A Does QED Originate from Non-Relativistic Systems?

[dextercioby]

To be precise, there is no known one. This is a small but very important difference.

The relevant particle physics as we know strongly relies on the (universal cover of the) restricted Poincare group, this is - as per Wikipedia's listing - involved in the W0, W1, W2 and W3 axioms, thus in all of them. We're pretty sure that there's no other theoretical model other than the SM of particle physics to describe the 3 currently relevant interactions at particle level. We can't do the following things:

- Check that the Standard Model fields obey the Wightman axioms.
- Replace Wightman axioms with another set of axioms in agreement with special relativity and check that the Standard Model fields obey the new set. We can replace them with the Haag-Ruelle ones, but they are equivalent, not better.
- Replace the Minkowski spacetime of special relativity (known to be superseded by the general theory of relativity) and recast Wightman's axioms in terms of a curved spacetime background.

 

[A. Neumaier]

to what should such a model lead. I always thought the microcausality condition is very important to get a Poinare covariant S-matrix?

There is a tension between the mainstream and what atyy argues.

atyy argues on purist grounds that a theory that doesn't satisfy the Wightman axioms isn't even quantum, so I countered (though in an answer to dextercioby's remark) that there are lots of quantum field theories that don't satisfy all Wightman axioms but are as quantum as his lattice theories.

You argue the mainstream theme, and then there it is no question that QED is a quantum theory and that the Wightman axioms are satisfied approximately, formally to infinite order and in practice to a few loops accuracy. This is enough in practice for highly accurate results.

The reconciliation is that the few-loop theories satisfy W3 approximately (satisfying you). But they violate W3 when taken as exact statement.

Indeed, the fields constructed in few-loop QED are only approximately causal because causality is violated at the first neglected order. atyy uses this as argument to claim that 1-loop QED is not a quantum theory. But although it does not satisfy W3 it satisfies W3 to 1-loop. Pointing out that W3 is not necessary for the construction of the Hilbert space therefore refutes atyy's ridiculous position

that without either lattice or a theory satisfying the Wightman axioms (or equivalent), QED cannot be said to be a quantum theory.

 

[A. Neumaier]

The relevant particle physics as we know strongly relies on the (universal cover of the) restricted Poincare group, this is - as per Wikipedia's listing - involved in the W0, W1, W2 and W3 axioms, thus in all of them. We're pretty sure that there's no other theoretical model other than the SM of particle physics to describe the 3 currently relevant interactions at particle level.

If we want to achieve the same level of agreement with experiment with respect to all these interactions then you may be right.

However, QED does not aim to describe more than the electromagnetic interaction. It only claims to be a quantum field theory decribing the electromagnetic interaction. Therefore it is known not to be the correct physical theory. But it is still a theory that one can investigate in itself, and many quantum field texts do it. They do it without any recourse to the Wightman axioms, which are simply irrelevant for QED as practices by the main stream in all textbooks on quantum field theory. Thus one cannot take (as atyy wants to have it) the Wightman axioms as the criterion that decides whether something in quantum field theory is quantum physics.

Even the mathematical physicists don't think that the Wightman axioms are the right framework for describing gauge theories like QED or QCD. They even replace the Hilbert space by a Krein space with an indefinite inner product - a horror that puts their efforts outside the quantum realm in atyy's strange world. But everything is still Poincare invariant!

Moreover, indefinite inner products are also needed for QCD, which atyy thinks exists as a Poincare invariant quantum field theory; thus his position is self-contradictory.

 

[vanhees71]

I also don't understand, why postings where deleted :-(.

 

[A. Neumaier]

We can't do the following things: [...] Replace Wightman axioms with another set of axioms in agreement with special relativity and check that the Standard Model fields obey the new set.

This is a very strong claim. I'd like to see your proof for this no-go theorem. I don't think you have more than a mere opinion in support of this!

 

[A. Neumaier]

I also don't understand, why postings where deleted :-(.

I do, though I haven't seen their content. But I have seen enough of R-M's posts to guess what kind of stuff it contained. None of his posts in this thread was constructive; all went of on irrelevant sidelines. Some of them apparently so much that the mentors found them obviously off-topic.

