Are There Viable Alternatives to Quantum Field Theory and Second Quantization?

[waterfall]

To people familiar with QFT. You know quantum fields are non-interacting and they use perturbations methods. Is there other studies or programme that would replace conventional QFT with full fledged interacting quantum fields?

Also about Second Quantization where they treat the Klein-Gorden and Dirac equations acting like classical equations like Maxwell Equations and quantize them to create field quantas such as matter or fermionic fields. Is there any studies or programme about alternative to this? Or are you certain 100% that Second Quantization is fully correct?

And if QFT being not yet perfect due to the non-interacting fields for example. Why are physicists convinced they an arrive at the Theory Of Everything when the foundations are faulty... or maybe they are just contended for now to arrive at Quantum Gravity? And can one even reach it with a possibily faulty QFT foundations? Maybe there is no theory of quantum gravity precisely because QFT is faulty? How possible is this?

 

[martinbn]

You impression of QFT is very inaccurate.

 

[waterfall]

You impression of QFT is very inaccurate.

I learned it from M.Y. Han's book "A Story Of Light: A Short Introduction To Quantum Field Theory Of Quarks And Leptons"

https://www.amazon.com/dp/9812560343/?tag=pfamazon01-20

Which part of the following do you think is inaccurate and why?

The first leap of faith is the introduction of the concept of matter fields, as discussed in Chapter 7. The quantization of the electromagentic field successfully incorporated photons as the quanta of that field and - this is critical - the electromagnetic field (the four-vector potential) satisfied a classical wave equation identical to the Klein-Gordon equation for zero-mass case. A classical wave equation of the 19th century turned out to be the same as the defining wave equation of relativistic quantum mechanics of the 20th century! This then led to the first leap of faith - the grandest emulation of radiation by matter - that all matter particles, electrons and positrons initially and now extended to all matter particles, quarks and leptons, should be considered as quanta of their own quantized fields, each to its own. The wavefunctions of the relativistic quantum mechanics morphed into classical fields. This conceptual transition from relativistic quantum mechanical wavefunctions to classical fields was the first necesary step toward quantized matter fields. Whether such emulation of radiation by matter is totally justifiable remains an open question. It will remain an open question until we successfully achive completely satisfactory quantum field theory of matter, a goal not yet fully achieved.

 

[Dickfore]

So, where does he say matter fields are non-interacting?

 

[tom.stoer]

there are of course fully non-perturbative methods in QFT

 

[waterfall]

So, where does he say matter fields are non-interacting?

It's in another chapter but I learned it first in the quantum physics forum by Fredrik who says Fock Space in QFT is non-interacting (which of the following is inaccurate, please correct it):

"A Fock space is constructed from the Hilbert space associated with the single-particle theory. You use the single-particle space to construct a space of 2-particle states, a space of 3-particle states, and so on, and then you combine them all into a Hilbert space that contains all the 1-particle states, all the 2-particle states, and so on. This Hilbert space is called a Fock space. So it's just an algebraic construction. You need nothing more than the Hilbert space from the single-particle theory to define it, and the single-particle theory can be defined using a Lagrangian with no products of more than two field components or derivatives of field components.

However, in non-rigorous QFT, I think the idea is just to ignore that the interacting Hilbert space is really a different Hilbert space, and just introduce operators that can take n-particle states to (n+1)-particle states for example. In this context, Fock space is, as you put it, "pretending to have interaction when it doesn't really". I really suck at QFT beyond the most basic stuff, so I can't explain it better, and I might even be wrong (about the stuff in this paragraph)."

 

[Dickfore]

It's in another chapter but I learned it first in the quantum physics forum by Fredrik who says Fock Space in QFT is non-interacting (which of the following is inaccurate, please correct it):

"A Fock space is constructed from the Hilbert space associated with the single-particle theory. You use the single-particle space to construct a space of 2-particle states, a space of 3-particle states, and so on, and then you combine them all into a Hilbert space that contains all the 1-particle states, all the 2-particle states, and so on. This Hilbert space is called a Fock space. So it's just an algebraic construction. You need nothing more than the Hilbert space from the single-particle theory to define it, and the single-particle theory can be defined using a Lagrangian with no products of more than two field components or derivatives of field components.

However, in non-rigorous QFT, I think the idea is just to ignore that the interacting Hilbert space is really a different Hilbert space, and just introduce operators that can take n-particle states to (n+1)-particle states for example. In this context, Fock space is, as you put it, "pretending to have interaction when it doesn't really". I really suck at QFT beyond the most basic stuff, so I can't explain it better, and I might even be wrong (about the stuff in this paragraph)."

I must admit this is the first time I hear of a Hilbert space being interacting or not. You may claim that the basis vectors constructed as a direct product of single-particle kets of arbitrary power are eigenkets of the Hamiltonian of the system only when the theory is non-interacting, but the space spanned by them is independent of the basis, and, at least in principle, one should be able to diagonalize even the interacting Hamiltonian acting on kets in this Fock space.

In my opinion, the most important sentence in your post is the bolded one. If there are products of more than 2 field operators in the Lagrangian, then this is necessarily an interction.

 

[waterfall]

I must admit this is the first time I hear of a Hilbert space being interacting or not. You may claim that the basis vectors constructed as a direct product of single-particle kets of arbitrary power are eigenkets of the Hamiltonian of the system only when the theory is non-interacting, but the space spanned by them is independent of the basis, and, at least in principle, one should be able to diagonalize even the interacting Hamiltonian acting on kets in this Fock space.

In my opinion, the most important sentence in your post is the bolded one. If there are products of more than 2 field operators in the Lagrangian, then this is necessarily an interction.

What? Let's go to the context used by M.Y. Han book "A Story Of Light: A Short Introduction To Quantum Field Theory Of Quarks And Leptons".

I'll quote only the relevant passages and omit the math and other detailing part:

"The quantization of fields and the emergence of particles as quanta of the quantized fields discussed in Chapter 9 represent the very essence of quantum field theory. The fields mentioned so far - Klein-Gorden, electromagnetic as well as Dirac fields - are, however, only for the non-interacting cases, that is, for free fields devoid of any interactions, the forces. The theory of free fields by itself is devoid of any physical content: there is no such thing in the real world as a free, non-interacting electron that exerts no force on an adjacnet electron. The theory of free fields provides the foundations upon which one can build the framwork for introducing real physics, namely, the interaction among particles."

