Interacting theory lives in a different Hilbert space [ ]

[strangerep]

in my opinion, your examples have little relevance to the problem of renormalization in QFT.

Well, they are not "my" examples. :-)

Your examples are related to ordinary quantum mechanics, where the number of particles is conserved (actually, this number is 1).

I'm pretty sure Klauder's stuff with quartic interaction qualifies as a QFT.

 

[meopemuk]

I'm pretty sure Klauder's stuff with quartic interaction qualifies as a QFT.

Then we use different terminologies. In my opinion, the characteristic feature of QFT is the presence of interactions changing the number of particles. So, QFT requires the Fock space, in which the number of particles is allowed to vary from 0 to infinity. (Even the presence of "quantum fields" is not essential, as I tried to explain in one of my previous posts.).

If the number of particles does not change, then we have ordinary quantum mechanics, which can be formulated in an N-particle Hilbert space. The examples of Bob and Klauber belong to the latter category.

 

[DrFaustus]

When I quoted Bogoliubov in one of my previous posts I was really trying to emphasize the following point. There is one simple question everyone discussing QFT should answer and which is crucial to the whole renormalizability issue.

What is the mathematical nature of a quantum field? In other words, what kind of mathematical object is a quantum field?

You might try to say that it's an operator, but that is not correct. It cannot bean operator because with operators you simply cannot satisfy the canonical commutation relations. Up to now, no one has found a better way of describing quantum fields mathematically than using the concept of "operator valued distribution". This means that if \varphi(x) is a quantum field, then \int \textrm{d} x \varphi(x) f(x) is a well defined operator on some Hilbert space (with f(x) a compactly supported function). And that's the "generalized function" that Bogoliubov talks about (see my previous post). Simply stated, quantum field = operator valued distribution. And the emphasis here is on "distribution".

Now, no matter how much you dislike renormalization, the problem with distributions is that pointwise multiplication is in general ill defined. (Convince yourself by multiplying two Dirac deltas by approximating them with Gaussians and taking the limit - you get infinity.) And this is the true origin of divergences in QFT - pointwise multiplication of distributions. In other words, an object as simple as \varphi^2(x) is ill defined and needs an actual proper defnition. (A mathematically correct definition for the square is :\varphi^2(x): = \lim_{x \to y}\Big[ \varphi(x) \varphi(y) - \langle \varphi(x)\varphi(y)\rangle_0 \Big] , where the colon denotes normal ordering and the expectation value is taken in the vacuum.) Such an object is a smooth function of x and the reason for this is that \varphi(x)\varphi(y) and \langle\varphi(x)\varphi(y) \rangle_0 have the same "singularity structure". In other words, by subracting them you are removing the singularity in \varphi(x)\varphi(y) and afterwards you can take the limit. But note that this is already a first example of renormalization! In this sense you could talk about renormalizing a squared Dirac delta. (As an aside, strangerep and Bob_for_short, if in that paper appears a squared delta, then the equation is mathematically ill defined and all the conclusions that one might draw are dubious to say the least. It's similar to dividing by zero - you simply cannot do it without some modification, like considering a limit.) And obviously, when you start multiplying Feynman diagrams, which are distributions themselves, things cannot be any better and in fact things actually get worse. For Christ's sake, they wouldn't be giving you a million if properly defining a QFT would be that easy.

Bottom line, as long as quantum fields are to be regarded as operator valued distributions, and they have to be some sort of nontrivial mathematical object if we want them to satisfy canonical commutation relations, then renormalization is unavoidable. That's the meaning of Bogoliubov's words.

Now, please find a way around that and I'll be happy to listen. But first answer to my question above. Because that'll be the starting point.