 

[vanhees71]

If we want to achieve the same level of agreement with experiment with respect to all these interactions then you may be right.

However, QED does not aim to describe more than the electromagnetic interaction. It only claims to be a quantum field theory decribing the electromagnetic interaction. Therefore it is known not to be the correct physical theory. But it is still a theory that one can investigate in itself, and many quantum field texts do it. They do it without any recourse to the Wightman axioms, which are simply irrelevant for QED as practices by the main stream in all textbooks on quantum field theory. Thus one cannot take (as atyy wants to have it) the Wightman axioms as the criterion that decides whether something in quantum field theory is quantum physics.

Even the mathematical physicists don't think that the Wightman axioms are the right framework for describing gauge theories like QED or QCD. They even replace the Hilbert space by a Krein space with an indefinite inner product - a horror that puts their efforts outside the quantum realm in atyy's strange world. But everything is still Poincare invariant!

Moreover, indefinite inner products are also needed for QCD, which atyy thinks exists as a Poincare invariant quantum field theory; thus his position is self-contradictory.

The aim of HEP theory (phenomenology) nowadays is to find "physics beyond the standard model". Besides the research on the high-energy frontier, one way is high-precision physis at low energies. Among them are naturally many that are QED in leading order. One example is the anomalous magnetic moment of the muon, where there seems to be a slight discrepancy between the SM predictions and the measured value (but it's below the $5 \sigma$ confidence level). At the moment, as far as I know, the problem is theory, and it is indeed the contribution from QCD corrections (or hadronic corrections if you work with effective hadronic models). It's of course not pure QED in the sense of 1948, but it's still QED. At the present demand of accuracy pure QED is simply not enough, and you need the corrections from all known interactions in the SM to adequately confront theory with experiment to maybe find "physics beyond the SM" or to find just another confirmation of it.

Of course, this thread is about a purely theoretical/mathematical issue, and I think it's fair to say that the issue is not settled. There is no mathematically strict foundation of QED and also surely not the SM as a whole. I think, everything has been said what can be said in this direction at the moment.

 

[rubi]

I think what atyy means when he says that QED is not a quantum theory is the following: A quantum theory usually requires Hilbert space over the complex numbers $\mathbb C$. If you have a QFT that can compute n-point functions that satisfy a certain positivity condition (you don't even need the full Wightman axioms), you can reconstruct the Hilbert space from them. However, in perturbative QFT, you don't get true n-point functions, but only n-point functions defined by a formal power series. If you only take the first few orders, it is unlikely that the functions will satisfy the positivity conditions and hence, you can't reconstruct a Hilbert space. If you take the full formal power series, you may get (something like) a Hilbert space, but over the ring of formal power series $\mathbb C\left[\left[e^2\right]\right]$ instead of the complex numbers $\mathbb C$. It's certainly debatable, whether this still counts as a bona fide quantum theory.

 

[A. Neumaier]

If you only take the first few orders, it is unlikely that the functions will satisfy the positivity conditions and hence, you can't reconstruct a Hilbert space.

Formally, you are perhaps right. But the slight errors can be accommodated in the same way as small errors in numerical computations. If one evaluates numerically a matrix that in exact arithmetic would be positive definite it can well be numerically indefinite, but by adding a tiny diagonal correction one can make it positive definite. Of course, one can do the same with the kernels of integral operators on a function space, and hence with approximate Wightman functions (which are a generalization of these), and it can be done in a covariant way with corrections of the first neglected order. Only W3 cannot be maintained, so locality isn't guaranteed.

Thus finite-loop QED is simply a slightly nonlocal approximation to the fully local QED, which I believe exists but whose existence has not yet been settled.

The deviations form nonlocality can be made small to very high order. The loop expansion of QED is believed to improve in accuracy until loop order around 137. As vanhees71 just remarked, the corrections due ot the presence of fields not included in the textbook QED framework make QED an inaccurate theory already at loop order 5 or 6, so that it is pointless from a physical point of view to require locality to be correct to higher order.

All this is independent of the question whether a local covariant QED exists in a mathematically rigorous sense. This problem is unsettled but doesn't influence the statements one can make at any finite loop order.

