Renormalizable quantum field theories

[DarMM]

So, previously in this thread where "the Fock space" has
been mentioned, one must understand that there is not one
Fock space mathematically, but rather an uncountably
infinite number of disjoint Fock-like spaces. The unitary
dressing transformations form part of a technique to find
which one is physically correct.

I should also add that unlike general reps of the CCR, these Fock spaces have been completely categorised. That is, there has been shown to be a certain number of basic families and all spaces within these families can be indexed by one continuous parameter. All other Fock spaces are then direct sums or tensor products of these basic Fock spaces.
For instance "scalar" Fock spaces are one family and they're indexed by mass.

So I should tidy up my language a bit. When I say a non-Fock space I mean a representation of the canonical commutation relations (CCR) which is not any of these uncountably infinite Fock spaces.
For instance $\phi^{4}_{3}$ in orthodox QFT, lives in a non-Fock space. There is no number operator defined over all of this space and there is no state that all annihilation operators acting on it gives zero. Hence the vacuum has no particular relation to the creation and annihilation operators and there are states which cannot be understood as being composed of particles.

 

[Bob_for_short]

Coulomb gauge doesn't keep Lorentz invariant ( Lorentz gauge does.)
And Coulomb gauge violates causality.

Apparently you understand the Lorentz invariance as an explicit one. But the same theory can be transformed, by the variable change, to an implicit Lorentz invariant form. That is what happens while gauge fixing if the gauge is no the Lorentz' one. The theory results remain relativistic whatever gauge is used.

The explicit Lorentz invariance was extremely useful to correctly discarding the perturbative corrections to the masses and charges in the renormalization prescription.

I do not obtain such a corrections to the fundamental constants, so I can work with an implicitly Lorentz invariant theory.

The Coulomb gauge "violates" the causality to the same extent as the other gauges. Read about Feynman propagator - it is different from zero in the space-like region.

In fact, this "violation" is due to too narrow and shallow (factually classical) understanding causality. In QM there is no point-like particles but waves existing in the whole volume. The QM interaction term takes into account their mutual influence. You can understand it as a wave interaction due to non linearity of the wave equation.

Bob_for_short.

 

[ytuab]

I do not obtain such a corrections to the fundamental constants, so I can work with an implicitly Lorentz invariant theory.

The Coulomb gauge "violates" the causality to the same extent as the other gauges. Read about Feynman propagator - it is different from zero in the space-like region.

In fact, this "violation" is due to too narrow and shallow (factually classical) understanding causality. In QM there is no point-like particles but waves existing in the whole volume. The QM interaction term takes into account their mutual influence. You can understand it as a wave interaction due to non linearity of the wave equation.
Bob_for_short.

This "violation" of Coulomb gauge is due to narrow and shallow ?
(Do you know the meaning of causality ?)

The causality violation of the Coulomb gauge is due to the independent scalar poteintial.
When the charge changes, scalar potential in all space will change at the same time.
Please read the upper site as I showed .

"The Coulomb gauge "violates" the causality to the same extent as the other gauges" is not correct.
Please read the part of "Coulomb gauge , Lorentz gauge" in your textbook.

In your paper, you use the distance r1-r2 (ex Eq(23)).
The absolute value r1-r2 or r will change, when the direction and velocity of Lorentz boosts changes. So this is not Lorentz invariant.
(Do you know the meaning of Lorentz boosts?)

If you have QFT textbook by Peskin, please read page 253.
"In the nonrelativistic limit it make sense to compute the potential V(r) ( q^2 << m^2)"
You use the potential energy ( ex E= q/r). This form is not basically Lorentz invariant.

(If you do not have Peskin textbook, please check your textbook .)

 

[Bob_for_short]

The causality violation of the gauge is due to the independent scalar poteintial.

The Coulomb gauge is not my invention. It is widely used in the relativistic QFT formulations. Apart from boosts one has to perform a specific gauge transformation to return to the Coulomb gauge in a new reference frame, so this formulation is Lorentz and gauge invariant. Read Schwinger's and Dirac's papers on this subject, for example. I wish you good luck in fighting against this gauge.

Bob.

 

[ytuab]

The Coulomb gauge is not my invention. It is widely used in the relativistic QFT formulations. Apart from boosts one has to perform a specific gauge transformation to return to the Coulomb gauge in a new reference frame, so this formulation is Lorentz and gauge invariant. Read Schwinger's and Dirac's papers on this subject, for example. I wish you good luck in fighting against this gauge.
Bob.

The Coulomb gauge is NOT widely used in the RELATIVISTIC QFT formulations.
The Lorentz gauge is used.

Please read the part of "Coulomb gauge" and "Lorentz gauge" in the textbook.

"Apart from boosts "? what do you mean?
Please read the part of "the Lorentz boosts" in your textbook.