[omitting 2 pages of calculations and details]

"Quantum field theory for interacting particles would have been completely solved, and we could have moved beyond it. Well, not exactly. Not exactly, because no one can solve the highly nonlinear copuled equations for interacting fields that result from the interacting Lagrangian density obtained by the subtitution rule. Exact and analytical solutions for interacting fields have never been obtained: we ended up with the Lagrangian that we could not solve!"

[omitting a page]

"At this point, the quantum field theory of interacting particles proceeded towards the only other alternative left: when so justified, treat the interaction part of the Lagrangian as a small perturbatoin to the free part of the Lagrangian"

[I won't quote other paragraphs anymore. Just see it in amazon free page preview if necessary]

Do you know the part about "subtitution rule" he was talking about? Any relation to it that you are talking about? He basically said the subtitution rule couldn't be solved. And we are left only with perturbation, and we know it is seems ad hoc. Therefore Quantum Field Theory seems to be flawed. How then could they arrive at the right theory of Quantum Gravity with such a flawed foundation?!

 

[atyy]

For a simple analogy, a linear oscillator has sinusoidal oscillations.

A nonlinear oscillator does not have sinusoidal oscillations.

Can the solution to the nonlinear oscillator be expressed as a sum of sinusoidal oscillations? Yes - that's what Fourier decomposition is.

For QFT, the analogy is:
linear -> non-interacting
nonlinear -> interacting
sinusoidal -> Fock space.

 

[tom.stoer]

The Fock space states are "blind" for interactions. The interactions are represented by
operators acting on Fock states. It's true that for some questions Fock states are not the best calculational tool, but they are not a foundational problem.

 

[juanrga]

To people familiar with QFT. You know quantum fields are non-interacting and they use perturbations methods. Is there other studies or programme that would replace conventional QFT with full fledged interacting quantum fields?

Also about Second Quantization where they treat the Klein-Gorden and Dirac equations acting like classical equations like Maxwell Equations and quantize them to create field quantas such as matter or fermionic fields. Is there any studies or programme about alternative to this? Or are you certain 100% that Second Quantization is fully correct?

And if QFT being not yet perfect due to the non-interacting fields for example. Why are physicists convinced they an arrive at the Theory Of Everything when the foundations are faulty... or maybe they are just contended for now to arrive at Quantum Gravity? And can one even reach it with a possibily faulty QFT foundations? Maybe there is no theory of quantum gravity precisely because QFT is faulty? How possible is this?

i)
Only free fields are well-defined in QFT, but there is not a replacement for «fully fledged interacting quantum fields» because the concept of field is not defined there.

ii)
«Second Quantization» is a misnomer. There is nothing that is quantized twice as Weinberg often remarks. 'Second' quantization is a formalism for dealing with creating/destruction and creation/destruction is also used in ordinary QM.

iii)
Only some naive physicists as string (brane and M) theorists believed that they could obtain a «Theory Of Everything» over the basis of QFT.

Others are working in more general theories, including far reaching generalizations of string, brane, and M theory.

iv)
The fiasco with quantum gravity has little to see with the limitations of QFT, and more with misunderstandings about general relativity.

 

[tom.stoer]

i)
Only free fields are well-defined in QFT, but there is not a replacement for «fully fledged interacting quantum fields» because the concept of field is not defined there.

Agreed -
- with a minor comment or question: is it really the concept of a "field" or a "field operator" that makes problems, or the concept for "interaction of fields".

ii)
«Second Quantization» is a misnomer. There is nothing that is quantized twice as Weinberg often remarks. 'Second' quantization is a formalism for dealing with creating/destruction and creation/destruction is also used in ordinary QM.

Agreed

iii)
Only some naive physicists as string (brane and M) theorists believed that they could obtain a «Theory Of Everything» over the basis of QFT.

Personally I agree that string- or M-theory may not the final answer, but I would not dare to call them naive, as long as I have nothing else to offer.

iv)
The fiasco with quantum gravity has little to see with the limitations of QFT, and more with misunderstandings about general relativity.

Here I don't agree; I think that in many approaches to quantum gravity one takes general relativity quite seriously (e.g. LQG); and I think that even reserach programs that are more inspired by QFT methods (string theory, asymptotic safety) do take GR seriously.

 

[Dickfore]

Any relation to it that you are talking about? He basically said the subtitution rule couldn't be solved. And we are left only with perturbation, and we know it is seems ad hoc. Therefore Quantum Field Theory seems to be flawed. How then could they arrive at the right theory of Quantum Gravity with such a flawed foundation?!

I'm afraid you had performed a logical fallacy here. Namely, your conclusion, "QFT seems to be flawed", does not follow from the premises you gave. Namely, "we know it seems ad hoc" does not count as logical reasoning.

Then, your last question is a false contradiction
(
It is equivalent to the line of reasoning:
1. If we can arrive at Quantum Gravity with the current formalism, then we know the current formalism is correct.
2. QFT is part of the current formalism.
-----------------------------------------------------------------------------------------
If we know the current formalism is correct, then we know QFT is correct.
If we can arrive at QG with the current formalism, then QFT is correct.
3. QFT is incorrect.
-------------------------------------------------------------------------------------
We cannot arrive at QG with the current formalism.
)
because premise 3 is the wrong conclusion that you drew from the above wrong analysis.

Also, relating to my previous post, see tom.stoer's post #10:

The Fock space states are "blind" for interactions. The interactions are represented by
operators acting on Fock states. It's true that for some questions Fock states are not the best calculational tool, but they are not a foundational problem.

 

[juanrga]

Here I don't agree; I think that in many approaches to quantum gravity one takes general relativity quite seriously (e.g. LQG); and I think that even research programs that are more inspired by QFT methods (string theory, asymptotic safety) do take GR seriously.

They take GR seriously, but what I said is that they misunderstand GR.