 

[rubi]

Formally, you are perhaps right. But the slight errors can be accommodated in the same way as small errors in numerical computations. If one evaluates numerically a matrix that in exact arithmetic would be positive definite it can well be numerically indefinite, but by adding a tiny diagonal correction one can make it positive definite. Of course, one can do the same with the kernels of integral operators on a function space, and hence with approximate Wightman functions (which are a generalization of these), and it can be done in a covariant way with corrections of the first neglected order. Only W3 cannot be maintained, so locality isn't guaranteed.

Thus finite-loop QED is simply a slightly nonlocal approximation to the fully local QED, whose existence has not yet been settled.

The deviations form nonlocality can be made small to very high order. The loop expansion of QED is believed to improve in accuracy until loop order around 137.

I agree that up to order around 137, the approximate Wightman functions are so good that they supposedly almost satisfy the positivity conditions up to small errors. But after that, the errors become bigger and a small correction won't suffice. But since nobody ever computes QED to such orders, you are right that one can probably reconstruct some approximate Hilbert space with tiny positivity corrections in all practically relevant situations.

 

[atyy]

rubi gave one reason why one cannot just choose an arbitrary place to cutoff the formal power series.

Another reason is why stop at 1 loop, 2 loops or 137 loops etc, why not just stop at tree level? In that case, we do get a Poincare invariant theory, and we happen to know it is not a quantum theory. So one cannot just pick an arbitrary term at which to cutoff the formal power series and say it is a quantum theory.

 

[A. Neumaier]

why not just stop at tree level? In that case, we do get a Poincare invariant theory, and we happen to know it is not a quantum theory. So one cannot just pick an arbitrary term at which to cutoff the formal power series and say it is a quantum theory.

Even at tree level, the smeared electron field operators do not commute but anticommute, so that we still have a quantum theory.

More realistically, we can stop at k-loops with $k>0$, compute the Wightman functions to k-loop, and add tiny higher order terms to remove terms that are slightly indefinite due to the approximation (should such terms arise). Then the positivity conditions holds, we can construct the associated Hilbert space, and we have a quantum theory. Something similar (though less rigorous) is indeed done in practice, where the Hilbert space is constructed through the CTP approach. One calculates in the latter matrix elements and expectations precisely as in quantum mechanics 1, but everything is manifestly covariant.

For example, pair production from a strong electric field or hot QED plasmas are discussed in this way. If that is not quantum theory then nothing is.

 

[PeterDonis]

None of his posts in this thread was constructive; all went of on irrelevant sidelines. Some of them apparently so much that the mentors found them obviously off-topic.

Correct.

 

[dextercioby]

This is a very strong claim. I'd like to see your proof for this no-go theorem. I don't think you have more than a mere opinion in support of this!

Yes, an opinion based on the following facts:

- QM of pointlike particles (assume Copenhagen, or don't worry on the collapse upon measurement) has from the axiomatic/formulation perspective at least 4 that we know of and work: Dirac-von Neumann (vectors and states), von Neumann (only operators - states represented through the von Neumann density operator), Feynman (path integral) and IE Segal algebraic axiomatization. (there could be others mentioned, such as the Weyl-Wigner-Barut symmetry based approach).
- From the axiomatization of quantum field theory that I know of, we've got the Wightman axiomatization (this would correspond roughly to a mix of the Weyl-Wigner-Barut and the Dirac-von Neumann) and the Haag-Ruelle algebraic approach (a prolongation of the IE Segal formulation based on the GNS theorem). There may be others (I think John Baez and IE Segal published a book at the end of the '80s (?) on "constructive quantum field theory" which should encompass more work on axiomatic field theory than one typically finds in BLT or BLOT - see below).

With all this in mind, the Millennium Problem is not finding an equivalent set of mathematically precise and physically reasonable axioms of QFT to replace the known ones and force QCD to satisfy them, but taking the Wightman ones for granted and showing that QCD satisfies them.

