv.org/abs/1305.1045

 

[A. Neumaier]

How can one violate microcausality and still be relativistic?

How does the relativistic Diirac equation satisfy microcausality?

There is research to try to build a lattice standard model. There are open problems, but fundamental conceptual problems with lattice QED.

Thus there is no research about lattice QED, in spite, or because, of its known poor approximation properties.

This thread is not about the standard model.

 

[atyy]

How does the relativistic Diirac equation satisfy microcausality?

Are you saying that causal perturbation theory truncated at one loop does not satisfy microcausality, yet it is a relativistic quantum theory?

 

[A. Neumaier]

Are you saying that causal perturbation theory truncated at one loop does not satisfy microcausality, yet it is a relativistic quantum theory?

Yes. I had stated this explicitly in post #41.

 

[atyy]

Yes. I had stated this explicitly in post #41.

And the Dirac equation violates microcausality?

 

[vanhees71]

But usual renormalized perturbation theory leads to a unitary Poincare-covariant S-matrix, fulfilling the linked-cluster principle (order by order).

I always understood that the Epstein-Glaser approach as in Scharff's book is equivalent to the usual perturbation theory using standard counterterm subtraction, and it's also called "causal". It's also in a sense very physical since it "smears" the distribution valued field operators, introducing a scale, which is necessary to define QFT as an effective theory in the first place.

QED may or may not be plagued by a Landau pole, but as an effective theory at least up to the energies available today in experiments it's still among the most successful theories ever.

 

[Tendex]

How does the relativistic Diirac equation satisfy microcausality?

In QFT the equation obtains Dirac fields that anticommute for spacelike intervals, how is this not satisfying microcausality?

 

[A. Neumaier]

And the Dirac equation violates microcausality?

Of course. The spacetime dependence of arbitrary operators ##A## is as given in my post #43 with the constant sequence ##\Phi_j=A##. (The limit is only needed to allow for distribution-valued fields, which cannot be obtained with constant sequences.)

Thus there are many quantum field theories that are just covariant, i.e., relativistic. Since the construction can be applied to any unitary representation of the Poincare group, it produces covariant fields from the positive energy sector of the Dirac equation.

Microcausality (a better name is microlocality) is the highly restrictive additional condition (required by the Wightman axioms) that the resulting operators commute at spacelike arguments. This requires a highly redundant representation, the positive energy sector of the Dirac equation is far too small.

Fock spaces over a covariant 1-particle Hilbert space wih their standard representation of the Poincare group give examples; they correspond to quasi-free QFTs. In the 4-dimensional case, these are the only presently known examples.

 

[A. Neumaier]

In QFT the equation obtains Dirac fields that anticommute for spacelike intervals, how is this not satisfying microcausality?

I talked about the (physical, positive energy sector) of the Dirac equation as counterexample to atyy's claim that not satisfying microcausality should imply not being relativistic. The Dirac equationis a covariant, hence relativistic equation for a 1-particle Hilbert space. It gives a covariant QFT without microcausality by the construction in post #43. The fields obtained are not the Dirac fields of the textbooks since the 1-particle Hilbert space is far too small.

On the other hand, the Dirac fields you talk about are fields over its (much bigger) Fock space. They satisfy microcausality.

 

[A. Neumaier]

I always understood that the Epstein-Glaser approach as in Scharff's book is equivalent to the usual perturbation theory using standard counterterm subtraction

... in the BPHS version, yes. But a lot of time passed since Epstein and Glaser.

Today's causal perturbation theory not only constructs (perturbatively) the S-matrix but also the field operators; see the second edition of Scharf's QED book. This is something techniques based on counterterms cannot achieve since they start with the ill-defined Dyson expansion and lose the connection with the Hilbert space and the operators. Thus they only get the S-matrix.

 

[vanhees71]

I talked about the (physical, positive energy sector) of the Dirac equation as counterexample to atyy's claim that not satisfying microcausality should imply not being relativistic. The Dirac equationis a covariant, hence relativistic equation for a 1-particle Hilbert space. It gives a covariant QFT without microcausality by the construction in post #43. The fields obtained are not the Dirac fields of the textbooks.

On the other hand, the Dirac fields you talk about are fields over its Fock space. They satsfy microcausality.

But that's why we use a field quantization and solve the problem of modes with negative frequency by writing a creation instead of an annihilation operator in front of the corresponding mode decomposition, and that's why in the standard construction of a local and microcausal QFT we have particles and antiparticles with positive energy. That's also so in Scharf's book, though hidden in quite unusual and overcomplicated looking notation, but I guess that's the price you have to pay for "more rigor".

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?

 

[A. Neumaier]

in the standard construction of a local and microcausal QFT we have particles and antiparticles with positive energy.

The problem is that the standard construction only produces free (or quasifree) fields.

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.)

That's also so in Scharf's book, though hidden in quite unusual and overcomplicated looking notation, but I guess that's the price you have to pay for "more rigor".

If you have access to the second edition, I can give you an Ariadne's thread through the book so that you recognize the standard stuff. Then you'll be able to profit from the remainder.

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?

It proceeds entirely covariantly from the scratch without ever introducing physically meaningless bare constants or cutoffs that would have to be remedied by subtractions. The mass and charge appearing is always the physical mass and charge of the electron. This reduces the confusion generated by the traditional approach.

But more importantly, the formalism constructs not only the S-matrix but, as mentioned already, also all other machinery one usually finds in quantum theory, which is completely lost in the traditional approach. This is a big plus even from a nonrigorous point of view.

 

[atyy]

I talked about the (physical, positive energy sector) of the Dirac equation as counterexample to atyy's claim that not satisfying microcausality should imply not being relativistic. The Dirac equationis a covariant, hence relativistic equation for a 1-particle Hilbert space. It gives a covariant QFT without microcausality by the construction in post #43. The fields obtained are not the Dirac fields of the textbooks since the 1-particle Hilbert space is far too small.

On the other hand, the Dirac fields you talk about are fields over its (much bigger) Fock space. They satisfy microcausality.

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

 

[vanhees71]

It proceeds entirely covariantly from the scratch without ever introducing physically meaningless bare constants or cutoffs that would have to be remedied by subtractions. The mass and charge appearing is always the physical mass and charge of the electron. This reduces the confusion generated by the traditional approach.

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.

One merit of Scharf's approach of course is that the "smearing" is a quite physically motivated strategy, though in this book it's somewhat hidden under a lot of formalism. Of course from a mathematical point of view this is necessary to have a rigorous foundation of perturbative QFT.

 

[atyy]

It proceeds entirely covariantly from the scratch without ever introducing physically meaningless bare constants or cutoffs that would have to be remedied by subtractions. The mass and charge appearing is always the physical mass and charge of the electron. This reduces the confusion generated by the traditional approach.

So is the causal perturbation theory philosophy different from the Wilsonian viewpoint that maybe QED is just an effective field theory, and the cutoff is meaningful?

It seems to me that if you don't want a cutoff, then this is pushing towards asymptotic safety of QED.

 

[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?