String theorists were notorious for believing that GR is equivalent to a spin-2 field theory over a flat background. And claimed that string theory was the final theory. String theorists did need about 40 years to understand that they would begin to search a background-less version (M-theory), but no string theorist has serious ideas about what M-theory is (M is somewhat used for Mistery).

LQG community is also rather confused but in a somewhat complementary way.

 

[tom.stoer]

String theorists were notorious for believing that GR is equivalent to a spin-2 field theory over a flat background. And claimed that string theory was the final theory. String theorists did need about 40 years to understand that they would begin to search a background-less version (M-theory), but no string theorist has serious ideas about what M-theory is (M is somewhat used for Mistery).

I can agree to that view.

LQG community is also rather confused but in a somewhat complementary way.

I commented on this confusion (from my persepctive) in some other threads.

 

[waterfall]

I learned from M.Y. Han book that there are 3 phases of development of quantum field theory and how they deal with non-interacting fields. I'll summarize it.

First phase (Early 1950s) - Langrangian Field Theory - based on canonical quantization, success in QED followed by non-expandability in the case of strong nuclear force and by non-renomalizability in the case of weak nuclear force.

Second phase (1950s-1960s) - Axiomatic QFT - for example S-Matrix theories and other axiomatic approaches, however they did not bring solutions to quantum field theories any closer than the Lagrangian field theories.

Third phase (1970s) - (Lagrangian) gauge field theory - ongoing

My question is. Can you make use of Gauge Theory without using Quantum Field Theory? Or the two completely related? But noether theorem can be applied to Newtonian physics so can the gauge symmetry concept of electromagnetism U(1), electroweak U(1)xSU(2), Strong SU(3) can be developed without using the concept of quantum field theory?

 

[Dickfore]

I learned from M.Y. Han book that there are 3 phases of development of quantum field theory and how they deal with non-interacting fields. I'll summarize it.

First phase (Early 1950s) - Langrangian Field Theory - based on canonical quantization, success in QED followed by non-expandability in the case of strong nuclear force and by non-renomalizability in the case of weak nuclear force.

Second phase (1950s-1960s) - Axiomatic QFT - for example S-Matrix theories and other axiomatic approaches, however they did not bring solutions to quantum field theories any closer than the Lagrangian field theories.

Third phase (1970s) - (Lagrangian) gauge field theory - ongoing

My question is. Can you make use of Gauge Theory without using Quantum Field Theory? Or the two completely related? But noether theorem can be applied to Newtonian physics so can the gauge symmetry concept of electromagnetism U(1), electroweak U(1)xSU(2), Strong SU(3) can be developed without using the concept of quantum field theory?

So, if I understood your exposition correctly, the third phase of the development of quantum field theory is gauge field theory. And, then, you ask if we can use gauge theory without using quantum field theory? Does this even make sense?

 

[strangerep]

I must admit this is the first time I hear of a Hilbert space being interacting or not.

A more accurate phrase is "supports an interacting representation of the Poincare group".

[Waterfall] may claim that the basis vectors constructed as a direct product of single-particle kets of arbitrary power are eigenkets of the Hamiltonian of the system only when the theory is non-interacting, but the space spanned by them is independent of the basis, and, at least in principle, one should be able to diagonalize even the interacting Hamiltonian acting on kets in this Fock space.

No, actually. This is the content of Haag's theorem. The basis eigenstates of a free Fock space fail (in general) to span the interacting Fock space. (In this sense, they are indeed a foundational problem.) That's one of the reasons why infinite renormalizations are necessary: they kinda "push" you toward the correct space (in perturbative steps). It's also part of the reason why rigorous proof of convergence for 4D QFTs remains problematic.

 

[strangerep]

[...] we are left only with perturbation, and we know it is seems ad hoc.

Actually, perturbation techniques in QFT are essentially a version of the Poincare-Lindstedt method in classical dynamics. E.g., if one tries to solve the quartic anharmonic oscillator using naive perturbation theory in terms of harmonic oscillator solutions, one gets solutions that escape to infinity. But the quartic anharmonic oscillator potential is clearly confining, so this cannot be correct. However, if instead of perturbation as a sequence of solutions of the harmonic oscillator, we regard it as a sequence of similar theories -- in which the coupling constants like mass and stiffness are also expanded perturbatively, then we get a much better solution involving higher harmonics. (The classical dynamics text of Jose & Saletan explains this reasonably well.)

In perturbative QFT with renormalization, we do something similar: the mass and other "constants" are considered as series expansions which we adjust at each perturbative order to eliminate any unphysical nonsense. It is, of course, remarkable that this technique of perturbation approximation as a "sequence of similar theories" yields results agreeing with experiment to extraordinary accuracy.

Therefore Quantum Field Theory seems to be flawed.

"Flawed" is far too harsh a word. More accurate is that "convergence of the perturbation series in 4D QFT has not been rigorously established". Also note that there is such a thing as "asymptotic series" in which the first few terms are excellent approximations, but the approximations then get worse for higher orders.

Also remember that QFTs are among the most accurate theories in all of physics, especially QED, in terms of agreement between theory and experiment.

 

[ParticleGrl]

there are of course fully non-perturbative methods in QFT

Generally? I've seen large N methods, but that's just perturbation theory in 1/N. Even the lattice is sort of an expansion in the lattice constant.

 

[atyy]

No, actually. This is the content of Haag's theorem. The basis eigenstates of a free Fock space fail (in general) to span the interacting Fock space. (In this sense, they are indeed a foundational problem.) That's one of the reasons why infinite renormalizations are necessary: they kinda "push" you toward the correct space (in perturbative steps). It's also part of the reason why rigorous proof of convergence for 4D QFTs remains problematic.

I've generally heard that it just says the interaction picture doesn't exist.

So why not just use the Heisenberg picture to define the theory?

Then use the interaction picture for approximation.

Is it any different from condensed matter, which is just Schroedinger's equation, so the problems are not foundational, just a matter of approximation? There the single-particle states are often hoped to span the solution space. There's no guarantee of that, but it's ok as long as they give a good approximation to the interacting ground state. And sometimes one just has to guess a trial variational wavefunction like BCS or Laughlin.

It is, of course, remarkable that this technique of perturbation approximation as a "sequence of similar theories" yields results agreeing with experiment to extraordinary accuracy.