So the focus is shifted from - show QED is in agreement with Wightman's work to show QCD is in agreement with Wightman's work. [Landau pole vs. Asymptotic freedom]. If nobody in the mathematics community is thinking of replacing Wightman's axioms with a better set and solve QCD (if they do that, they may not win 1
mio. $) , then most certainly the whole Standard Model won't be shown to be mathematically accurate, unless QCD within Wightman's axiomatization is solved first.

https://en.wikipedia.org/wiki/Yang–Mills_existence_and_mass_gap

 

[A. Neumaier]

an opinion based on the following facts:

But these facts have little to do with your claim,

We can't do the following things:

- Check that the Standard Model fields obey the Wightman axioms.
- Replace Wightman axioms with another set of axioms in agreement with special relativity and check that the Standard Model fields obey the new set. We can replace them with the Haag-Ruelle ones, but they are equivalent, not better.

- It is quite possible that the vacuum sectors of the standard model obeys the Wightman axioms. The problem is open at present, and the lack of techniques to do it is a poor argument for claiming that it can't be done. Not a single interacting 4D local relativistic theory is ruled out so far to exist!
- The Wightman axioms and the Haag-Kastler axioms (I assume you mean these) are related but not equivalent. I do not know a single argument why there cannot be other axiomatic schemes that other related axiomatic schemes could work (and could work better). But I do know that all work done in algebraic quantum field theory on gauge theories points to the fact that these must be described by a scheme different from the Wightman axioms, which are completely unable to describe charged gauge fields. Thus it is very likely that there is another set of axioms adapted to gaunge theories, and the standard model could well fall into the collection of theories described by these.

In particular, all this applies as possibility for QED. Note that the most-used operator version of QED, the Gupta-Bleuler formalism, is not even in its formal power series version (known to be rigorously well-defined) covered by the corresponding analogue of the Wightman axioms!

Baez and IE Segal published a book at the end of the '80s (?) on "constructive quantum field theory"

This is about satisfying the Wightman axioms for polynomial interactions in 2D QFT.

the Millennium Problem is not finding an equivalent set of mathematically precise and physically reasonable axioms of QFT to replace the known ones and force QCD to satisfy them, but taking the Wightman ones for granted and showing that QCD satisfies them.

The millennium problem is not about QCD as you claim, but about quantum Yang-Mills theory for arbitrary compact semisimple nonabelian gauge group. (No quarks, and not restricted to the $SU(3)$ gauge group of gluons). It is a mathematical challenge, and was chosen based on the belief that it is the mathematically most tractable one among all. The fact that it has a big prize on its solution doesn't mean that it is the only worthwhile goal in current constructive quantum field theory.

So the focus is shifted

Only in the millennium problem, not in the community of mathematical physicists!

If nobody in the mathematics community is thinking of replacing Wightman's axioms with a better set

Your hypothesis does not apply; people are thinking about that!

- There is a lot of work done by Strocchi and his collaborators on modifications of Wightman axioms for gauge theory/.

- There is also important work by Hollands on founding quantum field theory on the operator product expansion rather than on vacuum expectations - a completely different approach made necessary by the attempts to go beyond flat space. Wightman axioms cannot apply in curved space since Poincare invariance forces spacetime to be flat.

 

[A. Neumaier]

the Millennium Problem

Also, please take notice that I am very well informed about the challenges involved here.

 

[A. Neumaier]

Apparently, Kasia has written a book by now. Here's the link: http://www.springer.com/de/book/9783319258997 (I don't know if it's good)

The book mentioned, ''Perturbative Algebraic Quantum Field Theory. An Introduction for Mathematicians'' by Kasia Rejzner is excellent for those who have some mathematical maturity and don't mind a fast pace without too much prior physical motivation. She is like me both a mathematician and a theoretical physicist, and works like me in a mathematics department. She writes in the introduction,

As opposed to some other textbooks on the subject, I will not use the excuse that “physicists often do something that is not well defined”, so as mathematicians we don’t need to bother and just turn around for a while, until it’s over. Instead, I will jump straight into the lion’s den and will try to make mathematical sense of perturbative QFT all the way from the initial definition of the model to the interpretation of the results.

In the book, she features an introduction to classical mechanics in terms of modern differential geometry, the Haag-Kaster axioms for local quantum field theory in terms of nets of algebras, deformation quantization, Kaehler geometry and its quantization, Moeller operators and the S-matrix, the (Bogoliubov-)Epstein-Glaser axioms for the causal approach to quantum field theory, the renormalization group, and distribution splitting. Then she goes on to introduce classical gauge theories, their Batalin-Vilkovisky quantization, and she ends the book with an outlook on quantum field theory in curved space.