 

[Bob_for_short]

The Coulomb gauge is NOT widely used in the RELATIVISTIC QFT formulations. The Lorentz gauge is used.

See about practised gauges S. Weinberg's textbook, Volume 1.

"Apart from boosts "? what do you mean? Please read the part of "the Lorentz boosts" in your textbook.

If one makes only a boost, the Coulomb gauge Hamiltonian changes. If you make in addition a specific gauge transformation, the transformed Hamiltonian restores Coulomb gauge form in a new reference frame. I repeat this to you to explain how Lorentz invariance of the Coulomb gauge Hamiltonian can be preserved.

The Coulomb gauge is used in the fundamental quantization, especially in the Dirac's variables (gauge invariant formulation). It is as valid as the others. Your attacks on it are groundless.

The last: please, do not tell me what I shall read.

Bob.

 

[ytuab]

If one makes only a boost, the Coulomb gauge Hamiltonian changes. If you make in addition a specific gauge transformation, the transformed Hamiltonian restores Coulomb gauge form in a new reference frame. I repeat this to you to explain how Lorentz invariance of the Coulomb gauge Hamiltonian can be preserved.

The Coulomb gauge is used in the fundamental quantization, especially in the Dirac's variables (gauge invariant formulation). It is as valid as the others. Your attacks on it are groundless.

The last: please, do not tell me what I shall read.

Bob.

You understand what you say?

Lagrangian (Hamiltonian) must NOT be changed by a boost (relativistic). Do you know this meaning?

What do you mean "in adition specific gauge transformation" ?
(Do you know the meaning of the gauge transformation?)
Lagrangian(Hamiltonian) must NOT be changed by a gauge transformation.

The gauge transformation has NO relation here ( and in your paper).

Please do not say ridiculous things, Bob.

 

[Bob_for_short]

What do you mean "in addition specific gauge transformation" ?
(Do you know the meaning of the gauge transformation?)
Lagrangian(Hamiltonian) must NOT be changed by a gauge transformation.
The gauge transformation has NO relation here ( and in your paper).
Please do not say ridiculous things, Bob.

You apparently take me for a novice. I explain you for the third time: if you apply only a boost to the Coulomb gauge Hamiltonian, it changes - it becomes non Coulomb gauge one. New terms appear. One can restore the Coulomb gauge form in a new reference frame by applying an additional gauge transformation. Is it clear? (see Johnson K. // Ann. of Phys. 1960. V. 10. P. 536.).

I propose you not to comment my works anymore as well as the Coulomb gauge and Lorentz invariance. You yourself look ridiculous.

Bob.

 

[ytuab]

You apparently take me for a novice. I explain you for the third time: if you apply only a boost to the Coulomb gauge Hamiltonian, it changes - it becomes non Coulomb gauge one. New terms appear. One can restore the Coulomb gauge form in a new reference frame by applying an additional gauge transformation. Is it clear? (see Johnson K. // Ann. of Phys. 1960. V. 10. P. 536.).

I propose you not to comment my works anymore as well as the Coulomb gauge and Lorentz invariance. You yourself look ridiculous.

Bob.

What you say is not relevant here.

If Lagrangian (Hamiltonian) is changed by a boost, you must make such "specific gauge transformation methods" with each boost.
We usually choose "Lorentz gauge" if we do such an almost impossible thing.
It is much easier.

And everyone except you know that QED(relativistic) Lagrangian(Hamiltonian) must NOT be
changed under the boost and gauge transformation.

 

[Bob_for_short]

I propose you again not to comment my works anymore as well as what I know and what I do not know.

 

[ExactlySolved]

I just want to chime in that in the real physics community, renormalization has not been at all confusing or problematic since the mid 1970s. Building on some physical insights of Leo Kadanoff in the 1960s, Kenneth Wilson introduced the modern framework for field theoretic renormalization in 1973. This is when renormalization ceased to be a poorly understood necessity, and instead became a hugely valuable tool for understanding field theories. Wilson received the Nobel prize for this work in 1982.

An important part of the modern understanding is the relation between propragators and path integrals in a quantum field theory being mathematically identical to correlators and partition functions in a statistical field theory. In SFT it is favored to define field theories on a discrete lattice, while QFT prefers continuum field theories. Wilson's framework explains the 'infinities that plague QFT' by examining the continuum limit in detail, to see well behaved continuum quantum field theories as existing only at the scale invariant critical points of statistical field theories (scale invariance makes it possible to take the limit of zero lattice spacing).

Of course, there are lots of textbooks out there that do not discuss lattice field theories, and most laymen who study QFT don't care about condensed matter theory and statistical mechanics because these aren't as apparently 'sexy' as particle or string physics. The problem is that studying continuum field theory without knowing about lattice field theory is like doing calculus without a proper definition of limits: you are going to run into infinities that don't make sense.