In the case of QED, isn't this explained by the infrared fixed point in the renormalization group flow? But maybe that's the same, I recently read that the KAM theorem is actually a case of RG flow - have no idea how that's the case.

"Flawed" is far too harsh a word. More accurate is that "convergence of the perturbation series in 4D QFT has not been rigorously established". Also note that there is such a thing as "asymptotic series" in which the first few terms are excellent approximations, but the approximations then get worse for higher orders.

Isn't it thought that it shouldn't converge for QED, because of the Landau pole, unless the theory is asymptotically safe? QCD on the other hand is supposed to be UV complete. Since it's still a Clay problem, it obviously hasn't been proved, but I've seen references to bits of constructive field theory in QCD, such as Osterwalder-Schrader reflection positivity, which is one condition to establish the Wightman axioms.

 

[strangerep]

I've generally heard that it just says the interaction picture doesn't exist.

Yes, that's another of the heuristic ways to describe the importance of the theorem. The rigorous statements of it are a bit different of course. (The Wiki page is rather poor, imho.)

So why not just use the Heisenberg picture to define the theory?

Dirac advocated something related to this. Have you read his Yeshiva QFT lecture notes?

P.A.M. Dirac, "Lectures on Quantum Field Theory",
Belfer Graduate School of Science, Yeshiva Univ., NY, 1966

I found them very thought-provoking. The Heisenberg picture is indeed superior to the Schrodinger picture in some aspects, but doesn't solve all the problems.

Then use the interaction picture for approximation.

The difficulty (iiuc!) is that one still needs time-dependent vacuum states and associated GNS representations (which are unitarily inequivalent in general for different values of t). Thus, one must work in something larger than Fock space.

Is it any different from condensed matter, which is just Schroedinger's equation, so the problems are not foundational, just a matter of approximation? There the single-particle states are often hoped to span the solution space. There's no guarantee of that, but it's ok as long as they give a good approximation to the interacting ground state. And sometimes one just has to guess a trial variational wavefunction like BCS or Laughlin.

In condensed matter theory, Bogoliubov transformations are often lurking about, mapping between unitarily inequivalent representations. If one can find sufficiently nice Bogoliubov transformations then one may hope to diagonalize the Hamiltonian by successive such transformations. For the QFT case one must find "sufficiently nice" time-dependent B-transforms (to solve the full dynamics -- not merely the scattering problem).

In the case of QED, isn't this explained by the infrared fixed point in the renormalization group flow?

Hmm. I haven't thought about it in that context. But the IR problems in QED are known to be eliminated at all orders of perturbation by a dressing of the fermions by certain photon coherent states (a la Chung, Kibble, and others). A closer examination of that dressing reveals that it's a Bogoliubov-like transformation to a unitarily-inequivalent representation.

Isn't it thought that it shouldn't converge for QED [...]

:-) That's why I included the safe phrase "asymptotic series" -- series which look like they're converging nicely for a while but then go haywire.

 

[juanrga]

I learned from M.Y. Han book that there are 3 phases of development of quantum field theory and how they deal with non-interacting fields. I'll summarize it.

First phase (Early 1950s) - Langrangian Field Theory - based on canonical quantization, success in QED [...]

Except when you want to study a system so complex as two fully interacting electrons... then as Weinberg remarks in his QFT book, the QFT theory of bound states is not still in a satisfactory shape. And for three electrons the situations is still poor.

 

[atyy]

In condensed matter theory, Bogoliubov transformations are often lurking about, mapping between unitarily inequivalent representations. If one can find sufficiently nice Bogoliubov transformations then one may hope to diagonalize the Hamiltonian by successive such transformations. For the QFT case one must find "sufficiently nice" time-dependent B-transforms (to solve the full dynamics -- not merely the scattering problem).

I'm trying to learn what Haag's theorem is, and googling brings up articles by Fraser, and Earman and Fraser. It looks as if Haag's theorem only needs Euclidean invariance, so it would seem to apply to non-relativistic QFT. Does Haag's theorem apply in the non-relativistic QFT used in condensed matter? If Haag's theorem doesn't apply, is it because Euclidean invariance is broken by the lattice?

 

[Oudeis Eimi]

I learned it from M.Y. Han's book "A Story Of Light: A Short Introduction To Quantum Field Theory Of Quarks And Leptons"

https://www.amazon.com/dp/9812560343/?tag=pfamazon01-20

Which part of the following do you think is inaccurate and why?

[STRIKE]Waterfall, that is a pop-sci book. You may learn about science from such texts, but never
the science itself. That's totally beyond their scope.[/STRIKE]

EDIT: That's me talking out of my posterior. After taking a closer look to the above text, it
looks more like a graduate-level introductory work. My humblest apologies.

Still, the specific way you frame your questions and your replies seems to indicate your
understanding of the basics is both insufficient and plagued with misconceptions. What's
your physics background? Your public profile doesn't say. The reason I'm asking is, as
harsh as this may sound, you need a knowledge of physics (and maths) equivalent to a
bachelor's degree in physics before having a chance to actually being able to learn QFT (from
actual science textbooks, that is). Note that you don't need an actual degree - just the
equivalent level of knowledge.

 

[waterfall]

[STRIKE]Waterfall, that is a pop-sci book. You may learn about science from such texts, but never
the science itself. That's totally beyond their scope.[/STRIKE]

EDIT: That's me talking out of my posterior. After taking a closer look to the above text, it
looks more like a graduate-level introductory work. My humblest apologies.

Still, the specific way you frame your questions and your replies seems to indicate your
understanding of the basics is both insufficient and plagued with misconceptions. What's
your physics background? Your public profile doesn't say. The reason I'm asking is, as
harsh as this may sound, you need a knowledge of physics (and maths) equivalent to a
bachelor's degree in physics before having a chance to actually being able to learn QFT (from
actual science textbooks, that is). Note that you don't need an actual degree - just the
equivalent level of knowledge.

My physics background is Quantum Gravity, Superstrings Theory, etc. but on a conceptual level ala Brian Greene books. I know math is very important. And I have background in basics like Hamiltonian, Lagrangian, Gauge Theory, Special Relativity. We the public just want a birds eye view of it especially consider we pay taxes so we can't let you physicists spend billions at will without our monetary support and approval.