All in all, an introduction to the state of the art, featuring a much richer variety of algebraic quantum physics than @atyy and @dextercioby are dreaming of. No Wightman axioms at all! (They should write her an email asking her why she forgot to say how it all relates to what atyy thinks quantum theory is!)

 

[A. Neumaier]

Wightman axioms cannot apply in curved space since Poincare invariance forces spacetime to be flat.

The Haag-Kastler framework is more flexible:
Quantum field theory on curved spacetimes: Axiomatic framework and examples
https://arxiv.org/abs/1412.5125

 

[atyy]

The book mentioned, ''Perturbative Algebraic Quantum Field Theory. An Introduction for Mathematicians'' by Kasia Rejzner is excellent for those who have some mathematical maturity and don't mind a fast pace without too much prior physical motivation. She is like me both a mathematician and a theoretical physicist, and works like me in a mathematics department. She writes in the introduction,

In the book, she features an introduction to classical mechanics in terms of modern differential geometry, the Haag-Kaster axioms for local quantum field theory in terms of nets of algebras, deformation quantization, Kaehler geometry and its quantization, Moeller operators and the S-matrix, the (Bogoliubov-)Epstein-Glaser axioms for the causal approach to quantum field theory, the renormalization group, and distribution splitting. Then she goes on to introduce classical gauge theories, their Batalin-Vilkovisky quantization, and she ends the book with an outlook on quantum field theory in curved space.

All in all, an introduction to the state of the art, featuring a much richer variety of algebraic quantum physics than @atyy and @dextercioby are dreaming of. No Wightman axioms at all! (They should write her an email asking her why she forgot to say how it all relates to what atyy thinks quantum theory is!)

It's up to you to promote what you like. I cannot agree. Formal power series alone are just that, and are not much the state of the art in physically meaningful rigour.

 

[atyy]

Why should we keep only terms to one loop. In QED Kinoshita et al have done the calculation to 5 or even more loops. The asymptotic series tells you where to stop, namely at the order, where the apparent corrections get larger than the previous order. The proper vertex functions and thus also the connected Green's functions used to calculate approximations to S-matrix elements in perturbation theory are manifestly Lorentz covariant.

I'm well aware of that. I'm responding with skepticism to A. Neumaier's claim that 1 loop QED constructs a rigourous Poincare invariant quantum theory.

The regularization has been taken to the physical limit after renormalization. That's the point of renormalization. The RG equations tell you, when the perturbative approach breaks down, namely when the running couplings get large (at low energies for QCD at (very) high energies for QED).

In the Wilsonian view, the explanation is different. We start with a regularization such as Hamiltonian lattice QED, which is a well-defined quantum theory. Then the physical, "continuum" limit does not correspond to making the lattice finer, but comes from coarse graining, which gives us QED as a low energy effective theory. To make the lattice spacing zero would correspond to running the renormalization group to high energies and seeing whether one can get asymptotic freedom or asymptotic safety. Since at present, asymptotic freedom and asymptotic safety for QED are unknown, we cannot take the lattice spacing to zero.

 

[atyy]

OK, I'm pretty satisfied that A. Neumaier's is wrong about tree-level, 1 loop, 2 loops etc QED being rigourous Poincare invariant quantum theories. I'm sure he disagrees, but we shall have to agree to disagree.

Let's see if we can go back to the point about lattice QED being a conceptually good starting point for Wilsonian renormalization to obtain QED as a low energy effective field theory. Here are specific comments along this line of thinking.

https://arxiv.org/abs/hep-lat/0211036
Lattice Perturbation Theory
Stefano Capitani
"In principle all known perturbative results of continuum QED and QCD can also be reproduced using a lattice regularization instead of the more popular ones. However, calculating in such a way the correction to the magnetic moment of the muon (to make an example) would be quite laborious. A lattice cutoff would not be the best choice in most cases, for which instead regularizations like Pauli-Villars or dimensional regularization are more suited and much easier to employ. The main virtue of the lattice regularization is instead to allow nonperturbative investigations, which usually need some perturbative calculations to be interpreted properly. As we have already mentioned, the connection from Monte Carlo results of matrix elements to their corresponding physical numbers, that is the matching with the continuum physical theory, has to be realized by performing a lattice renormalization."

