Exploring Alternatives to QFT: A Critique of Non-Interacting Quantum Fields

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In summary, the conversation discusses various aspects of quantum field theory (QFT) and its foundations. There is a question about whether there are other studies or programs that could potentially replace conventional QFT with fully interacting quantum fields. The conversation also touches on the concept of Second Quantization, where classical equations are quantized to create field quanta, and whether there are alternative theories to this. The speaker also questions the accuracy of the impression of QFT and how physicists can confidently arrive at a Theory of Everything when the foundations of QFT may be faulty. Finally, there is a discussion about Fock space in QFT and whether it is non-interacting, with some conflicting opinions on the matter.
  • #1
waterfall
381
1
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?
 
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  • #2
You impression of QFT is very inaccurate.
 
  • #3
martinbn said:
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.
 
  • #4
So, where does he say matter fields are non-interacting?
 
  • #5
there are of course fully non-perturbative methods in QFT
 
  • #6
Dickfore said:
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)."
 
  • #7
waterfall said:
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.
 
  • #8
Dickfore said:
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?!
 
  • #9
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.
 
  • #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.
 
  • #11
waterfall said:
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.
 
  • #12
juanrga said:
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".

juanrga said:
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

juanrga said:
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.

juanrga said:
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.
 
  • #13
waterfall said:
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:

tom.stoer said:
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.
 
  • #14
tom.stoer said:
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.
 
  • #15
juanrga said:
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.

juanrga said:
LQG community is also rather confused but in a somewhat complementary way.
I commented on this confusion (from my persepctive) in some other threads.
 
  • #16
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?
 
  • #17
waterfall said:
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?
 
  • #18
Dickfore said:
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.
 
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  • #19
waterfall said:
[...] 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.
 
  • #20
tom.stoer said:
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.
 
  • #21
strangerep said:
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.

strangerep said:
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.

strangerep said:
"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.
 
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  • #22
atyy said:
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.
 
  • #23
waterfall said:
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 :rolleyes: 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.
 
  • #24
strangerep said:
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?
 
  • #25
waterfall said:
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.
 
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  • #26
Oudeis Eimi said:
[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?
 
  • #27
waterfall said:
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)!
 
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  • #28
atyy said:
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)!
 
  • #29
waterfall said:
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.
 
  • #30
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.
 
  • #31
Dickfore said:
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?
 
  • #32
waterfall said:
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?
 
  • #33
waterfall said:
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)
 
Last edited:
  • #34
atyy said:
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?
 
  • #35
waterfall said:
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?
 
<h2>What is QFT?</h2><p>QFT stands for Quantum Field Theory, which is a theoretical framework used to describe the behavior of subatomic particles and their interactions.</p><h2>What are the limitations of QFT?</h2><p>One of the main limitations of QFT is that it does not take into account gravity, making it incompatible with Einstein's theory of general relativity. Additionally, QFT has difficulty explaining certain phenomena, such as the Higgs mechanism and the hierarchy problem.</p><h2>What are the alternatives to QFT?</h2><p>Some alternatives to QFT include string theory, loop quantum gravity, and non-local hidden variable theories. These theories attempt to address the limitations of QFT and provide a more complete understanding of the fundamental laws of nature.</p><h2>What is the critique of non-interacting quantum fields?</h2><p>The critique of non-interacting quantum fields suggests that the concept of point particles, which is central to QFT, may not accurately describe the behavior of subatomic particles. This critique also questions the validity of using infinities in calculations, which is a common practice in QFT.</p><h2>What are the implications of exploring alternatives to QFT?</h2><p>Exploring alternatives to QFT can lead to a better understanding of the fundamental laws of nature and potentially reconcile the discrepancies between QFT and general relativity. It may also open up new avenues for research and potentially lead to new technologies and advancements in our understanding of the universe.</p>

What is QFT?

QFT stands for Quantum Field Theory, which is a theoretical framework used to describe the behavior of subatomic particles and their interactions.

What are the limitations of QFT?

One of the main limitations of QFT is that it does not take into account gravity, making it incompatible with Einstein's theory of general relativity. Additionally, QFT has difficulty explaining certain phenomena, such as the Higgs mechanism and the hierarchy problem.

What are the alternatives to QFT?

Some alternatives to QFT include string theory, loop quantum gravity, and non-local hidden variable theories. These theories attempt to address the limitations of QFT and provide a more complete understanding of the fundamental laws of nature.

What is the critique of non-interacting quantum fields?

The critique of non-interacting quantum fields suggests that the concept of point particles, which is central to QFT, may not accurately describe the behavior of subatomic particles. This critique also questions the validity of using infinities in calculations, which is a common practice in QFT.

What are the implications of exploring alternatives to QFT?

Exploring alternatives to QFT can lead to a better understanding of the fundamental laws of nature and potentially reconcile the discrepancies between QFT and general relativity. It may also open up new avenues for research and potentially lead to new technologies and advancements in our understanding of the universe.

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