Speaking of birds eye view. Brian Greene talked a lot about perturbation theory, infinities and renormalization. So I guess we use them because we assume spacetime is continuous at small scale so infinities like that occurring in GR singularity occurs? My questions are. What would it take to have a fully interacting theory without the use of perturbation theory? If spacetime is discrete, would it remove the need for this as well as renormalization? I think we use this not because of the limitations of math but because of lack of complete understanding of the physical aspects or physics of the situation?

 

[atyy]

Speaking of birds eye view. Brian Greene talked a lot about perturbation theory, infinities and renormalization. So I guess we use them because we assume spacetime is continuous at small scale so infinities like that occurring in GR singularity occurs? My questions are. What would it take to have a fully interacting theory without the use of perturbation theory? If spacetime is discrete, would it remove the need for this as well as renormalization? I think we use this not because of the limitations of math but because of lack of complete understanding of the physical aspects or physics of the situation?

Quantum field theory does begin with a very fine lattice of points in continuous spacetime. Because the lattice is fine, it looks continuous at low energies. Renormalization is the procedure of figuring out how a quantum field theory with a given symmetry looks like at low energies. Some quantum field theories like QCD have lattices whose spacing can be made as small as one wishes, and are therefore consistent theories even at very high energies. Other quantum field theories like QED appear to be inconsistent at high energies. http://www.staff.science.uu.nl/~hooft101/lectures/basisqft.pdf

Gravity is perturbatively inconsistent at high energies. Searches for quantum gravity either try to see if the theory is non-perturbatively consistent at high energy (Asymptotic Safety), or if new degrees of freedom like strings enter at high energies (String Theory). String theory has led to the remarkable conjecture (AdS/CFT) that quantum field theory in 3 dimensional spacetime may provide a consistent theory of quantum gravity in some 4 dimensional spacetimes. So that is a case of being simultaneously being (in 3D) and not being quantum field theory (in 4D)!

 

[waterfall]

Quantum field theory does begin with a very fine lattice of points in continuous spacetime. Because the lattice is fine, it looks continuous at low energies. Renormalization is the procedure of figuring out how a quantum field theory with a given symmetry looks like at low energies. Some quantum field theories like QCD have lattices whose spacing can be made as small as one wishes, and are therefore consistent theories even at very high energies. Other quantum field theories like QED appear to be inconsistent at high energies. http://www.staff.science.uu.nl/~hooft101/lectures/basisqft.pdf

How is this idea of lattice of points related to the commonly used concept of cancelling out infinities in the calculations? According to M.Y. Han in his graduate level book "Story of Light":

"The requirements of mass and charge renormalizations, on the one hand, and the inescapable appearance of infinities in perturbative calculations, on the other hand, are actually quite separate issues; they trace their origins to different sources. In practice, however, the two become inseparably intertwined in that we utilize the procedures to renormalize mass and charge to absorb, and thus get rid of, the unwanted apperance of infinities in calculations."

"... It can then be shown in the pertubation calculations that certain types of infinities that occur can all be lumped into the mass counter term. With the bare mass also taken to be of infinite value, the two infinities - the infinities coming out of the pertubation calculations and the infinity of the bare mass - cancel each other out leaving us with a finite value for the actual, physical mass of an electron. The difference between two different infinities can certainly be finite. This process, quite fancy indeed, is called mass renormalization"


Now what has this got to do with your lattice idea of spacetime? Maybe you mean at low energies, perturbation is not needed in the calculations? But without perturbation, the magnetic charge of an electron would be different than the measured value. Also how is your statement that "Renormalization is the procedure of figuring out how a quantum field theory with a given symmetry looks like at low energies" fit to the idea of cancelling out infinities?

Gravity is perturbatively inconsistent at high energies. Searches for quantum gravity either try to see if the theory is non-perturbatively consistent at high energy (Asymptotic Safety), or if new degrees of freedom like strings enter at high energies (String Theory). String theory has led to the remarkable conjecture (AdS/CFT) that quantum field theory in 3 dimensional spacetime may provide a consistent theory of quantum gravity in some 4 dimensional spacetimes. So that is a case of being simultaneously being (in 3D) and not being quantum field theory (in 4D)!

 

[Dickfore]

We the public just want a birds eye view of it especially consider we pay taxes so we can't let you physicists spend billions at will without our monetary support and approval.

This means you had paid 1/307,000,000 millionth of my salary. Considering the average working load is 2000 hours per year (8 hours per day, 5 days per weeks, 52 weeks per year - 10 days of holidays and furloughs), it means you have a share of 0.23 s per physicist per year. Considering the American Physical Society has 50,055 registered members, you can interview physicists for 20 minutes per year.

I think you spent your allowed time. Please give us our refund.

 

[Physics Monkey]

The thing about Haag's theorem is that it relies on the assumption of infinite degrees of freedom. For any finite lattice model in finite volume there is no issue of inequivalence. Instead, one is faced with the very physical question of how, in the large system size limit, various states become orthogonal or various timescales go to infinity.

It was Haag himself who emphasized in one of his earlier articles that his theorem really didn't have anything profound to say about a finite number of experiments done in a finite volume of spacetime. Unfortunately, I don't have the reference handy. I would argue that the last section of the wikipedia article "Ignorance on the part of the QFT practitioner" is quite misleading in this regard. I think we understand very well what the issues are (although I would love to discover otherwise).

There is also Fell's theorem that roughly speaking says we can approximate the physical state of an interacting system using conventional Fock states even given Haag's theorem.

 

[waterfall]

This means you had paid 1/307,000,000 millionth of my salary. Considering the average working load is 2000 hours per year (8 hours per day, 5 days per weeks, 52 weeks per year - 10 days of holidays and furloughs), it means you have a share of 0.23 s per physicist per year. Considering the American Physical Society has 50,055 registered members, you can interview physicists for 20 minutes per year.

I think you spent your allowed time. Please give us our refund.

Peter Woit and Lee Smolin who wrote "Not Even Wrong" and "Physics Wrong Turn" mentioned how half of the billions of dollars of funding were wasted by physicists doing "Recreational Mathematical Theology" in Superstrings theory and how they were sidetracked by "symmetries". There is a third book written by another, does anyone remember or know the title?