 

[Bob_for_short]

Thank you, Strangerep, for mentioning my paper. It shows that I did manage to reformulate and find a finite perturbative solution of that particular problem. I would also add that the the precision of my solution is very good and quite sufficient for practical goals.Dear Dr. Faustus,

Nobody argues about the nature of fields in QFTs. Nobody argues that they come in product in the perturbation theory. You yourself say that taking a finite (Gaussian or so) distribution makes the product well defined. What I found is a natural mechanism of keeping the product well defined due to quantum mechanical smearing. There is no need to "remove it". It follows from a new initial approximation and a new interaction Hamiltonian. It is natural rather than artificial. No corrections to the fundamental constants appear and the matrix elements are finite and small. The perturbation series thus are different from what you imply as inevitable.

 

[Bob_for_short]

When I quoted Bogoliubov in one of my previous posts I was really trying to emphasize the following point. There is one simple question everyone discussing QFT should answer and which is crucial to the whole renormalizability issue.

What is the mathematical nature of a quantum field? In other words, what kind of mathematical object is a quantum field?

...Up to now, no one has found a better way of describing quantum fields mathematically than using the concept of "operator valued distribution".

What we encounter in QFTs is not just a distribution product but a T-ordered product, i.e., a propagator, i.e., a solution of equation with a point-like source which is reduced to the Coulomb field in a non-relativistic case, i.e., a banal particle interaction. Make this interaction naturally smeared and no ill-definiteness arise. Roughly speaking, interaction of charge with the quantized EMF, taken into account in the first turn, brings this smearing in a natural way. The interaction reminder for the perturbation theory is then not so dangerous.

 

[Bob_for_short]

In other words, an object as simple as \varphi^2(x) is ill defined and needs an actual proper defnition.

When I encountered a delta-function product δ²(z-z₁) in my integral, I had fun while thinking that this product should "actually" be determined with experimental data.

 

[strangerep]

I'm pretty sure Klauder's stuff with quartic interaction qualifies as a QFT.

Then we use different terminologies. In my opinion, the characteristic feature of QFT is the presence of interactions changing the number of particles.

I meant all the rest of Klauder's work, not merely the small subset I mentioned
in my earlier post.

 

[strangerep]

You might try to say that [a quantum field] is an operator, but that is not correct. It cannot be an operator because with operators you simply cannot satisfy the canonical commutation relations.

OK, so let's review that in a bit more detail before continuing...

In a unitary rep of the usual CCRs, at least one of the P,Q operators must be
unbounded. (The proof can be found in Reed & Simon.) Therefore at least one of
the operators cannot be defined on the whole Hilbert space.

In a standard field theory, we want an uncountably infinite collection of such
operators, parameterized by points of Minkowski space. If one tried to write (naively):
<br /> [a_x, a^*_y] ~=~ \delta_{xy}<br />
then the "Kronecker delta" operator on the rhs is not trace class, right?
But is that the only difficulty? Or is it more problematic that such a
generalized "Kronecker delta" with continuous-valued indices (ie not a
Dirac delta distribution) is only nonzero on a set of Lebesgue measure zero?
That causes problems in spectral representation theorems, right?
Anything else?

Generalizing the rhs to a Dirac delta distribution still has the problem
that such things only make sense when integrated against a test function
from (eg) the Schwarz space.

Since test functions are square-integrable, one then adopts a different
form for the CCRs:
<br /> [a_f, a^*_g] ~=~ (f,g)<br />
(where f,g are test functions). I.e., one parameterizes the set of operators
using functions from Schwarz space instead. In this way, one constructs
an infinite set of bona fide operators on a Hilbert space, but then must
confront the fact that in the definition
\phi_f ~:=~ \int \textrm{d}x \; \varphi(x) f(x)
\varphi(x) is only a distribution, but we really want to construct
interactions by multiplying \varphi(x) terms. Hence the problem.

 

[meopemuk]

... but we really want to construct
interactions by multiplying \varphi(x) terms. Hence the problem.

I still don't see the problem.