 

[vanhees71]

In the Wilsonian view, the explanation is different. We start with a regularization such as Hamiltonian lattice QED, which is a well-defined quantum theory. Then the physical, "continuum" limit does not correspond to making the lattice finer, but comes from coarse graining, which gives us QED as a low energy effective theory. To make the lattice spacing zero would correspond to running the renormalization group to high energies and seeing whether one can get asymptotic freedom or asymptotic safety. Since at present, asymptotic freedom and asymptotic safety for QED are unknown, we cannot take the lattice spacing to zero.

Yes, the Wilsonian point of view is about "coarse graining", but it's not different in content from the older RG approaches, which are just more practical in calculations leading to phenomenology. As I said, the running of the coupling tells you, where PT breaks down. For QED that's at high energies since the coupling becomes large at high energies.

Of course, I can only agree with the quote of Capitani, you cited in #171.

 

[A. Neumaier]

are not much the state of the art in physically meaningful rigour.

Since the state of the art in physically meaningful rigor is empty according to your criteria, but quantum field theory is very alive, your peculiar standards are simply too narrow to be useful.

 

[A. Neumaier]

we cannot take the lattice spacing to zero.

So there is no continuum limit, so lattice QED is not a good approximation of QED. As the lack of papers done on the topic amply demonstrates.

Hamiltonian lattice QED [...] coarse graining, which gives us QED as a low energy effective theory.

You always claim this, but haven't given a single reference. You cannot, because nobody proved it; it is false. Wilson's RG does not prove this. It only shows how to go from one theory to a coarser one. To claim a connection with experimental low energy QED one must do some work, and nobody has done it.

Stefano Capitani
"In principle all known perturbative results of continuum QED and QCD can also be reproduced using a lattice regularization

This is also only a claim without a proof. No reference is given, and the topic is not considered later, though a 221 page treatise should have enough space for explaining such matters of principle. None of the 72 papers by Capitani in the arxiv list you linked to is on lattice QED. All are about lattice QCD which, because of asymptotic freedom, has no obvious barrier in taking the continuum limit, so going to finer and finer lattices indeed recovers informally Poincare invariant QCD. To get results that can be compared with experiment, this extrapolation to the continuum limit is essential!

Thus lattice QCD has a very different character than lattice QED for which this (the only!) connection breaks down!

Rather than only repeat an unproved mantra, try to find a reference that actually gives a cogent argument for Capitani's claim, so that one can inspect the silent assumptions made! You'll see that there is no valid argument! The only arguments along these lines all depend on asymptotic freedom, and this argument is not valid for QED!

 

[A. Neumaier]

Of course, I can only agree with the quote of Capitani, you cited in #171.

Why, of course? QED is very different from QCD, as regards lattice approximations!

 

[vanhees71]

I meant more the part that for QED the standard techniques of perturbation theory in the continuum is more applicable. Of course, I also agree with the statement that one needs to make contact with the continuum limit in lattice QCD and to achieve this one needs renormalized perturbative lQCD. As I stressed several times, I don't think that lattice QED is of any practical significance.

 

[atyy]

Yes, the Wilsonian point of view is about "coarse graining", but it's not different in content from the older RG approaches, which are just more practical in calculations leading to phenomenology. As I said, the running of the coupling tells you, where PT breaks down. For QED that's at high energies since the coupling becomes large at high energies.

Of course, I can only agree with the quote of Capitani, you cited in #171.

Ok, so we agree then.

 

[A. Neumaier]

so we agree then.

except on the part involving lattice QED; see #175 and #176!

 

[dextercioby]

As also privately mitgeteilt, thank you, Arnold, for providing references to the state of the art work on rigorous QFT.

 

[atyy]

Of course, I can only agree with the quote of Capitani, you cited in #171.

So if one agrees with Capitani, would one agree that it is alright to consider the high energy theory (at some high but finite energy cutoff) from which QED is derived to be non-relativistic?

Specifically, one starts with lattice QED at fine but finite spacing, which is a non-relativistic theory. Then the usual covariant perturbative continuum QED is derived as a low energy effective theory. Obviously, this is only in principle, as one would not use this method for practical calculations.