 

[Dickfore]

Peter Woit and Lee Smolin who wrote "Not Even Wrong" and "Physics Wrong Turn" mentioned how half of the billions of dollars of funding were wasted by physicists doing "Recreational Mathematical Theology" in Superstrings theory and how they were sidetracked by "symmetries". There is a third book written by another, does anyone remember or know the title?

What's your point?

 

[atyy]

Now what has this got to do with your lattice idea of spacetime? Maybe you mean at low energies, perturbation is not needed in the calculations? But without perturbation, the magnetic charge of an electron would be different than the measured value. Also how is your statement that "Renormalization is the procedure of figuring out how a quantum field theory with a given symmetry looks like at low energies" fit to the idea of cancelling out infinities?

Great question! The idea of figuring out how a quantum field theory with a given symmetry looks like at low energies makes sense - let's call this the Wilson-Kadanoff renormalization group. The idea of cancelling infinities is nonsensical. So the latter is simply a calculational trick, while the former provides the conceptual foundation. Historically, the trick was discovered first, and was accepted even though it was nonsensical because of its successful experimental predictions. However, Feynman, Dirac and many physicists continued to worry about the nonsensical subtraction of infinities. Around 1970, the discovery of the Wilson-Kadanoff renormalization group gave a conceptual basis to the calculational trick (ie. no infinities are actually subtracted), and physicists stopped worrying about the subtraction of infinities.

A slightly technical, but quite readable if you are patient, history is given in the http://fds.oup.com/www.oup.co.uk/pdf/0-19-922719-5.pdf.

More technical details are found in
http://arxiv.org/abs/hep-th/9210046v2
http://web.physics.ucsb.edu/~mark/ms-qft-DRAFT.pdf (chapter 29)

 

[waterfall]

Great question! The idea of figuring out how a quantum field theory with a given symmetry looks like at low energies makes sense - let's call this the Wilson-Kadanoff renormalization group. The idea of cancelling infinities is nonsensical. So the latter is simply a calculational trick, while the former provides the conceptual foundation. Historically, the trick was discovered first, and was accepted even though it was nonsensical because of its successful experimental predictions. However, Feynman, Dirac and many physicists continued to worry about the nonsensical subtraction of infinities. Around 1970, the discovery of the Wilson-Kadanoff renormalization group gave a conceptual basis to the calculational trick (ie. no infinities are actually subtracted), and physicists stopped worrying about the subtraction of infinities.

A slightly technical, but quite readable if you are patient, history is given in the http://fds.oup.com/www.oup.co.uk/pdf/0-19-922719-5.pdf.

More technical details are found in
http://arxiv.org/abs/hep-th/9210046v2
http://web.physics.ucsb.edu/~mark/ms-qft-DRAFT.pdf (chapter 29)

I have read Zinn-Justins' first chapter as you suggested. It ends with this and producing new questions:

"This modern viewpoint, deeply based on RG ideas and the notion of scale-dependent effective interactions, not only provides a more consistent picture of QFT, but also a framework in which new physics phenomena can be discussed.
It implies that QFTs are somewhat temporary constructions. Due to an essential coupling of very different physical scales, renormalizable QFTs have a consistency limited to low-energy (or large-distance) physics. One uses the terminology of effective QFT, approximations of an as yet unknown more fundamental theory of a radically different nature."

My questions are. First are you 100% certain the Renormalization Group arguments are totally valid? Do 100% of physicists believe in it? Or are there some doubts?

Second about this fundamental theory of a radically different nature? Does it include Superstrings? Or something beyond Superstrings?

 

[waterfall]

I have read Zinn-Justins' first chapter as you suggested. It ends with this and producing new questions:

"This modern viewpoint, deeply based on RG ideas and the notion of scale-dependent effective interactions, not only provides a more consistent picture of QFT, but also a framework in which new physics phenomena can be discussed.
It implies that QFTs are somewhat temporary constructions. Due to an essential coupling of very different physical scales, renormalizable QFTs have a consistency limited to low-energy (or large-distance) physics. One uses the terminology of effective QFT, approximations of an as yet unknown more fundamental theory of a radically different nature."

My questions are. First are you 100% certain the Renormalization Group arguments are totally valid? Do 100% of physicists believe in it? Or are there some doubts?

Second about this fundamental theory of a radically different nature? Does it include Superstrings? Or something beyond Superstrings?

I remember reading in Lisa Randall's "Warped Passages" about the Hierarchy Problem and how one of the purposes of supersymmetry was to render the Higgs particle not equivalent to the Planck Mass sort of by making the masses of the superparticles cancel out with them. How come Renormalization Group theory was not applied in this case? Why do physicists have to propose Supersymmetry to handle the infinities issues?

 

[strangerep]

I'm trying to learn what Haag's theorem is, and googling brings up articles by Fraser, and Earman and Fraser. It looks as if Haag's theorem only needs Euclidean invariance, so it would seem to apply to non-relativistic QFT. Does Haag's theorem apply in the non-relativistic QFT used in condensed matter? If Haag's theorem doesn't apply, is it because Euclidean invariance is broken by the lattice?

I looked at some of the Fraser/Earman papers several years ago and got the impression that they're more philosophers than physicists (being in the Dept. of History and Philosophy of Science at Pittsburgh). They seemed to be most interested in exploring the fact that, in infinite dimensions, there can exist unitarily inequivalent representations of the CCRs -- and one certainly doesn't need full Poincare relativity to explore that. The textbooks of Umezawa et al ("Thermofield Dynamics & Condensed States" and "Advanced Field Theory") contain useful introductions to inequivalent reps.

For Haag's theorem in a relativistic context, there's always Streater & Wightman's "PCT, Spin, Statistics, and all that". But the first exposition of Haag's theorem that I could actually follow (including the proof) was in Barton's little-known book:

G. Barton,
Introduction to Advanced Field Theory,
Interscience, 1963.

He also has a chapter near the end with some interesting remarks and speculations about the role of unitarily inequivalent representations in full QFT.