For example, in QED interaction is built as a product of three \varphi(x) terms. I can represent each term as a linear combination of creation and annihilation operators, perform the product and obtain a collection of quite regular terms like aaa, a^{\dag}aa, a^{\dag}a^{\dag}a, a^{\dag}a^{\dag}a^{\dag}. There are no bad singularities. The only problem is that these terms act non-trivially on the vacuum and 1-particle states.

 

[Bob_for_short]

I see we speak completely different languages. Eugene is concentrated on a⁺ and a, Dr. Faustus and Dr. Strangerep operate in terms of fields φ, and I talk about different interacting species (electroniums) involved in a new rather than fixed-once-and-forever Hamiltonian with self-action.

I agree, in a narrow sense of "different Hilbert spaces" your reasoning may be sufficient but what we finally seek is how to get a working theory with physically meaningful degrees of freedom, don't we?

 

[DarMM]

I still don't see the problem.

For example, in QED interaction is built as a product of three \varphi(x) terms. I can represent each term as a linear combination of creation and annihilation operators, perform the product and obtain a collection of quite regular terms like aaa, a^{\dag}aa, a^{\dag}a^{\dag}a, a^{\dag}a^{\dag}a^{\dag}. There are no bad singularities. The only problem is that these terms act non-trivially on the vacuum and 1-particle states.

The interaction density in QED is supposed to be an operator valued distribution (OVD). Upon integration to obtain the actual interaction it is supposed to be an operator.
The problem is that the interaction density is not an OVD. It's produced by multiplying three well defined OVDs, namely \psi, \bar\psi and A_{\mu}, however as pointed out by DrFaustus, the product of OVDs cannot be defined in general and the resulting integrated object is not an operator and is indeed quite singular. To show this for yourself test it by making it act on a few states and check the norm of the resulting state, you will find it is infinite.

Renormalization is the method of defining powers of OVDs.

Now for \phi^{4} in two and three dimensions Glimm and Jaffe (with simpler proofs being provided later by others), showed that a part of of this method involves choosing the correct representation of the canonical commutation relations, which means choosing the correct Hilbert space. A Hilbert space which must be different from Fock space.
In fact you can show that in Fock space the most you can "soften" a product of OVDs (in the hopes of making it well defined) is by using Wick ordering on it. If Wick ordering doesn't work then it cannot be defined and you must leave Fock space.

All other QFTs which have been shown to nonperturbativley exist (which includes gauge theories and Yukawa models in two and three dimensions) do in fact live in another Hilbert space which is not Fock space. The work done so far on four dimensional theories by Balaban, Magnen, Seneor and Rivasseau shows that this is also the case in four dimensions.

 

[Bob_for_short]

- Excuse me! I am lost! Please, tell me, where I am?
- You are in a car, Sir.

... however as pointed out by DrFaustus, the product of OVDs cannot be defined in general and the resulting integrated object is not an operator and is indeed quite singular. To show this for yourself test it by making it act on a few states and check the norm of the resulting state, you will find it is infinite.

Infinite corrections, to be short.

Renormalization is the method of defining powers of OVDs.

From experimental data? Are you serious?

Where from do those "interactions" come like φ⁴ and jA ? Don't we put in Lagrangians some trial stuff ourselves and then forget about "trial character" of our activity?

 

[DarMM]

Infinite corrections, to be short.

No, the norm of a state created by the interaction Lagrangian is infinite. I'm just talking about operators and states, it has nothing to do with infinite corrections.

Renormalization is the method of defining powers of OVDs.

From experimental data? Are you serious?

Where from do those "interactions" come like φ⁴ and jA ? Don't we put in Lagrangians some trial stuff ourselves and then forget about "trial character" of our activity?

I'm not sure what you are talking about. I'm just saying the renormalization is a way of defining powers of operator valued distributions. Just like the Gram-Schmidt procedure is a way of producing an orthonormal basis. Nothing to do with experiment, it's just mathematical statement.
I'm talking about how you define \phi^{4} and \psi \bar\psi A_{\mu} mathematically, if they are the experimentally correct interactions is a different question. However it is a question answered in the affirmative for \psi \bar\psi A_{\mu}.