 

[Bob_for_short]

Thank you, buddy, for your popular explanation, but I know that.

Let me also make a popular explanation of my view point.

An old married couple of Europeans takes a car voyage over the United States. They visit different sites in the country and enter a big city. Soon they get lost. They stop their car and ask a pedestrian: "Excuse us, Sir, we got lost here. Tell us where we are, please?" The pedestrian answers: "You are in a car".

Needless to say to what extent such an answer is useless.

The same useless statements are:

The infinities (divergences) are due to ill-defined product of distributions (x-space).
The infinities (divergences) are due to divergent integrals in the momentum space (p-space).


These statements are correct but are in fact a tautology to a great extent.

The worst one is the following:

The infinities (divergences) are due to some unknown physics at short distances.

The latter is the most misleading.

My opinion, based on my solid experience, is the following:

Mathematically infinite corrections are due to too bad (too distant) initial approximation (free and "point-like" particles).

Physically it means the physicists did not guess (or pick up) a correct physical picture for the initial states.

I showed that the corrections to the fundamental constants appear due to wrong self-action ansatz. It leads to kinetic perturbative terms that add some kinetic constants to the initial ones. I underline that the self-action ansatz was introduced in order to preserve the energy-momentum conservation law. It never worked properly but always with difficulties. In fact, it had always to be abandoned with help of exact (in CED) or perturbative (in QED) renormalizations.

The energy-momentum conservation law can be preserved in another way: by considering a compound system with the center of inertia and relative degrees of freedom. This approach is based on a potential rather than on kinetic "perturbative" (or better interaction) terms, and it does not lead to mathematical and conceptual difficulties. At the same time it describes naturally all physical phenomena. Now, knowing all that, why should we neglect this physically and mathematically correct approach, make masses and charges guilty of our bad understanding of nature, and whisper about probable unknown phenomena at short distances?

The quantum mechanical charge smearing, always existing in the nature, should be taken into account exactly rather than perturbatively. That is the right solution of these problems. Then no divergences appear, not corrections to the fundamental constants arise, no renormalization is necessary. It is a short-cut, if you like, to the final finite results.

I underline - the charge smearing size is always much larger than any possible lattice or Plank distances, or other artificial "space-time grains". The charge form-factors serve as natural regularizators (cut-off mechanisms). There is no problem at short distances at all! What can be simpler?

Study my works carefully. I simplified everything to reveal the most explicitly the point where we make a mistake with the energy-momentum conservation law in particle-field interaction.

Bob.

 

[meopemuk]

Building on some physical insights of Leo Kadanoff in the 1960s, Kenneth Wilson introduced the modern framework for field theoretic renormalization in 1973.

Hi ExactlySolved,

I admit that I don't know Wilson's framerwork well enough. I would appreciate if you can clarify how this framework is used in the particular example of renormalized QED. In particular, I would like to know what it tells us about the cutoff-dependence of the QED Hamiltonian. Can we obtain a finite Hamiltonian in the limit of infinite cutoff?

From what I've read, my understanding is that QED is considered to be an "effective field theory", which means that it makes sense only at limited momenta/energies (or large enough distances). For small distances or large momenta, QED must be replaced by some (yet unknown) theory, which takes into account "space-time granularity" or some other (yet unknown) small distance effects. Basically, this means that we are not allowed to take the infinite cutoff limit in QED. The Hamiltonian defined at the allowed finite cutoff remains finite, and this is the Hamiltonian, which should be used if one wants to study the time evolution of states and observables. Is this description correct?

Thanks.
Eugene.

 

[LaserMind]

Well, what a surprise - I was the original guy who created this
thread - and am shocked by the huge reaction.

I have read the comments (some over my head) and will attempt a
'board room' overview (from my wisdom through great age perspective!):

1) There is a need for ultra violet and infra red cut-offs in these theorems.
- interestingly many views support this for seemingly different reasons.

2) There is a view (Bob's) that 'a quantum smearing of charge' provides a cut-off naturally,
- but this is not widely accepted yet.

3) Other views where operators are applied artificially to impose a cut off, through
renormalization and other devices.

4) There are worries about Lorentz Invariance, particles or free fields,
peturbations and infinities that are preventing a fully self consistent theory,
and these issues can be over complex.

7) It appears that the QFT approach itself is posing questions. Is there another approach?

8) The 'correct' approach is dependant from what perspective we view from (low high r, high or low energy etc)
Experimental results confirm these approaches.

It appears to me at this stage, that we need different approaches for different situations and that
there is no overall fit-all solution - at least as yet.


a) We are discussing cut-off but its ramifications are too large to keep the discussion bounded.
b) The discussions open up interesting avenues to underlying truths.
c) Complexity is an issue preventing clear solutions (or is this just me?)