An analogy from condensed matter is the non-relativistic theory theory of the graphene lattice which gives rise to relativistic massless Dirac fermions as a low energy effective theory.

 

[vanhees71]

There's not the slightest hint of the violation of Poincare symmetry. Only recently the measurement of antihydrogen energy levels as well as the magnetic moment of the antiproton once more indicate the precise fulfillment of the CPT theorem, following from Poincare invariance and locality of QFT underlying the Standard Model.

Of course, this doesn't rule out possible violations of Poincare symmetry at higher energies, where the Standard Model becomes invalid. However, the low-energy effective theory, defined by perturbative continuum QED, is obviously a very accurate description which is Poincare symmetric for all practical purposes.

 

[atyy]

There's not the slightest hint of the violation of Poincare symmetry. Only recently the measurement of antihydrogen energy levels as well as the magnetic moment of the antiproton once more indicate the precise fulfillment of the CPT theorem, following from Poincare invariance and locality of QFT underlying the Standard Model.

Of course, this doesn't rule out possible violations of Poincare symmetry at higher energies, where the Standard Model becomes invalid. However, the low-energy effective theory, defined by perturbative continuum QED, is obviously a very accurate description which is Poincare symmetric for all practical purposes.

Here you are answering with real data. In real life, lattice QED will fail way below its cutoff because it doesn't incorporate the weak and strong interactions.

Would you agree that a theory with an energy cutoff cannot be truly Poincare invariant?

 

[A. Neumaier]

ne starts with lattice QED at fine but finite spacing, which is a non-relativistic theory. Then the usual covariant perturbative continuum QED is derived as a low energy effective theory. Obviously, this is only in principle

This is not even in principle, only in your imagination.

Please point to a paper where the in principle proof is given that the usual covariant perturbative continuum QED is derivable as a low energy effective theory.

An analogy from condensed matter is the non-relativistic theory theory of the graphene lattice which gives rise to relativistic massless Dirac fermions as a low energy effective theory.

This is only a hoped-for analogy - until someone proves your claim that one can actually construct low energy continuum QED is in 3 space dimensions from a lattice!

The starting point is not a discretized version of the theory one ends up with (as you propose it for getting low energy continuum QED from lattice QED), but a quite different lattice theory! We discussed this at length in https://www.physicsforums.com/posts/5294008/ and later posts there.

Note also that graphene a much simpler situation. Graphene and the resulting relativistic Dirac fermions are in 2 space dimensions only, and the Fermions resulting are massless. In comparison, relativistic interacting quantum field theories in 2 space-dimensions satisfying the Wightman axioms have been constructed, even in the massive case; the problem is here much simpler because of superrenormalizability. We discussed already much of this earlier: https://www.physicsforums.com/posts/5443402/ and other posts in that thread.

 

[A. Neumaier]

Would you agree that a theory with an energy cutoff cannot be truly Poincare invariant?

As long as the cutoff is fixed the theory is not Poincare invariant. But perturbative QED can be constructed without any cutoff at all!

Even in lattice QCD, which you so like, the extrapolation to the continuum limit must be done to compare with experiment, and Poincare invariance is believed to emerge in this limit. For some lower-dimensional lattice theories this can even be proved!

 

[A. Neumaier]

Specifically, one starts with lattice QED at fine but finite spacing, which is a non-relativistic theory. Then the usual covariant perturbative continuum QED is derived as a low energy effective theory.

You should first tell us how this can be done without performing the continuum limit (extrapolating for lattice spacing to zero)! This is needed if one wants to recover the theory with whose discretized action one started!

 

[Elias1960]

Lattice QED was specifically designed to have the right continuum limit. So it's not a good example if you're wanting to show that Lorentz invariance can arise natural as a continuum approximation to a non-invariant theory. To be convincing you would have to have an independent motivation for lattice QED that did not rely on having the right continuum limit.

What about a quite arbitrary atomic model which, in the large distance limit, gives a standard wave equation for its sound waves of type $(\partial_t^2 - c^2\partial_i^2) u = 0$? Such things exist everywhere in a quite natural way, without any humans inventing them. But the wave equation has Lorentz symmetry too.