 

[meopemuk]

... as pointed out by DrFaustus, the product of OVDs cannot be defined in general and the resulting integrated object is not an operator and is indeed quite singular. To show this for yourself test it by making it act on a few states and check the norm of the resulting state, you will find it is infinite.

I've calculated such products in the case of QED interaction. The resulting expressions in terms of creation and annihilation operators are long and boring, but they are not singular. At least, they are no more singular than the usual Coulomb potential. You can check the details in Appendix L of http://arxiv.org/abs/physics/0504062

Eugene.

 

[Fredrik]

DarMM, I'm pretty impressed by the knowledge you seem to have about these things. I would appreciate if you could give us a few tips about the best books and articles that one would have to study to get closer to your level of awesomeness.

 

[Bob_for_short]

No, the norm of a state created by the interaction Lagrangian is infinite. I'm just talking about operators and states, it has nothing to do with infinite corrections.

But in physics we do not seek the state norms! Especially those from interaction Lagrangians. We calculate corrections.

You can say it is the "norm" which is infinite, or the "momentum integral" at height momenta, or a "distribution product" is infinite, whatever. Such statements or observations lead to nothing. "You are in a car, Sir". It is not an answer. It is a restatement of the same useless statement. The right answer is: "You made a mistake at this and that places. That is why you have problems".

I'm not sure what you are talking about. I'm just saying the renormalization is a way of defining powers of operator valued distributions. Just like the Gram-Schmidt procedure is a way of producing an orthonormal basis. Nothing to do with experiment, it's just mathematical statement.

Then why to do the renormalizations? We calculated the norm, it is infinite, the job is done, everybody is happy, let us go home. No? You want to compare something with experiment, don't you? Isn't it the true motivation for renormalizations - patching a failed theory?

Concerning L_int = jA, it works for j_extA and jA_ext where the subscript "ext" means an external, known function. Extrapolating this form to the self-consistent case has failed and the conceptual and mathematical difficulties testify it. Now the theory is something that starts from non-physical things, patched with non-physical counter-terms, and free (!?) constants are used to fit the experiment, as if a theory were a fitting curve. What a shame!

 

[DarMM]

I've calculated such products in the case of QED interaction. The resulting expressions in terms of creation and annihilation operators are long and boring, but they are not singular. At least, they are no more singular than the usual Coulomb potential. You can check the details in Appendix L of http://arxiv.org/abs/physics/0504062

Eugene.

Hey,
It's great to see such expressions documented somewhere. However I'm not referring to such expressions, because you can't see from the expressions themselves how singular the objects are. What I'm talking about is the following:
Take the interaction Langrangian operator B = \int{\psi \bar\psi A_{\mu}(x)dx} and take any Fock state \lambda. Then create the state B\lambda = \phi, you will find that (\phi,\phi) = \infty, \forall \lambda \neq 0. So B is undefined for all states.

 

[DarMM]

DarMM, I'm pretty impressed by the knowledge you seem to have about these things. I would appreciate if you could give us a few tips about the best books and articles that one would have to study to get closer to your level of awesomeness.

Hey Fredrik,
Absolutely, let me give you some "must reads" first. Then I'll have a think about more specific articles.

PCT, Spin and Statistics and all that by Streater and Wightman, it's basically a good summary of the general properties of QFTs and also describes what exactly fields are mathematically.
To get a handle on the fact that QFT uses different reps try starting off with either Strocchi's Spontaneous Symmetry Breaking book, publisher Springer Berlin/Heidelberg. A free alternative which is a fun read is Stephen Summer's paper http://arxiv.org/abs/0802.1854" Particularly check out note 5 on page 4 and the references there in.
Also check out Stephen Summer's webpage.