 

[atyy]

@waterfall, supersymmetry isn't meant to solve the problem of infinities, it's meant to solve the "naturalness" problem. It's not primarily a mathematical problem, more a problem of explaining why some parameters in the standard model have to specified so precisely to match experimental observations. There's a discussion of this on p6 of the Zinn-Justin chapter. The Wilson-Kadanoff viewpoint that non-renormalizable theories are acceptable effective field theories, and that renormalization is just a way to see how they look like at low energies, underlies two different approaches to quantum gravity: string theory and asymptotic safety. A further argument for the Wilson-Kadanoff viewpoint is the gauge/gravity conjecture in which the renormalization flow is transformed into a spatial dimension.

@strangerep, thanks for the references! I came across an interesting comment in Rivasseau's "From Perturbative to Constructive Renormalization" in which he says the same formal series can be derived in spite of Haag's theorem, by a method given by Epstein and Glaser, but also further indicates that actual meaning should be given by Euclidean field theory, checking if the Osterwalder-Schrader axioms are satisfied, and analytically continuing to Minkowski space. I think one of the problems in LQG is choosing between unitarily inequivalent representations due to Haag's theorem. Apparently Thiemann's master constraint programme tries to use dynamics to choose the appropriate representation. There seems to be an analogy with a particle on a circle.

 

[strangerep]

I came across an interesting comment in Rivasseau's "From Perturbative to Constructive Renormalization" in which he says the same formal series can be derived in spite of Haag's theorem, by a method given by Epstein and Glaser,

If you're not familiar with the Epstein-Glaser method, try Scharf's book:

G. Scharf,
Finite Quantum Electrodynamics -- The Causal Approach,
Springer, 2nd Ed., 1995. ISBN 3-540-60142-2

(Make sure you get the 2nd edition -- it has a lot more stuff than the first.)

But the basic idea of Epstein-Glaser-Scharf is that QFT infinities arise from multiplying distributions by \Theta(t) (step-function) in the time-ordered products. The discontinuity in the step function means that the product is no longer a tempered distribution. The method then revolves around inserting correction terms perturbatively to fix it -- using causality as a guide. But it's quite a few years since I went through the 1st edition of Scharf's book, back when I knew far less QFT and math than now. I really should read the 2nd edition thoroughly some time. :-(

[...] use dynamics to choose the appropriate representation.

Haag also makes a brief mention in his book about how choosing the representation is a "dynamical problem". I guess that means choosing an appropriate time-dependent Bogoliubov transformation, but I don't understand that stuff very well -- and modern LQG even less. :-(

 

[waterfall]

@waterfall, supersymmetry isn't meant to solve the problem of infinities, it's meant to solve the "naturalness" problem. It's not primarily a mathematical problem, more a problem of explaining why some parameters in the standard model have to specified so precisely to match experimental observations. There's a discussion of this on p6 of the Zinn-Justin chapter. The Wilson-Kadanoff viewpoint that non-renormalizable theories are acceptable effective field theories, and that renormalization is just a way to see how they look like at low energies, underlies two different approaches to quantum gravity: string theory and asymptotic safety. A further argument for the Wilson-Kadanoff viewpoint is the gauge/gravity conjecture in which the renormalization flow is transformed into a spatial dimension.

@strangerep, thanks for the references! I came across an interesting comment in Rivasseau's "From Perturbative to Constructive Renormalization" in which he says the same formal series can be derived in spite of Haag's theorem, by a method given by Epstein and Glaser, but also further indicates that actual meaning should be given by Euclidean field theory, checking if the Osterwalder-Schrader axioms are satisfied, and analytically continuing to Minkowski space. I think one of the problems in LQG is choosing between unitarily inequivalent representations due to Haag's theorem. Apparently Thiemann's master constraint programme tries to use dynamics to choose the appropriate representation. There seems to be an analogy with a particle on a circle.

From all these infinities and renormalization thing. It looks like our physics is mainly about interactions between particles. So I think it's true they are just lower limit or classical limit of a completely radical theory. Remember Zinn-Justin's last sentence in the book you shared where it is quoted "One uses the terminology of effective QFT, approximations of an as yet unknown more fundamental theory of a radically different nature."

The radical theory would make possible for example the holodeck in Star Trek where one can manifest any object or make them physical. This is engineering beyond the vacuum. It seems our present physics just focus on the interactions of particles, they don't even know how spacetime is connected to quantum particles. So spacetime could be just a temporary construction, and if we can have access to the more fundamental theory, then we can reprogram spacetime and matter to make possible the idea of Holodeck in Star Trek. This is possible isn't it? You can't make arguments about our mere physics of interactions to judge the limit of what is possible. Do you accept this (and others too)?

 

[strangerep]

[...] the holodeck in Star Trek [...] This is possible isn't it?

You just crossed over into the twilight zone of crackpot speculation.

(Moderators: maybe it's time to close this thread?)

 

[waterfall]

You just crossed over into the twilight zone of crackpot speculation.

(Moderators: maybe it's time to close this thread?)

I'm just asking if our physics is the final.. but I noticed they are mostly based on interactions... on non-interacting quantum fields and renormalization group that is ad hoc. Don't worry. I'm not a star trek fan. But it's just asking if our physics is really the final.. or just the beginning to another chapter like from Newtonian to einsteinian or quantum...

 

[martinbn]

I'm just asking if our physics is the final.. but I noticed they are mostly based on interactions... on non-interacting quantum fields and renormalization group that is ad hoc. Don't worry. I'm not a star trek fan. But it's just asking if our physics is really the final.. or just the beginning to another chapter like from Newtonian to einsteinian or quantum...

What if it is not the final theory? In fact, I don't think that anybody thinks that it is.

 

[waterfall]

What if it is not the final theory? In fact, I don't think that anybody thinks that it is.

It's reported in many news and magazines that when the Higgs will be found found. Physics will be almost complete. But it may be just the beginning.. perhaps we are like starting in Newton stage comparatively and physics would continue to develope the next 400 years...

With non-positive results in Loop quantum gravity and Superstrings, we may be on a wrong foundation and quantum gravity may be more than a century away... you think we can solve it before year 2100?

 

[tom.stoer]

It's reported in many news and magazines that when the Higgs will be found found. Physics will be almost complete.