Some good general overview articles are:
http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.bams/1183550013" David Brydges
http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.bams/1183553963" Paul Federbush
They basically run through what rigorous field theory is trying to do and what kind of mathematical problem constructive field theory is. Brydges says its the study of very general diffusion processes and showing such processes exist. Federbush says its a problem in measure theory.

Now most people recommend Glimm and Jaffe's "Quantum Physics, A Functional Integral point of view", but I would actually say not to read it until much later for two reasons:
(1)One third of the book is basically one giant proof, which is that \phi^{4} exists in two dimensions. This is only necessary reading if you want to see how these things are proved.
(2)It doesn't provide an overview of rigorous field theory like most people think it does. Rather it provides an overview of rigorous Statistical Mechanics and how techniques from Statistical Mechanics, combined with measure theory can be used to analytically control a quantum field theory enough to prove its existence and some of its properties rigorously. A big mistake of mine was trying to understand this book too early.

That's just a start, I'll have a think and come back with a more extensive list.

 

[DrFaustus]

Finally the heavy artillery has arrived: DarMM. (Am I allowed to so blatantly admire the knowledge of a user on here?) Anyway, it's good to see you're here to help us, me included, understand this stuff better :)

Bob_for_short ->

Isn't it the true motivation for renormalizations - patching a failed theory?

This is not true. In lower dimensions, 2 and 3, QFT is all but a failed theory. A number of models has been constructed rigorously and non perturbatively. Moreover, for these models the usual perturbative expansion has been shown to be an asymptotic expansion to the non-perturbative solution. (Hence no problems with the non convergence of the perturbative series.) And of course, it requires renormalization. And when a QFT will be constructed rigorously in 4D, then you will be able to say the same. Clearly, given the impressive experimental accuracy, I dare you to say it's a failed theory.

Renormalization is a well understood procedure. I know it has the feeling of "ad hoc" subtractions. And it might indeed have been like that in the early days of QED. But that is not so anymore. And as I said some time ago, it need not be done at all by infinite subtractions - you might do it by "extending the product of distributions to coinciding points", and if done in this way there are no infinities arising. (See book by Scharf - "Finite QED". As the title suggests, there is not a single infinity.)

 

[Bob_for_short]

...What I'm talking about is the following:
Take the interaction Langrangian operator B = \int{\psi \bar\psi A_{\mu}(x)dx} and take any Fock state \lambda. Then create the state B\lambda = \phi, you will find that (\phi,\phi) = \infty, \forall \lambda \neq 0. So B is undefined for all states.

What the use of your expression if it never appears in calculations? Take any other expression, make a right statement about it and it will have the same relation to calculations as the previous one, namely no relation at all. We speak of practical things.

Yes, corrections are divergent too. Now, who is guilty? Fields? Interactions at short distances? State norms? Or physicist who failed to construct something reasonable and trying to justify obvious patches?

 

[DarMM]

Apologies Fredrik, I forgot the best book of all! If you can get your hand on it, try Barry Simon's gem of a book "The P(\phi)_{2} Euclidean Quantum Field Theory".
It's full of lovely details about quantum field theories and their relations to stochastic processes and probability theory in general.

 

[Bob_for_short]

... And when a QFT will be constructed rigorously in 4D, then you will be able to say the same.

I am afraid that 2D and 3D cases are just traps rather than a right direction. Otherwise the 4D case would have been resolved by now.

Clearly, given the impressive experimental accuracy, I dare you to say it's a failed theory.

Look, take any reasonable function and expand it in a power series. Then calculate it with the small parameter of about 0.001 (≈ α/2π) to the fourth order. You will obtain even better accuracy than in QED.

Moreover, the perturbative series in QED, considered as correct, serve to determine the value of the small parameter (alpha) from some experiments. It is not really correct. In my practice I encountered a case where seemingly correct series turned out to be wrong (due to the phenomenon of perturbative-spectral non-commutativity). So using it for finding the small parameter is erroneous. We are still unsure about correctness of the renormalized solutions.

...See book by Scharf - "Finite QED". As the title suggests, there is not a single infinity.

Do you have this book at hand? I have a question on its results.

 

[meopemuk]

Hey,
It's great to see such expressions documented somewhere. However I'm not referring to such expressions, because you can't see from the expressions themselves how singular the objects are. What I'm talking about is the following:
Take the interaction Langrangian operator B = \int{\psi \bar\psi A_{\mu}(x)dx} and take any Fock state \lambda. Then create the state B\lambda = \phi, you will find that (\phi,\phi) = \infty, \forall \lambda \neq 0. So B is undefined for all states.

I basically agree with you, but I would like to choose a slightly different wording to formulate the problem. Let me consider an example in which B is the interaction operator of QED and \lambda = a^{\dag}|0 \rangle is an 1-electron state. Suppose that I want to calculate the scattering matrix element in which both the initial and the final states have just one electron. I chose this example, because from physical intuition it is easy to guess what the result should be. A single electron cannot experience any scattering. It should propagate freely, and the S-operator should be just the identity operator, which is what we get already in the 0-th order S_0 = 1. This means, that all higher order contributions must be zero. However, this is not what we obtain in QED.

The second order S-matrix element is obtained (roughly) as

S_2 = \langle 0|a BB a^{dag} |0\rangle......(1)

(which is basically the same as your formula (\phi,\phi)). After normal ordering of the product BB one can easily show that the result is infinite. In the language of Feynman diagrams this expression is represented by the famous electron self-energy diagram, where the infinity comes from the loop integral. Of course, we can avoid the infinity by cutting artificially the loop integral at high momentum. But, still the physical requirement S_2 = 0 cannot be satisfied.

In the standard QED this problem if fixed as follows. It is admitted that the interaction B is not correct. The "correct" interaction is obtained by adding certain "renormalization counterterms" C to the original Hamiltonian. The new "renormalized" interaction is then B+C. If the counterterms are carefully chosen, then in each perturbation order S-matrix elements become finite and physically acceptable. For example, in the 2nd order we choose the counterterm C_2 such as to cancel exacly the infinite contribution (1)

\langle 0|a C_2 a^{dag}|0\rangle= -\langle 0|a BB a^{dag} |0\rangle

to make sure that the "renormalized" S-matrix satisfies the physical condition S_2 = 0. This is my understanding of how the renormalization program works in QED and in any other quantum field theory.

The renormalization works great if our only goal is to calculate scattering amplitudes and cross-sections. However, the full "renormalized" interaction Hamiltonian B+C has lost any physical meaning, because the counterterms C involve explicitly infinite constants, such as mass and charge "corrections". It would be impossible to use this Hamiltonian in any practical calculations of, e.g., time evolution of states and observables.

Do you agree with this assessment of renormalization difficulties in QFT? If yes, then we can talk about how these difficulties can be avoided. If no, then what is your proposal?

Eugene.

 

[Fredrik]

Thanks for the tips DarMM. Keep 'em coming if you think of anything else to add. I'm going to read the review articles now and save the rest for later. I just started with Summers. I have never heard of the books by Strocchi and Simon. Streater & Wightman has been on my bookshelf for a while, but I have only studied a small part of it carefully so far.

Have any of you read https://www.amazon.com/dp/079230540X/?tag=pfamazon01-20 by Bogolubov, Logunov, Todorov & Oksak? I've been curious about that one, but not curious enough to pay what the sellers are asking for. (You have to see it to believe it, so I suggest you click on the link).

Do any of you know the differences between the two editions of Glimm & Jaffe?

 

[Fra]

I didn't follow this discussion from start but I just want to throw in a comment here.