Standard model (including Higgs) + classical gravity (general relativity) is by no means complete

1) the perturbation series of standard model QFTs does not converge (here I do not mean the infinities in each term but the series as a whole
2) there are problems in the UV, especially for the Higgs
3) gravity is not quantized, but we know that QFT + classical gravity is incomplete
4) gravity itself is incomplete (singularities)

Of course there are additional physical issues like unification, reason for SU(3)*SU(2)*U(1), coupling constants, particles, fermion generations etc.; but even w/o taking these questions into account, the mathematical structure "standard model + classical gravity" is ill-defined.

 

[martinbn]

tom.stoer, a side question about your 1), 4).

1) the perturbation series is an asymptotic series, so the non-covergence is normal. Just like in classical mechanics, say in the work of Poincare, so this by itself is not a problem. Of course there is a difference, in QFT there is no non-perturbative formulation (if i understand correctly).

4) why do singularities mean that gravity is incomplete?

waterfall,

if I understand you correctly you afraid that physicists think that physics is almost complete, and you disagree, but i don't think that is the case, dispite of what some news and magazines may say. Also I get the feeling that you believe that about 80 years ago physics took a wrong turn with QFT and now it is in a dead end street, so they should all stop what they are doing and go back to the begining. That is misunderstanding what physics is and what it has done. Of course I may be completely missing you point.

 

[waterfall]

Standard model (including Higgs) + classical gravity (general relativity) is by no means complete

1) the perturbation series of standard model QFTs does not converge (here I do not mean the infinities in each term but the series as a whole
2) there are problems in the UV, especially for the Higgs
3) gravity is not quantized, but we know that QFT + classical gravity is incomplete
4) gravity itself is incomplete (singularities)

Of course there are additional physical issues like unification, reason for SU(3)*SU(2)*U(1), coupling constants, particles, fermion generations etc.; but even w/o taking these questions into account, the mathematical structure "standard model + classical gravity" is ill-defined.

After a week of understanding the essence of QFT, I just realized how badly is our situation. It's like we were back in the days of Newton. When you don't know QFT. You think i'ts very impressive and we are near to the solution of everything. Isn't it that Steven Weinberg wrote how we soon would have a theory of everything. See:

http://www.math.vt.edu/people/gao/physics/weinberg.html

What would it take to create interacting fields. Maybe we need to find alternatives for the fock space which doesn't even interact. It's quiet bad. We have quantum field theory, but the fields don't interact and we have to use artificial means and ad hoc pertubation series.

I think it's time I should reread Lee Smolin Not Even Wrong and Peter Woit Physics Wrong Turn.. because there is a possibility they may be right and String theory and even Loop Quantum Gravity are just "Recreational Mathematical Theology". I forgot all their arguments.

 

[waterfall]

tom.stoer, a side question about your 1), 4).

1) the perturbation series is an asymptotic series, so the non-covergence is normal. Just like in classical mechanics, say in the work of Poincare, so this by itself is not a problem. Of course there is a difference, in QFT there is no non-perturbative formulation (if i understand correctly).

4) why do singularities mean that gravity is incomplete?

waterfall,

if I understand you correctly you afraid that physicists think that physics is almost complete, and you disagree, but i don't think that is the case, dispite of what some news and magazines may say. Also I get the feeling that you believe that about 80 years ago physics took a wrong turn with QFT and now it is in a dead end street, so they should all stop what they are doing and go back to the begining. That is misunderstanding what physics is and what it has done. Of course I may be completely missing you point.

I'm not saying it's a dead end street. I think it's similar to what happened in General Relativity. Had Einstein not discover certain math technique (I forget if it's differential geometry or tensor calculus), he won't be able to perfect GR and make things lorentz covariant. So I think we have missed the right mathematical tool or language for true QFT instead of the sporatic Fock space that may be just child's play. Now I wonder what is the right mathematicals for fields that indeed interact. It means we have to replace or rather enchance Hilbert Space too with a more superior mathematics. Anyone has any idea what kind of math algorithm for it and if I make sense at all?

 

[tom.stoer]

1) the perturbation series is an asymptotic series, so the non-covergence is normal. Just like in classical mechanics, say in the work of Poincare, so this by itself is not a problem. Of course there is a difference, in QFT there is no non-perturbative formulation (if i understand correctly).

I agree; this is perhaps not a fundamental issue.

It's not true that there are no non-perturbative tools, but one cannot say that there is a fully developed non-perturbative approach applicable to all questions in QFT; it strongly depeds on the use case.

4) why do singularities mean that gravity is incomplete?

b/c GR is formulated for smooth manifolds w/o boundary and w/o defects; at singularities the theory is no longer predictive; you cannot formulate boundary or initial conditions; you don't know where all the matter goes in a black hole (the Schwarzschild metric is a vacuum solution with a point-like singularity); b/c when combined with QFT a black hole it violates unitarity; ...

 

[waterfall]

I was rereading Lee Smolin "Trouble with Physics". He was saying in the following in page 249 that Loop Quantum Gravity was trying to reinvent QFT?? I thought LQG is all about gravity. How come I don't hear about QFT being redone in LQG formulation?

"This work was made possible by Ashtekar's great discovery that general relativity could be expressed in language like that of a gauge field. The metric of spacetime, then, turns out to be something like an electric field. When we tried to treat the corresponding field lines quantum-mechanically, we were forced to treat them without a background because there was none - the field lines already described the geometry of space. Once we made them quantum-mechanical, there was no classical geometry left. So we had to reinvent quantum field theory in order to work without a background metric. To make a long story short, it took the input of many people, with a variety of skills from physics and mathematics, but we succeeded. The result is loop quantum gravity."

Do you agree we have to reinvent quantum field theory in order to work without a background metric? Btw.. why hasn't anyone told me the answer to the "Alternative to QFT " in my thread question is nothing but Loop Quantum Gravity as Smolin mentioned?

 

[suprised]

I think it's time I should reread Lee Smolin Not Even Wrong and Peter Woit Physics Wrong Turn..

It's amazing how one can mess up every little detail. Sorry, don't be surprised about not gettting answers, it's just too far off.