Clearly, given the impressive experimental accuracy, I dare you to say it's a failed theory.
...
Renormalization is a well understood procedure. I know it has the feeling of "ad hoc" subtractions. And it might indeed have been like that in the early days of QED. But that is not so anymore.

While I do not share Bob_for_short's way of reasoning, I share and appreciate his courage to keep looking for something better.

I have a feeling som of the discussion here, regards to what extent renormalization is mathematically well defined. But as has often been the case in history, a mathematically reasonably well defined model, can still be ambigous from the physical point of view.

I do not understand how people can have such confidence in the current framework given that we have a large set of unsolved issues. Before I am ready to setttle, I would like to see how this framework solves

gravity successfully included in the framework, hierarchy problem, BH information paradoxes, cosmological constant, unification

Until then, I find it speculative to not consider the option that our current framework of QFT/renormalizations will need revision.

What the specifics will be is more difficult. But some of the comment to Bob seems to convey and idea there is nothing wrong with the current renorm/qft framework whatsoever. On that point I disagree.

I might question Bob's specific suggestions, but I do not question his quest.

/Fredrik

 

[DarMM]

Have any of you read https://www.amazon.com/dp/079230540X/?tag=pfamazon01-20 by Bogolubov, Logunov, Todorov & Oksak? I've been curious about that one, but not curious enough to pay what the sellers are asking for. (You have to see it to believe it, so I suggest you click on the link).

Well those prices are ridiculous!
I have read the book, my opinion would be that it was well a head of its time, you'll see here for the first time the renormalization group, the problems with multiplying distributions as the source of the UV divergences, ideas on how to construct theories nonperturbatively, e.t.c.
However usually you can find most of its ideas distilled into a better form after a few years of improvement in other books and articles. You see Bogolubov, Wightman and Symanzik basically started trying to get a handle on what field theories actually were back in the 1950s and tried to prove things nonperturbatively. However since then their discoveries have been refined and placed in a broader context, so I would see it as a book mainly for historical interest or to get "inspiration from a master", rather than a book to learn from.

Do any of you know the differences between the two editions of Glimm & Jaffe?

Yes, the second edition includes extended comments on three things:
(1)More on rigorous gauge theories, since a few gauge theories had been shown to exist between the two editions.
(2)More on theories in dimensions other than two. This was because the proofs of the existence of most three-dimensional theories were horribly long when the first edition came out, so they didn't really go into them.
(3)Different methods of proving the things from the first edition and more development of Hilbert space theory.

To be honest the first edition is as good as the second, since the main draw is that the book contains the best complete proof of the existence of an interacting quantum field theory \phi^{4}_{2}, which is in both editions.
Besides that, the best other thing in the book is the first six chapters, which offer an overview of rigorous statistical mechanics (chapters 4,5) and the axioms of quantum field theory (chapter 6).
I think if you want some good general reading, as opposed to wading through a huge proof, the main draw of Glimm and Jaffe is chapters 3 and 6. These chapters will give you a good idea what the path integral is supposed to be mathematically.

In fact if I were to recommend Glimm and Jaffe's book to a beginner, I would say read chapter 3 and 6 to see what the path integral actually is as a mathematical object.

 

[DarMM]

This is my understanding of how the renormalization program works in QED and in any other quantum field theory.

I would view your understanding as correct.

The renormalization works great if our only goal is to calculate scattering amplitudes and cross-sections. However, the full "renormalized" interaction Hamiltonian B+C has lost any physical meaning, because the counterterms C involve explicitly infinite constants, such as mass and charge "corrections". It would be impossible to use this Hamiltonian in any practical calculations of, e.g., time evolution of states and observables.

Yes, exactly. In fact it is hard to show, but it can be shown that such a Hamiltonian can not generate unitary time evolution.

Do you agree with this assessment of renormalization difficulties in QFT? If yes, then we can talk about how these difficulties can be avoided. If no, then what is your proposal?

Yes, I agree with this assessment. Allow me to take a quantum field theory with these problems which we actually fully understand mathematically, namely \phi^{4}_{3} (subscript is the dimension of spacetime).
Exactly the same thing happens there, the S-matrix converges as an operator on Fock space so scattering is fine. However finite time evolution is not. James Glimm proved (following ideas by Rudolf Haag) in 1968 that the renormalized Hamiltonian converges as an operator not on Fock space, but on another Hilbert space, which I will call Glimm space. It is only on this space that the Hamiltonian with counterterms is self-adjoint (proven later) and bounded below and can generate unitary time evolution.
This why you need another Hilbert space:
(1)To get a finite S-matrix the Hamiltonian needs counterterms.
(2)The counterterms mean the Hamiltonian doesn't work as an operator on Fock space.
(3)Move to another Hilber space, the S-matrix will remain finite and the Hamiltonian will be well-defined.

In fact you can show that if you just pick Glimm space from the start and never go near Fock space, then everything is finite and well-defined from the beginning. Things only go wrong because we pick the wrong representation of the canonical commutation relations to begin with.

 

[DrFaustus]

Bob_for_short ->

I am afraid that 2D and 3D cases are just traps rather than a right direction. Otherwise the 4D case would have been resolved by now.

This is just your own speculation and guessing. It is no argument at all. Still, if nothing else, that 2D and 3D QFT exist shows that renormalization is a non-issue in the sense that it is present even in mathematically rigorously defined theories.

Look, take any reasonable function and expand it in a power series. Then calculate it with the small parameter of about 0.001 (≈ α/2π) to the fourth order. You will obtain even better accuracy than in QED.

Please, show me how to do that. Maybe we could share a Nobel prize.

I have Scharf's book. Go on and ask.

Fra -> QFT is, first and foremost, a mathematical framework. Sure, physicists use it to compute real life quantities, but it is a mathematical framework that starts from some (physically motivated) axioms and goes on from there. That one can confidently use it to compute physical quantities is precisely because it has been under scrutiny for the past 80 or so years and it's been proved times and again that "everything fits". And it is within this framework that renormalization is understood. No one claims it is the final framework to do fundamental physics and the problems you point out are amongst the reasons to go beyond it. But that does not imply a thing about the mathematical structure of QFT. On the other hand, Bob_for_short is attacking renormalization within QFT. And all I've been trying to say is that as long as QFT is formulated in terms of operator valued distributions (which quantum fields are), then some sort of renormalization is unavoidable. It is an intrinsic feature of the theory. And the usual physical interpretation that considers it an "evil" property that one should try to eliminate apparently does nothing but confuse people. The bottom line here is that renormalization is a purely mathematical necessity arising out of multiplication of distributions. You want to get rid of renormalization? Find a way to express the principles of quantum physics and the finite speed of propagation of signals that does not use any kind of distribution.

 

[DarMM]

I am afraid that 2D and 3D cases are just traps rather than a right direction. Otherwise the 4D case would have been resolved by now.

In the sense we are talking about it is!
Balaban, as well as independantly Magnen, Seneor and Rivasseau, have shown that the ultraviolet limit of pure SU(2) Yang-Mills theory in four dimensions is well-defined with the renormalizations. What has yet to be rigorously studied is the infrared limit (infinite volume) and the mass-gap, but this isn't yet understood even on a physical non-rigorous level, hence the use of lattice gauge theory, e.t.c.

However renormalization has been shown to rigorously work in a four-dimensional gauge theory and give a finite ultraviolet limit.

 

[DarMM]

And all I've been trying to say is that as long as QFT is formulated in terms of operator valued distributions (which quantum fields are), then some sort of renormalization is unavoidable.

Indeed, in fact there is a theorem which proves that this is the case. See the paper of Glimm and Jaffe "Infinite Renormalization of the Hamiltonian Is Necessary" Journal of Mathematical Physics, Vol. 10, p.2213-2214.